text: Section 4.1 starts by providing a comparison of the stability and recovery guarantees of the convex formu- lation against traditional approaches such as nuclear norm minimization, nonlinear propagation, and ridge regression. Sections 4.2 through 4.5 discuss scalable numerical schemes. Simply listing the moments up to order 4 O has complexity (N4) where N = m + n, hence is not a scalable representation of the moments matrix. The traditional remedy is the factorized gradient approach due to Burer and Monteiro [9, 10], but our ﬁrst numerical observation will not be a surprise to the specialist: diﬃcult instances of matrix completion lead to the presence of spurious local minimizers. With adequate compression of the variables and constraints, and provided convergence is to the global minimizer, we show how the problem can be solved in an empirical O (N2) complexity. The conclusions of section 4 can be summarized as follows. • Factorization approaches sometimes introduce spurious minimizers for suﬃciently diﬃcult (small δ) problems. When convergence to such minimizers occur, it is sometimes possible to add an additional rounding step and to extract the solution from the second order block rather than considering the whole matrix. • O Factorizing the moment matrix in low rank form still has storage complexity (N2), hence is not fully scalable. We propose to instead view the moment matrix as a tensor, and upgrade to a more eﬃcient O hierarchical low-rank factorization with storage complexity (N). This factorization seems to always work when the simpler factorized gradient works. • The hierarchical factorization is in itself not suﬃcient to guarantee scalability, as formulation (2.1), and O in particular total symmetry, still requires encoding a combinatorial ( (N4)) number of constraints. Section 4.4 then introduces three diﬀerent trace relations, which are derived from the third and fourth order total symmetry constraints. Enforcing those relations in place of the original total symmetry O O constraints reduces the computational cost required to enforce these constraints from (N4) to (1) in the best case. This compression of the total symmetry constraint thus reduces the global complexity to O a factor (N3). Empirically, we again observe that those trace relations can be used as a substitute for the more expensive total symmetry constraints as soon as the traditional factorized gradient method works. • O Given the (1) trace relations and the hierarchical low rank factorization of the moments tensor, a O O last bottleneck that prevent reducing the global computational cost from (N3) to (N2) is given by the Higher Order Aﬃne constraints which enforce the moments constraints derived from multiplying any of the original constraint by any monomial of degree at most two, to be satisﬁed. Encoding those constraints requires storing matrices of size O(N3) (i.e product of b ∈ RN) by m ∈ RN2. We propose 0 O to encode these constraints through random sampling, minimizing over distinct batches of size (N) iteratively. Such formulation does not seem to modify the convergence properties and enables us to × apply the semideﬁnite program (2.1) to matrices X of sizes up to 100 100 without making use of the reduced sparsity based formulation of corollary 3. Dealing with such matrices is not practical in the original framework of the Lasserre hierarchy with two rounds of lifting. 4.1 Lipschitz stability To illustrate how the noise can aﬀect a nonlinear reconstruction in the propagation framework, we conduct (cid:107) (cid:107) ∼ N the following experiments. We consider a noise vector ε = γn/ n for n (0, I ). We gradually increase 3 the amplitude γ of the noise vector. For those noise vectors, we let M denote the solution obtained through P propagation and M the solution obtained through the stable semideﬁnite relaxation (1.16). We consider L 30

text: Section

text: starts

text: by

text: providing

text: a

text: comparison

text: of

text: the

text: stability

text: and

text: recovery

text: guarantees

text: of

text: the

text: convex

text: formu-

text: lation

text: against

text: traditional

text: approaches

text: such

text: as

text: nuclear

text: norm

text: minimization,

text: nonlinear

text: propagation,

text: and

text: ridge

text: regression.

text: Sections

text: through

text: discuss

text: scalable

text: numerical

text: schemes.

text: Simply

text: listing

text: the

text: moments

text: up

text: to

text: order

text: O

text: has

text: complexity

text: where

text: N

text: m

text: n,

text: hence

text: is

text: not

text: a

text: scalable

text: representation

text: of

text: the

text: moments

text: matrix.

text: The

text: traditional

text: remedy

text: is

text: the

text: factorized

text: gradient

text: approach

text: due

text: to

text: Burer

text: and

text: Monteiro

text: but

text: our

text: ﬁrst

text: numerical

text: observation

text: will

text: not

text: be

text: a

text: surprise

text: to

text: the

text: specialist:

text: diﬃcult

text: instances

text: of

text: matrix

text: completion

text: lead

text: to

text: the

text: presence

text: of

text: spurious

text: local

text: minimizers.

text: With

text: adequate

text: compression

text: of

text: the

text: variables

text: and

text: constraints,

text: and

text: provided

text: convergence

text: is

text: to

text: the

text: global

text: minimizer,

text: we

text: show

text: how

text: the

text: problem

text: can

text: be

text: solved

text: in

text: an

text: empirical

text: O

text: complexity.

text: The

text: conclusions

text: of

text: section

text: can

text: be

text: summarized

text: as

text: follows.

text: Factorization

text: approaches

text: sometimes

text: introduce

text: spurious

text: minimizers

text: for

text: suﬃciently

text: diﬃcult

text: (small

text: δ)

text: problems.

text: When

text: convergence

text: to

text: such

text: minimizers

text: occur,

text: it

text: is

text: sometimes

text: possible

text: to

text: add

text: an

text: additional

text: rounding

text: step

text: and

text: to

text: extract

text: the

text: solution

text: from

text: the

text: second

text: order

text: block

text: rather

text: than

text: considering

text: the

text: whole

text: matrix.

text: O

text: Factorizing

text: the

text: moment

text: matrix

text: in

text: low

text: rank

text: form

text: still

text: has

text: storage

text: complexity

text: hence

text: is

text: not

text: fully

text: scalable.

text: We

text: propose

text: to

text: instead

text: view

text: the

text: moment

text: matrix

text: as

text: a

text: tensor,

text: and

text: upgrade

text: to

text: a

text: more

text: eﬃcient

text: O

text: hierarchical

text: low-rank

text: factorization

text: with

text: storage

text: complexity

text: (N).

text: This

text: factorization

text: seems

text: to

text: always

text: work

text: when

text: the

text: simpler

text: factorized

text: gradient

text: works.

text: The

text: hierarchical

text: factorization

text: is

text: in

text: itself

text: not

text: suﬃcient

text: to

text: guarantee

text: scalability,

text: as

text: formulation

text: and

text: O

text: in

text: particular

text: total

text: symmetry,

text: still

text: requires

text: encoding

text: a

text: combinatorial

text: number

text: of

text: constraints.

text: Section

text: then

text: introduces

text: three

text: diﬀerent

text: trace

text: relations,

text: which

text: are

text: derived

text: from

text: the

text: third

text: and

text: fourth

text: order

text: total

text: symmetry

text: constraints.

text: Enforcing

text: those

text: relations

text: in

text: place

text: of

text: the

text: original

text: total

text: symmetry

text: O

text: O

text: constraints

text: reduces

text: the

text: computational

text: cost

text: required

text: to

text: enforce

text: these

text: constraints

text: from

text: to

text: in

text: the

text: best

text: case.

text: This

text: compression

text: of

text: the

text: total

text: symmetry

text: constraint

text: thus

text: reduces

text: the

text: global

text: complexity

text: to

text: O

text: a

text: factor

text: Empirically,

text: we

text: again

text: observe

text: that

text: those

text: trace

text: relations

text: can

text: be

text: used

text: as

text: a

text: substitute

text: for

text: the

text: more

text: expensive

text: total

text: symmetry

text: constraints

text: as

text: soon

text: as

text: the

text: traditional

text: factorized

text: gradient

text: method

text: works.

text: O

text: Given

text: the

text: trace

text: relations

text: and

text: the

text: hierarchical

text: low

text: rank

text: factorization

text: of

text: the

text: moments

text: tensor,

text: a

text: O

text: O

text: last

text: bottleneck

text: that

text: prevent

text: reducing

text: the

text: global

text: computational

text: cost

text: from

text: to

text: is

text: given

text: by

text: the

text: Higher

text: Order

text: Aﬃne

text: constraints

text: which

text: enforce

text: the

text: moments

text: constraints

text: derived

text: from

text: multiplying

text: any

text: of

text: the

text: original

text: constraint

text: by

text: any

text: monomial

text: of

text: degree

text: at

text: most

text: two,

text: to

text: be

text: satisﬁed.

text: Encoding

text: those

text: constraints

text: requires

text: storing

text: matrices

text: of

text: size

text: (i.e

text: product

text: of

text: b

text: RN)

text: by

text: m

text: We

text: propose

text: O

text: to

text: encode

text: these

text: constraints

text: through

text: random

text: sampling,

text: minimizing

text: over

text: distinct

text: batches

text: of

text: size

text: (N)

text: iteratively.

text: Such

text: formulation

text: does

text: not

text: seem

text: to

text: modify

text: the

text: convergence

text: properties

text: and

text: enables

text: us

text: to

text: apply

text: the

text: semideﬁnite

text: program

text: to

text: matrices

text: X

text: of

text: sizes

text: up

text: to

text: without

text: making

text: use

text: of

text: the

text: reduced

text: sparsity

text: based

text: formulation

text: of

text: corollary

text: Dealing

text: with

text: such

text: matrices

text: is

text: not

text: practical

text: in

text: the

text: original

text: framework

text: of

text: the

text: Lasserre

text: hierarchy

text: with

text: two

text: rounds

text: of

text: lifting.

text: Lipschitz

text: stability

text: To

text: illustrate

text: how

text: the

text: noise

text: can

text: aﬀect

text: a

text: nonlinear

text: reconstruction

text: in

text: the

text: propagation

text: framework,

text: we

text: conduct

text: N

text: the

text: following

text: experiments.

text: We

text: consider

text: a

text: noise

text: vector

text: ε

text: γn/

text: n

text: for

text: n

text: I

text: We

text: gradually

text: increase

text: the

text: amplitude

text: γ

text: of

text: the

text: noise

text: vector.

text: For

text: those

text: noise

text: vectors,

text: we

text: let

text: M

text: denote

text: the

text: solution

text: obtained

text: through

text: P

text: propagation

text: and

text: M

text: the

text: solution

text: obtained

text: through

text: the

text: stable

text: semideﬁnite

text: relaxation

text: We

text: consider

text: L


text: When minimizing the nuclear norm of rank one matrices, one only enforces constraints on monomials of degree at most two on the entries of the generating vectors x and y. The nuclear norm was shown in [23] to be equivalent to the following semideﬁnite program, (cid:18) (cid:19) minimize Tr(W ) X X subject to W = 11 12 X21 X22 (1.12) ∈ (X ) = (X ) , (i, j) Ω, 12 ij 0 ij (cid:23) W 0. When X = x yT , the matrix W is a proxy for the rank one matrix 0 0 0 (cid:18) (cid:19) x xT x yT W = 0 0 0 0 (1.13) 0 y xT y yT 0 0 0 0 The positive semideﬁnite constraint on X is thus used in combination with the trace norm, as a convex relaxation of the rank one constraint. It is interesting to note that formulation (1.12) only optimizes over monomials of bidegree (1, 1). The key idea of the ”second round of lifting” is to extend this type of formulation ⊗ to monomials of higher degree in the original unknowns. Introducing z = (x , y ) and z(2) = vec(z 2) = ⊗ 0 0 0 0 0 vec(z z ), we consider the larger matrix M deﬁned as 0 0  0      1 z z(2) ⊗0 ⊗0 M0 = z0 z0⊗2 z0 ⊗z(02) , (1.14) z(2) z(2) z z(2) z(2) 0 0 0 0 0 ⊗ 2 In (1.14), we thus have W = z . At order two, the semideﬁnite relaxation considers as unknowns all the 0 0 entries of a positive semi-deﬁnite proxy M of the same structure as M , but without the explicit link to a 0 vector z . Instead, two categories of linear constraints are intended to force the matrix M to inherit the 0 structure of M : 0 • Structural constraints/ total symmetry. Due to the additional monomials that appear in (1.14), there now exist corresponding additional relations between the entries of M (and thus M as well). In ⊗ 0 ⊗ particular, all monomials in z(2) ﬁnd an exact match in the block z 2. Within the block z(2) z(2), one 0 0 0 0 must also list all the total symmetry constraints of a tensor of order 4. More generally, the structural constraints enforce equality of the entries that are identical in the rank-one matrix M . 0 • Higher-order aﬃne constraints. Similarly, for any of the original aﬃne constraints applying on the ⊗ 2 elements of z , one can now deﬁne higher order constraints that are jointly enforced on the elements 0 ⊗ of z and the elements of the block z(2) z . As an example, consider that one is given the constraint 0 0 0 − X = x y = (X ) . It is now possible to enforce the constraints x y x (X ) x for any monomial ij i j 0 ij i i k 0 ij k x . More generally, the higher-order constraints are obtained by multiplying the original constraints k by any product of the entries of x and y of degree at most two. (cid:23) The second-order formulation in this paper consists in combining all these constraints with M 0, and with minimization of Tr(M). The point of this paper is to prove that this formulation, of order 4 in x and y, is 6

text: When

label: paragraph

text: minimizing

text: the

text: nuclear

text: norm

text: of

text: rank

text: one

text: matrices,

text: one

text: only

text: enforces

text: constraints

text: on

text: monomials

text: of

text: degree

text: at

text: most

text: two

text: on

text: the

text: entries

text: of

text: the

text: generating

text: vectors

text: x

text: and

text: y.

text: The

text: nuclear

text: norm

text: was

text: shown

text: in

text: to

text: be

text: equivalent

text: to

text: the

text: following

text: semideﬁnite

text: program,

label: equation

text: minimize

text: Tr(W

text: X

text: X

text: subject

text: to

text: W

text: (X

text: (X

text: (i,

text: j)

text: Ω,

text: ij

text: ij

text: W

text: When

text: X

text: x

text: yT

text: the

text: matrix

text: W

text: is

text: a

text: proxy

text: for

text: the

text: rank

text: one

text: matrix

text: x

text: xT

text: x

text: yT

text: W

text: y

text: xT

text: y

text: yT

text: The

text: positive

text: semideﬁnite

text: constraint

text: on

text: X

text: is

text: thus

text: used

text: in

text: combination

text: with

text: the

text: trace

text: norm,

text: as

text: a

text: convex

text: relaxation

text: of

text: the

text: rank

text: one

text: constraint.

text: It

text: is

text: interesting

text: to

text: note

text: that

text: formulation

text: only

text: optimizes

text: over

text: monomials

text: of

text: bidegree

text: The

text: key

text: idea

text: of

text: the

text: ”second

text: round

text: of

text: lifting”

text: is

text: to

text: extend

text: this

text: type

text: of

text: formulation

text: to

text: monomials

text: of

text: higher

text: degree

text: in

text: the

text: original

text: unknowns.

text: Introducing

text: z

text: (x

text: y

text: and

text: vec(z

text: vec(z

text: z

text: we

text: consider

text: the

text: larger

text: matrix

text: M

text: deﬁned

text: as

text: z

text: ⊗0

text: ⊗0

text: z

text: In

text: we

text: thus

text: have

text: W

text: z

text: At

text: order

text: two,

text: the

text: semideﬁnite

text: relaxation

text: considers

text: as

text: unknowns

text: all

text: the

text: entries

text: of

text: a

text: positive

text: semi-deﬁnite

text: proxy

text: M

text: of

text: the

text: same

text: structure

text: as

text: M

text: but

text: without

text: the

text: explicit

text: link

text: to

text: a

text: vector

text: z

text: Instead,

text: two

text: categories

text: of

text: linear

text: constraints

text: are

text: intended

text: to

text: force

text: the

text: matrix

text: M

text: to

text: inherit

text: the

text: structure

text: of

text: M

text: Structural

text: constraints/

text: total

text: symmetry.

text: Due

text: to

text: the

text: additional

text: monomials

text: that

text: appear

text: in

text: there

text: now

text: exist

text: corresponding

text: additional

text: relations

text: between

text: the

text: entries

text: of

text: M

text: (and

text: thus

text: M

text: as

text: well).

text: In

text: particular,

text: all

text: monomials

text: in

text: ﬁnd

text: an

text: exact

text: match

text: in

text: the

text: block

text: z

text: Within

text: the

text: block

text: one

text: must

text: also

text: list

text: all

text: the

text: total

text: symmetry

text: constraints

text: of

text: a

text: tensor

text: of

text: order

text: More

text: generally,

text: the

text: structural

text: constraints

text: enforce

text: equality

text: of

text: the

text: entries

text: that

text: are

text: identical

text: in

text: the

text: rank-one

text: matrix

text: M

text: Higher-order

text: aﬃne

text: constraints.

text: Similarly,

text: for

text: any

text: of

text: the

text: original

text: aﬃne

text: constraints

text: applying

text: on

text: the

text: elements

text: of

text: z

text: one

text: can

text: now

text: deﬁne

text: higher

text: order

text: constraints

text: that

text: are

text: jointly

text: enforced

text: on

text: the

text: elements

text: of

text: z

text: and

text: the

text: elements

text: of

text: the

text: block

text: z

text: As

text: an

text: example,

text: consider

text: that

text: one

text: is

text: given

text: the

text: constraint

text: X

text: x

text: y

text: (X

text: It

text: is

text: now

text: possible

text: to

text: enforce

text: the

text: constraints

text: x

text: y

text: x

text: (X

text: x

text: for

text: any

text: monomial

text: ij

text: i

text: j

text: ij

text: i

text: i

text: k

text: ij

text: k

text: x

text: More

text: generally,

text: the

text: higher-order

text: constraints

text: are

text: obtained

text: by

text: multiplying

text: the

text: original

text: constraints

text: k

text: by

text: any

text: product

text: of

text: the

text: entries

text: of

text: x

text: and

text: y

text: of

text: degree

text: at

text: most

text: two.

text: The

text: second-order

text: formulation

text: in

text: this

text: paper

text: consists

text: in

text: combining

text: all

text: these

text: constraints

text: with

text: M

text: and

text: with

text: minimization

text: of

text: Tr(M).

text: The

text: point

text: of

text: this

text: paper

text: is

text: to

text: prove

text: that

text: this

text: formulation,

text: of

text: order

text: in

text: x

text: and

text: y,

text: is


text: Noname manuscript No. (will be inserted by the editor) Data Science vs. Statistics: Two Cultures? · Iain Carmichael J.S. Marron Submitted to the Japanese Journal of Statistics and Data Science Abstract Data science is the business of learning from data, which is tra- ditionally the business of statistics. The term data science, however, is of- ten viewed as a broader, task-driven and computationally-oriented version of statistics. Examining and expanding upon the views of a number of statisti- cians, this paper encourages a big-tent view of data analysis. Data science, both the term and idea, has its origins in statistics, representing a reaction to a narrower view of data analysis. We examine how diﬀerent and evolving approaches to data analysis are related to broader trends in data science (e.g. exploratory analysis, machine learning, reproducibility, computation, etc). Fi- nally, we discuss how these trends relate to academic statistics, including future directions for communication, education and research. · · · Keywords Computation Literate Programming Machine Learning · Reproducibility Robustness 1 Introduction A simple deﬁnition of a data scientist is someone who uses data to solve problems. In the past few years this term has caused a lot of buzz1 in industry, I. Carmichael B30 Hanes Hall University of North Carolina at Chapel Hill E-mail: iain@unc.edu J.S. Marron 352 Hanes Hall University of North Carolina at Chapel Hill E-mail: marron@unc.edu 1 https://hbr.org/2012/10/data-scientist-the-sexiest-job-of-the-21st-century ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Noname

text: manuscript

text: No.

text: (will

text: be

text: inserted

text: by

text: the

text: editor)

text: Data

label: title

text: Science

text: vs.

text: Statistics:

text: Two

text: Cultures?

label: author

text: Iain

text: Carmichael

text: J.S.

text: Marron

text: Submitted

label: date

text: to

text: the

text: Japanese

text: Journal

text: of

text: Statistics

text: and

text: Data

text: Science

text: Abstract

text: Data

label: abstract

text: science

text: is

text: the

text: business

text: of

text: learning

text: from

text: data,

text: which

text: is

text: tra-

text: ditionally

text: the

text: business

text: of

text: statistics.

text: The

text: term

text: data

text: science,

text: however,

text: is

text: of-

text: ten

text: viewed

text: as

text: a

text: broader,

text: task-driven

text: and

text: computationally-oriented

text: version

text: of

text: statistics.

text: Examining

text: and

text: expanding

text: upon

text: the

text: views

text: of

text: a

text: number

text: of

text: statisti-

text: cians,

text: this

text: paper

text: encourages

text: a

text: big-tent

text: view

text: of

text: data

text: analysis.

text: Data

text: science,

text: both

text: the

text: term

text: and

text: idea,

text: has

text: its

text: origins

text: in

text: statistics,

text: representing

text: a

text: reaction

text: to

text: a

text: narrower

text: view

text: of

text: data

text: analysis.

text: We

text: examine

text: how

text: diﬀerent

text: and

text: evolving

text: approaches

text: to

text: data

text: analysis

text: are

text: related

text: to

text: broader

text: trends

text: in

text: data

text: science

text: (e.g.

text: exploratory

text: analysis,

text: machine

text: learning,

text: reproducibility,

text: computation,

text: etc).

text: Fi-

text: nally,

text: we

text: discuss

text: how

text: these

text: trends

text: relate

text: to

text: academic

text: statistics,

text: including

text: future

text: directions

text: for

text: communication,

text: education

text: and

text: research.

text: Keywords

text: Computation

text: Literate

text: Programming

text: Machine

text: Learning

text: Reproducibility

text: Robustness

text: Introduction

text: A

text: simple

text: deﬁnition

text: of

text: a

text: data

text: scientist

text: is

text: someone

text: who

text: uses

text: data

text: to

text: solve

text: problems.

text: In

text: the

text: past

text: few

text: years

text: this

text: term

text: has

text: caused

text: a

text: lot

text: of

text: in

text: industry,

text: I.

text: Carmichael

text: Hanes

text: Hall

text: University

text: of

text: North

text: Carolina

text: at

text: Chapel

text: Hill

text: E-mail:

text: iain@unc.edu

text: J.S.

text: Marron

text: Hanes

text: Hall

text: University

text: of

text: North

text: Carolina

text: at

text: Chapel

text: Hill

text: E-mail:

text: marron@unc.edu

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Data Science vs. Statistics: Two Cultures? 19 With the large number of technologies involved with data analysis (pro- gramming languages, visualization software, algorithms, etc) one often feels a bit overwhelmed at what one might be expected to know. It is infeasible to know everything. This is where updating the statistics education curriculum is critical. There is probably some rough core set of computational knowl- edge every trained statistician should have. Given the ﬁxed cost of learning the core computational curriculum, the marginal cost of learning additional computational skills will go down. 4.3.2 Pedagogy The references cited above in Section make a number of good recommen- dations for improving statistics education. We highlight a couple of points here. As many of the above references discuss, the current statistics curriculum often lacks data analysis (Tukey, 1962; Nolan and Temple Lang, 2010). Real data analysis makes the discipline more concrete to students. Focus on solving a real problem can be engaging to students who might otherwise ﬁnd the subject boring. Teaching data analysis is challenging52, but it’s challenging in the way that teaching the practice of engineering or using the scientiﬁc method is challenging. By not giving students practice doing data analysis for a real problem, the statistics curriculum may encourage students to view statistical methodology as a hammer to be procedurally applied to data. It’s well established in engineering and the physical sciences that students should get some practical experience doing the thing during their education: why does the same principle not apply more often statistics at the undergraduate level? When teaching statistical modeling53 it might be more eﬀective to intro- duce the model (e.g. linear/logistic regression) in terms of a predictive context instead of the traditional inferential context. While the math behind linear models may not be particularly sophisticated, the concept of randomness and relating it to the real world through data is non-trivial. Moreover, inferential modeling comes with a lot of baggage. For predictive purposes, models such as linear regression can be introduced as an optimization problem which can be heuristically motivated and analytically solved54. Once students are com- fortable with data modeling we can then introduce inferential/conﬁrmatory modeling. Finally, we suggest teaching data before statistics in introductory statistics courses; in other words, we should teach exploratory analysis before inferen- tial analysis. This would involve teaching programming, data visualization and manipulation before teaching hypothesis testing. Students are more likely to care about hypothesis testing if they have actually worked on a real prob- 52 https://simplystatistics.org/2017/12/20/thoughts-on-david-donoho-s-fifty- years-of-data-science/ 53 E.g. in an upper level undergraduate course such as UNC’s STOR 455: Statistical Meth- ods I. 54 See for example https://www.inferentialthinking.com/chapters/13/prediction. html. ##LTLine## ##LTLine##

text: Data

text: Science

text: vs.

text: Statistics:

text: Two

text: Cultures?

text: With

text: the

text: large

text: number

text: of

text: technologies

text: involved

text: with

text: data

text: analysis

text: (pro-

text: gramming

text: languages,

text: visualization

text: software,

text: algorithms,

text: etc)

text: one

text: often

text: feels

text: a

text: bit

text: overwhelmed

text: at

text: what

text: one

text: might

text: be

text: expected

text: to

text: know.

text: It

text: is

text: infeasible

text: to

text: know

text: everything.

text: This

text: is

text: where

text: updating

text: the

text: statistics

text: education

text: curriculum

text: is

text: critical.

text: There

text: is

text: probably

text: some

text: rough

text: core

text: set

text: of

text: computational

text: knowl-

text: edge

text: every

text: trained

text: statistician

text: should

text: have.

text: Given

text: the

text: ﬁxed

text: cost

text: of

text: learning

text: the

text: core

text: computational

text: curriculum,

text: the

text: marginal

text: cost

text: of

text: learning

text: additional

text: computational

text: skills

text: will

text: go

text: down.

label: section

text: Pedagogy

text: The

text: references

text: cited

text: above

text: in

text: Section

text: make

text: a

text: number

text: of

text: good

text: recommen-

text: dations

text: for

text: improving

text: statistics

text: education.

text: We

text: highlight

text: a

text: couple

text: of

text: points

text: here.

text: As

text: many

text: of

text: the

text: above

text: references

text: discuss,

text: the

text: current

text: statistics

text: curriculum

text: often

text: lacks

text: data

text: analysis

text: (Tukey,

text: Nolan

text: and

text: Temple

text: Lang,

text: Real

text: data

text: analysis

text: makes

text: the

text: discipline

text: more

text: concrete

text: to

text: students.

text: Focus

text: on

text: solving

text: a

text: real

text: problem

text: can

text: be

text: engaging

text: to

text: students

text: who

text: might

text: otherwise

text: ﬁnd

text: the

text: subject

text: boring.

text: Teaching

text: data

text: analysis

text: is

text: but

text: it’s

text: challenging

text: in

text: the

text: way

text: that

text: teaching

text: the

text: practice

text: of

text: engineering

text: or

text: using

text: the

text: scientiﬁc

text: method

text: is

text: challenging.

text: By

text: not

text: giving

text: students

text: practice

text: doing

text: data

text: analysis

text: for

text: a

text: real

text: problem,

text: the

text: statistics

text: curriculum

text: may

text: encourage

text: students

text: to

text: view

text: statistical

text: methodology

text: as

text: a

text: hammer

text: to

text: be

text: procedurally

text: applied

text: to

text: data.

text: It’s

text: well

text: established

text: in

text: engineering

text: and

text: the

text: physical

text: sciences

text: that

text: students

text: should

text: get

text: some

text: practical

text: experience

text: doing

text: the

text: thing

text: during

text: their

text: education:

text: why

text: does

text: the

text: same

text: principle

text: not

text: apply

text: more

text: often

text: statistics

text: at

text: the

text: undergraduate

text: level?

text: When

text: teaching

text: statistical

text: it

text: might

text: be

text: more

text: eﬀective

text: to

text: intro-

text: duce

text: the

text: model

text: (e.g.

text: linear/logistic

text: regression)

text: in

text: terms

text: of

text: a

text: predictive

text: context

text: instead

text: of

text: the

text: traditional

text: inferential

text: context.

text: While

text: the

text: math

text: behind

text: linear

text: models

text: may

text: not

text: be

text: particularly

text: sophisticated,

text: the

text: concept

text: of

text: randomness

text: and

text: relating

text: it

text: to

text: the

text: real

text: world

text: through

text: data

text: is

text: non-trivial.

text: Moreover,

text: inferential

text: modeling

text: comes

text: with

text: a

text: lot

text: of

text: baggage.

text: For

text: predictive

text: purposes,

text: models

text: such

text: as

text: linear

text: regression

text: can

text: be

text: introduced

text: as

text: an

text: optimization

text: problem

text: which

text: can

text: be

text: heuristically

text: motivated

text: and

text: analytically

text: Once

text: students

text: are

text: com-

text: fortable

text: with

text: data

text: modeling

text: we

text: can

text: then

text: introduce

text: inferential/conﬁrmatory

text: modeling.

text: Finally,

text: we

text: suggest

text: teaching

text: data

text: before

text: statistics

text: in

text: introductory

text: statistics

text: courses;

text: in

text: other

text: words,

text: we

text: should

text: teach

text: exploratory

text: analysis

text: before

text: inferen-

text: tial

text: analysis.

text: This

text: would

text: involve

text: teaching

text: programming,

text: data

text: visualization

text: and

text: manipulation

text: before

text: teaching

text: hypothesis

text: testing.

text: Students

text: are

text: more

text: likely

text: to

text: care

text: about

text: hypothesis

text: testing

text: if

text: they

text: have

text: actually

text: worked

text: on

text: a

text: real

text: prob-

text: years-of-data-science/

text: E.g.

text: in

text: an

text: upper

text: level

text: undergraduate

text: course

text: such

text: as

text: UNC’s

text: STOR

text: Statistical

text: Meth-

text: ods

text: I.

text: See

text: for

text: example

text: html.

text: ##LTLine##

text: ##LTLine##


text: 20 Iain Carmichael, J.S. Marron lem/dataset which motivates it (as opposed to a hypothetical or artiﬁcial one). Berkeley’s new course Data 8: The Foundations of Data Science appears to take this approach55. 5 Conclusion So far the arguments in this paper have been about providing value to society by broadening the discipline in technical ways. Equally as important is increasing diversity in statistics by encouraging women and underrepresented minorities to join and stay in the discipline. Programs, conferences and other eﬀorts such as: ASA Committee on Minorities in Statistics56, the Women in Machine Learning conference57 and the Women in Statistics and Data Science conference58 should be strongly encouraged and expanded upon. We return to the question of whether data science and statistics are really two diﬀerent disciplines. If statistics is deﬁned as the narrow discipline de- scribed by the quote from John Chambers in Section then the answer is yes. However, if statistics embraces the broader idea of greater data science (e.g. by putting more focus on computation in education, research and communication) then we argue the answer is no. Acknowledgements This research was supported in part by the National Science Foun- dation under Grant No. 1633074. We would like to thank Deborah Carmichael for editorial comments. References Alivisatos P (2017) Stem and computer science education: Preparing the 21st century workforce. Research and Technology Subcommittee House Commit- tee on Science, Space, and Technology Anderson C (2008) The end of theory: The data deluge makes the scientiﬁc method obsolete. Wired magazine 16(7):16–07 Aravkin A, Davis D (2016) A smart stochastic algorithm for nonconvex op- timization with applications to robust machine learning. arXiv preprint arXiv:161001101 Association AS, et al (2014) Curriculum guidelines for undergraduate pro- grams in statistical science. Retrieved March 3, 2009, from http://www amstat org/education/curriculumguidelines cfm Barnes N (2010) Publish your computer code: it is good enough. Nature News 467(7317):753–753 55 E.g. see the order of the chapters in the textbook: https://www.inferentialthinking. com/. 56 http://community.amstat.org/cmis/home 57 http://wimlworkshop.org/ 58 http://ww2.amstat.org/meetings/wsds/2018/ ##LTLine## ##LTLine##

text: Iain

text: Carmichael,

text: J.S.

text: Marron

text: lem/dataset

text: which

text: motivates

text: it

text: (as

text: opposed

text: to

text: a

text: hypothetical

text: or

text: artiﬁcial

text: one).

text: Berkeley’s

text: new

text: course

text: Data

text: The

text: Foundations

text: of

text: Data

text: Science

text: appears

text: to

text: take

text: this

text: Conclusion

text: So

text: far

text: the

text: arguments

text: in

text: this

text: paper

text: have

text: been

text: about

text: providing

text: value

text: to

text: society

text: by

text: broadening

text: the

text: discipline

text: in

text: technical

text: ways.

text: Equally

text: as

text: important

text: is

text: increasing

text: diversity

text: in

text: statistics

text: by

text: encouraging

text: women

text: and

text: underrepresented

text: minorities

text: to

text: join

text: and

text: stay

text: in

text: the

text: discipline.

text: Programs,

text: conferences

text: and

text: other

text: eﬀorts

text: such

text: as:

text: ASA

text: Committee

text: on

text: Minorities

text: in

text: the

text: Women

text: in

text: Machine

text: Learning

text: and

text: the

text: Women

text: in

text: Statistics

text: and

text: Data

text: Science

text: should

text: be

text: strongly

text: encouraged

text: and

text: expanded

text: upon.

text: We

text: return

text: to

text: the

text: question

text: of

text: whether

text: data

text: science

text: and

text: statistics

text: are

text: really

text: two

text: diﬀerent

text: disciplines.

text: If

text: statistics

text: is

text: deﬁned

text: as

text: the

text: narrow

text: discipline

text: de-

text: scribed

text: by

text: the

text: quote

text: from

text: John

text: Chambers

text: in

text: Section

text: then

text: the

text: answer

text: is

text: yes.

text: However,

text: if

text: statistics

text: embraces

text: the

text: broader

text: idea

text: of

text: greater

text: data

text: science

text: (e.g.

text: by

text: putting

text: more

text: focus

text: on

text: computation

text: in

text: education,

text: research

text: and

text: communication)

text: then

text: we

text: argue

text: the

text: answer

text: is

text: no.

text: Acknowledgements

text: This

text: research

text: was

text: supported

text: in

text: part

text: by

text: the

text: National

text: Science

text: Foun-

text: dation

text: under

text: Grant

text: No.

text: We

text: would

text: like

text: to

text: thank

text: Deborah

text: Carmichael

text: for

text: editorial

text: comments.

text: References

text: Alivisatos

text: P

text: Stem

text: and

text: computer

text: science

text: education:

text: Preparing

text: the

text: century

text: workforce.

text: Research

text: and

text: Technology

text: Subcommittee

text: House

text: Commit-

text: tee

text: on

text: Science,

text: Space,

text: and

text: Technology

text: Anderson

text: C

text: The

text: end

text: of

text: theory:

text: The

text: data

text: deluge

text: makes

text: the

text: scientiﬁc

text: method

text: obsolete.

text: Wired

text: magazine

text: Aravkin

text: A,

text: Davis

text: D

text: A

text: smart

text: stochastic

text: algorithm

text: for

text: nonconvex

text: op-

text: timization

text: with

text: applications

text: to

text: robust

text: machine

text: learning.

text: arXiv

text: preprint

text: Association

text: AS,

text: et

text: al

text: Curriculum

text: guidelines

text: for

text: undergraduate

text: pro-

text: grams

text: in

text: statistical

text: science.

text: Retrieved

text: March

text: from

text: amstat

text: org/education/curriculumguidelines

text: cfm

text: Barnes

text: N

text: Publish

text: your

text: computer

text: code:

text: it

text: is

text: good

text: enough.

text: Nature

text: News

text: E.g.

text: see

text: the

text: order

text: of

text: the

text: chapters

text: in

text: the

text: textbook:

text: com/.

text: ##LTLine##

text: ##LTLine##


text: 22 Iain Carmichael, J.S. Marron Felder RM, Brent R (2016) Teaching and learning STEM: A practical guide. John Wiley & Sons Freitas AA (2014) Comprehensible classiﬁcation models: a position paper. ACM SIGKDD explorations newsletter 15(1):1–10 Gentleman R, Carey V, Huber W, Irizarry R, Dudoit S (2006) Bioinformatics and computational biology solutions using R and Bioconductor. Springer Science & Business Media Goodfellow I, Bengio Y, Courville A (2016) Deep learning. book in preparation for mit press. URL¡ http://www deeplearningbook org Graves TL, Karr AF, Marron J, Siy H (2000) Predicting fault incidence us- ing software change history. IEEE Transactions on software engineering 26(7):653–661 Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA (2011) Robust statis- tics: the approach based on inﬂuence functions, vol 114. John Wiley & Sons Hand DJ, et al (2006) Classiﬁer technology and the illusion of progress. Sta- tistical science 21(1):1–14 Hardin J, Hoerl R, Horton NJ, Nolan D, Baumer B, Hall-Holt O, Murrell P, Peng R, Roback P, Temple Lang D, et al (2015) Data science in statistics curricula: Preparing students to “think with data”. The American Statisti- cian 69(4):343–353 Hicks SC, Irizarry RA (2017) A guide to teaching data science. The American Statistician (just-accepted):00–00 Hooker G, Hooker C (2017) Machine learning and the future of realism. arXiv preprint arXiv:170404688 Huber PJ (2011) Robust statistics. In: International Encyclopedia of Statisti- cal Science, Springer, pp 1248–1251 Kiar G, Bridgeford E, Chandrashekhar V, Mhembere D, Burns R, Roncal WG, Vogelstein J (2017) A comprehensive cloud framework for accurate and reliable human connectome estimation and meganalysis. bioRxiv p 188706 Knuth DE (1984) Literate programming. The Computer Journal 27(2):97–111 Kross S, Peng RD, Caﬀo BS, Gooding I, Leek JT (2017) The democratization of data science education. PeerJ PrePrints Leek JT, Peng RD (2015) Opinion: Reproducible research can still be wrong: Adopting a prevention approach. Proceedings of the National Academy of Sciences 112(6):1645–1646 Lipton ZC (2016) The mythos of model interpretability. arXiv preprint arXiv:160603490 Lu X, Marron J, Haaland P (2014) Object-oriented data analysis of cell images. Journal of the American Statistical Association 109(506):548–559 Maronna R, Martin RD, Yohai V (2006) Robust statistics, vol 1. John Wiley & Sons, Chichester. ISBN Marron J (1999) Eﬀective writing in mathematical statistics. Statistica neer- landica 53(1):68–75 Marron J (2017) Big data in context and robustness against heterogeneity. Econometrics and Statistics 2:73–80 ##LTLine##

text: Iain

text: Carmichael,

text: J.S.

text: Marron

text: Felder

label: reference

text: RM,

text: Brent

text: R

text: Teaching

text: and

text: learning

text: STEM:

text: A

text: practical

text: guide.

text: John

text: Wiley

text: Sons

text: Freitas

text: AA

text: Comprehensible

text: classiﬁcation

text: models:

text: a

text: position

text: paper.

text: ACM

text: SIGKDD

text: explorations

text: newsletter

text: Gentleman

text: R,

text: Carey

text: V,

text: Huber

text: W,

text: Irizarry

text: R,

text: Dudoit

text: S

text: Bioinformatics

text: and

text: computational

text: biology

text: solutions

text: using

text: R

text: and

text: Bioconductor.

text: Springer

text: Science

text: Business

text: Media

text: Goodfellow

text: I,

text: Bengio

text: Y,

text: Courville

text: A

text: Deep

text: learning.

text: book

text: in

text: preparation

text: for

text: mit

text: press.

text: URL¡

text: deeplearningbook

text: org

text: Graves

text: TL,

text: Karr

text: AF,

text: Marron

text: J,

text: Siy

text: H

text: Predicting

text: fault

text: incidence

text: us-

text: ing

text: software

text: change

text: history.

text: IEEE

text: Transactions

text: on

text: software

text: engineering

text: Hampel

text: FR,

text: Ronchetti

text: EM,

text: Rousseeuw

text: PJ,

text: Stahel

text: WA

text: Robust

text: statis-

text: tics:

text: the

text: approach

text: based

text: on

text: inﬂuence

text: functions,

text: vol

text: John

text: Wiley

text: Sons

text: Hand

text: DJ,

text: et

text: al

text: Classiﬁer

text: technology

text: and

text: the

text: illusion

text: of

text: progress.

text: Sta-

text: tistical

text: science

text: Hardin

text: J,

text: Hoerl

text: R,

text: Horton

text: NJ,

text: Nolan

text: D,

text: Baumer

text: B,

text: Hall-Holt

text: O,

text: Murrell

text: P,

text: Peng

text: R,

text: Roback

text: P,

text: Temple

text: Lang

text: D,

text: et

text: al

text: Data

text: science

text: in

text: statistics

text: curricula:

text: Preparing

text: students

text: to

text: “think

text: with

text: data”.

text: The

text: American

text: Statisti-

text: cian

text: Hicks

text: SC,

text: Irizarry

text: RA

text: A

text: guide

text: to

text: teaching

text: data

text: science.

text: The

text: American

text: Statistician

text: Hooker

text: G,

text: Hooker

text: C

text: Machine

text: learning

text: and

text: the

text: future

text: of

text: realism.

text: arXiv

text: preprint

text: Huber

text: PJ

text: Robust

text: statistics.

text: In:

text: International

text: Encyclopedia

text: of

text: Statisti-

text: cal

text: Science,

text: Springer,

text: pp

text: Kiar

text: G,

text: Bridgeford

text: E,

text: Chandrashekhar

text: V,

text: Mhembere

text: D,

text: Burns

text: R,

text: Roncal

text: WG,

text: Vogelstein

text: J

text: A

text: comprehensive

text: cloud

text: framework

text: for

text: accurate

text: and

text: reliable

text: human

text: connectome

text: estimation

text: and

text: meganalysis.

text: bioRxiv

text: p

text: Knuth

text: DE

text: Literate

text: programming.

text: The

text: Computer

text: Journal

text: Kross

text: S,

text: Peng

text: RD,

text: Caﬀo

text: BS,

text: Gooding

text: I,

text: Leek

text: JT

text: The

text: democratization

text: of

text: data

text: science

text: education.

text: PeerJ

text: PrePrints

text: Leek

text: JT,

text: Peng

text: RD

text: Opinion:

text: Reproducible

text: research

text: can

text: still

text: be

text: wrong:

text: Adopting

text: a

text: prevention

text: approach.

text: Proceedings

text: of

text: the

text: National

text: Academy

text: of

text: Sciences

text: Lipton

text: ZC

text: The

text: mythos

text: of

text: model

text: interpretability.

text: arXiv

text: preprint

text: Lu

text: X,

text: Marron

text: J,

text: Haaland

text: P

text: Object-oriented

text: data

text: analysis

text: of

text: cell

text: images.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association

text: Maronna

text: R,

text: Martin

text: RD,

text: Yohai

text: V

text: Robust

text: statistics,

text: vol

text: John

text: Wiley

text: Sons,

text: Chichester.

text: ISBN

text: Marron

text: J

text: Eﬀective

text: writing

text: in

text: mathematical

text: statistics.

text: Statistica

text: neer-

text: landica

text: Marron

text: J

text: Big

text: data

text: in

text: context

text: and

text: robustness

text: against

text: heterogeneity.

text: Econometrics

text: and

text: Statistics

text: ##LTLine##


text: Optimal Timing to Trade Along a Randomized Brownian Bridge ∗ † ‡ Tim Leung Jiao Li Xin Li May 30, 2020 Abstract This paper studies an optimal trading problem that incorporates the trader’s market view on the terminal asset price distribution and uninformative noise embedded in the asset price dy- namics. We model the underlying asset price evolution by an exponential randomized Brownian bridge (rBb) and consider various prior distributions for the random endpoint. We solve for the optimal strategies to sell a stock, call, or put, and analyze the associated delayed liquidation pre- mia. We solve for the optimal trading strategies numerically and compare them across diﬀerent prior beliefs. Among our results, we ﬁnd that disconnected continuation/exercise regions arise when the trader prescribe a two-point discrete distribution and double exponential distribution. Keywords: speculative trading, Brownian bridge, optimal stopping, variational inequality JEL Classiﬁcation: C41, G11, G13 Mathematics Subject Classiﬁcation (2010): 60G40, 62L15, 91G20, 91G80 ∗ Department of Applied Mathematics, University of Washington, Seattle WA 98195. E-mail: timleung@uw.edu. Corresponding author. † APAM Department, Columbia University, New York, NY 10027; email: jl4170@columbia.edu. ‡ Bank of America Merrill Lynch, One Bryant Park, New York, NY 10036; email: xinli.columbia@gmail.com. 1 ##LTLine##

text: Optimal

text: Timing

text: to

text: Trade

text: Along

text: a

text: Randomized

text: Brownian

text: Bridge

text: Tim

text: Leung

text: Jiao

text: Li

text: Xin

text: Li

text: May

text: Abstract

text: This

text: paper

text: studies

text: an

text: optimal

text: trading

text: problem

text: that

text: incorporates

text: the

text: trader’s

text: market

text: view

text: on

text: the

text: terminal

text: asset

text: price

text: distribution

text: and

text: uninformative

text: noise

text: embedded

text: in

text: the

text: asset

text: price

text: dy-

text: namics.

text: We

text: model

text: the

text: underlying

text: asset

text: price

text: evolution

text: by

text: an

text: exponential

text: randomized

text: Brownian

text: bridge

text: (rBb)

text: and

text: consider

text: various

text: prior

text: distributions

text: for

text: the

text: random

text: endpoint.

text: We

text: solve

text: for

text: the

text: optimal

text: strategies

text: to

text: sell

text: a

text: stock,

text: call,

text: or

text: put,

text: and

text: analyze

text: the

text: associated

text: delayed

text: liquidation

text: pre-

text: mia.

text: We

text: solve

text: for

text: the

text: optimal

text: trading

text: strategies

text: numerically

text: and

text: compare

text: them

text: across

text: diﬀerent

text: prior

text: beliefs.

text: Among

text: our

text: results,

text: we

text: ﬁnd

text: that

text: disconnected

text: continuation/exercise

text: regions

text: arise

text: when

text: the

text: trader

text: prescribe

text: a

text: two-point

text: discrete

text: distribution

text: and

text: double

text: exponential

text: distribution.

text: Keywords:

text: speculative

text: trading,

text: Brownian

text: bridge,

text: optimal

text: stopping,

text: variational

text: inequality

text: JEL

text: Classiﬁcation:

text: Mathematics

text: Subject

text: Classiﬁcation

text: Department

text: of

text: Applied

text: Mathematics,

text: University

text: of

text: Washington,

text: Seattle

text: WA

text: E-mail:

text: timleung@uw.edu.

text: Corresponding

text: author.

text: APAM

text: Department,

text: Columbia

text: University,

text: New

text: York,

text: NY

text: email:

text: Bank

text: of

text: America

text: Merrill

text: Lynch,

text: One

text: Bryant

text: Park,

text: New

text: York,

text: NY

text: email:

text: xinli.columbia@gmail.com.

text: ##LTLine##


text: The optimal stopping problem V is solved from the variational inequality: { L − } max V (t, x) , f(t, x) V (t, x) = 0, (3.11) ∈ × R ∈ R where (t, x) [0, T¯) , with the terminal condition V (T¯, x) = f(T¯, x) for all x . We will discuss in Section ?? our numerical scheme to solve for the optimal trading strategies. Next, we examine the trading strategies for stock and options, and study the varying eﬀects of the trader’s belief encoded in the random variable D. For each security type, we will derive the corresponding drive function. It will then be the inputs for the variational inequality (??), which will be solved numerically for the optimal trading strategies. We consider several combinations in securities and beliefs to see any major diﬀerences in the strategies. 3.1 Stocks For selling the stock S, the reward function is simply f(x) = ex. Then, the drive function, denoted by G (t, x), for selling the stock can be computed via (??). Precisely, we get stock − 1 G (t, x) = ( r + A(t, x) + σ2)ex, (3.12) stock 2 ∈ × R for (t, x) [0, T ] . As seen in Figure ??, the function A(t, x) and thus drive function depend heavily on the prescribed distribution of D, and may be nonlinear. In generally, it is diﬃcult to pinpoint the behavior of the drive function G (t, x). Nevertheless, under the normal distribution stock for D, we obtain the following properties for G in x and in t respectively. stock Proposition 5 Suppose that the trader’s belief on log-price follows a normal distribution as in Example 2. Then, the drive function G (t, x) given in (??) is stock (i) downward-sloping and concave in x if √ ≤ − σ > T σ and x q(t) 2, D √ or ≥ − σ < T σ and x q(t) 1, D (ii) downward-sloping and convex in x if √ − ≤ ≤ − σ > T σ and q(t) 2 x q(t) 1, D (iii) upward-sloping and concave in x if √ − ≤ ≤ − σ < T σ and q(t) 2 x q(t) 1, D (iv) upward-sloping and convex in x if √ ≥ − σ > T σ and x q(t) 1, D √ or ≤ − σ < T σ and x q(t) 2, D 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: optimal

text: stopping

text: problem

text: V

text: is

text: solved

text: from

text: the

text: variational

text: inequality:

text: L

text: max

text: V

text: (t,

text: x)

text: f(t,

text: x)

text: V

text: (t,

text: x)

text: R

text: R

text: where

text: (t,

text: x)

text: T¯)

text: with

text: the

text: terminal

text: condition

text: V

text: (T¯,

text: x)

text: f(T¯,

text: x)

text: for

text: all

text: x

text: We

text: will

text: discuss

text: in

text: Section

text: our

text: numerical

text: scheme

text: to

text: solve

text: for

text: the

text: optimal

text: trading

text: strategies.

text: Next,

text: we

text: examine

text: the

text: trading

text: strategies

text: for

text: stock

text: and

text: options,

text: and

text: study

text: the

text: varying

text: eﬀects

text: of

text: the

text: trader’s

text: belief

text: encoded

text: in

text: the

text: random

text: variable

text: D.

text: For

text: each

text: security

text: type,

text: we

text: will

text: derive

text: the

text: corresponding

text: drive

text: function.

text: It

text: will

text: then

text: be

text: the

text: inputs

text: for

text: the

text: variational

text: inequality

text: which

text: will

text: be

text: solved

text: numerically

text: for

text: the

text: optimal

text: trading

text: strategies.

text: We

text: consider

text: several

text: combinations

text: in

text: securities

text: and

text: beliefs

text: to

text: see

text: any

text: major

text: diﬀerences

text: in

text: the

text: strategies.

text: Stocks

text: For

text: selling

text: the

text: stock

text: S,

text: the

text: reward

text: function

text: is

text: simply

text: f(x)

text: ex.

text: Then,

text: the

text: drive

text: function,

text: denoted

text: by

text: G

text: (t,

text: x),

text: for

text: selling

text: the

text: stock

text: can

text: be

text: computed

text: via

text: Precisely,

text: we

text: get

text: stock

text: G

text: (t,

text: x)

text: r

text: A(t,

text: x)

text: stock

text: R

text: for

text: (t,

text: x)

text: T

text: As

text: seen

text: in

text: Figure

text: the

text: function

text: A(t,

text: x)

text: and

text: thus

text: drive

text: function

text: depend

text: heavily

text: on

text: the

text: prescribed

text: distribution

text: of

text: D,

text: and

text: may

text: be

text: nonlinear.

text: In

text: generally,

text: it

text: is

text: diﬃcult

text: to

text: pinpoint

text: the

text: behavior

text: of

text: the

text: drive

text: function

text: G

text: (t,

text: x).

text: Nevertheless,

text: under

text: the

text: normal

text: distribution

text: stock

text: for

text: D,

text: we

text: obtain

text: the

text: following

text: properties

text: for

text: G

text: in

text: x

text: and

text: in

text: t

text: respectively.

text: stock

text: Proposition

text: Suppose

text: that

text: the

text: trader’s

text: belief

text: on

text: log-price

text: follows

text: a

text: normal

text: distribution

text: as

text: in

text: Example

text: Then,

text: the

text: drive

text: function

text: G

text: (t,

text: x)

text: given

text: in

text: is

text: stock

text: (i)

text: downward-sloping

text: and

text: concave

text: in

text: x

text: if

text: σ

text: T

text: σ

text: and

text: x

text: q(t)

text: D

text: or

text: σ

text: T

text: σ

text: and

text: x

text: q(t)

text: D

text: (ii)

text: downward-sloping

text: and

text: convex

text: in

text: x

text: if

text: σ

text: T

text: σ

text: and

text: q(t)

text: x

text: q(t)

text: D

text: (iii)

text: upward-sloping

text: and

text: concave

text: in

text: x

text: if

text: σ

text: T

text: σ

text: and

text: q(t)

text: x

text: q(t)

text: D

text: (iv)

text: upward-sloping

text: and

text: convex

text: in

text: x

text: if

text: σ

text: T

text: σ

text: and

text: x

text: q(t)

text: D

text: or

text: σ

text: T

text: σ

text: and

text: x

text: q(t)

text: D

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3.6 2.7 3.4 Exercise Region 2.6 Continuation Region 3.2 2.5 3 2.4 Continuation Region Exercise Region 2.8 2.3 2.6 2.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 time time (a) (b) 3.4 3.4 3.3 3.3 Exercise Region Exercise Region 3.2 3.2 3.1 3.1 3 3 Continuation Region Continuation Region 2.9 2.9 2.8 2.8 2.7 2.7 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 time time (c) (d) Figure 3: Optimal boundaries for selling stock under two-point discrete distribution (a & b), normal distribution (c) and double exponential distribution (d). Parameters for each plot as follows: (a) − − δ = δ = 0.1, p = p = 0.5; (b) δ = δ = 2, p = p = 0.5; (c) µ = 0, σ = 0.2; (d) θ = 0, p = u d u d u d u d D 1 p = 0.5, λ = λ = 10. Common parameters: S = 2.72(X = 1), r = 0.1, T¯ = 1, T = 1.1, σ = 0.4. 2 1 2 0 0 16 ecirp kcots ecirp kcots ecirp kcots ecirp kcots ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: Exercise

text: Region

text: Continuation

text: Region

text: Continuation

text: Region

text: Exercise

text: Region

text: time

text: time

text: (a)

text: (b)

text: Exercise

text: Region

text: Exercise

text: Region

text: Continuation

text: Region

text: Continuation

text: Region

text: time

text: time

text: (c)

text: (d)

text: Figure

text: Optimal

text: boundaries

text: for

text: selling

text: stock

text: under

text: two-point

text: discrete

text: distribution

text: (a

text: b),

text: normal

text: distribution

text: (c)

text: and

text: double

text: exponential

text: distribution

text: (d).

text: Parameters

text: for

text: each

text: plot

text: as

text: follows:

text: (a)

text: δ

text: δ

text: p

text: p

text: (b)

text: δ

text: δ

text: p

text: p

text: (c)

text: µ

text: σ

text: (d)

text: θ

text: p

text: u

text: d

text: u

text: d

text: u

text: d

text: u

text: d

text: D

text: p

text: λ

text: λ

text: Common

text: parameters:

text: S

text: r

text: T¯

text: T

text: σ

text: ecirp

text: kcots

text: ecirp

text: kcots

text: ecirp

text: kcots

text: ecirp

text: kcots

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text: Figure 5: Optimal trading regions for stock liquidation under the two-point discrete distribution. The continuation regions are in yellow (light) color and the exercise regions are in blue (dark), − and they are disconnected. Parameters: δ = δ = 0.8, p = p = 0.5. Common parameters: u d u d X = 1, r = 0.1, T¯ = 1, T = 1.1, σ = 0.4. 0 (a) (b) Figure 6: Optimal trading regions for stock liquidation under the double exponential distribution. The continuation regions are in yellow (light) color and the exercise regions are in blue (dark), and they are disconnected. Parameters for (a): θ = 0, λ = λ = 4.714, p = p = 0.5; parameters for 1 2 1 2 (b): θ = 0, λ = λ = 3.536, p = p = 0.5. Common parameters: X = 1, r = 0.1, T¯ = 1, T = 1 2 1 2 0 1.1, σ = 0.4.. 18 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Figure

text: Optimal

label: caption

text: trading

text: regions

text: for

text: stock

text: liquidation

text: under

text: the

text: two-point

text: discrete

text: distribution.

text: The

text: continuation

text: regions

text: are

text: in

text: yellow

text: (light)

text: color

text: and

text: the

text: exercise

text: regions

text: are

text: in

text: blue

text: (dark),

text: and

text: they

text: are

text: disconnected.

text: Parameters:

text: δ

text: δ

text: p

text: p

text: Common

text: parameters:

text: u

text: d

text: u

text: d

text: X

text: r

text: T¯

text: T

text: σ

text: (a)

text: (b)

text: Figure

text: Optimal

text: trading

text: regions

text: for

text: stock

text: liquidation

text: under

text: the

text: double

text: exponential

text: distribution.

text: The

text: continuation

text: regions

text: are

text: in

text: yellow

text: (light)

text: color

text: and

text: the

text: exercise

text: regions

text: are

text: in

text: blue

text: (dark),

text: and

text: they

text: are

text: disconnected.

text: Parameters

text: for

text: (a):

text: θ

text: λ

text: λ

text: p

text: p

text: parameters

text: for

text: (b):

text: θ

text: λ

text: λ

text: p

text: p

text: Common

text: parameters:

text: X

text: r

text: T¯

text: T

text: σ

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text: Proof. The delayed liquidation premium associated with the long-call-short-put position is given by (cid:26)(cid:90) (cid:27) (cid:0) (cid:1) E (cid:26)(cid:90) τ − − − (cid:27) | L − (t, x) = sup e r(u t) G (u, X ) G (u, X ) du X = x call put ∈T call u put u t τ t,T¯ t E τ − − | = sup e r(u t)G (u, X ) du X = x (3.21) ∈T stock u t τ t,T¯ t The last term is delayed liquidation premium associated with selling the stock S. As an interesting consequence of Proposition ??, consider a pair of call and put with strike K ≤ { } 1 with maturity T , and another pair with strike K and maturity T , with T¯ min T , T . The 1 2 2 1 2 respective long-call-short-put positions will yield the same optimal timing strategy. This strategy is identical to that of selling the underlying stock S, which is independent of strike and maturity. 4 Numerical Implementation We now summarize the numerical scheme used to solve the variational inequality satisﬁed by the value function for the optimal trading problem (??). A ﬁnite diﬀerence scheme is used to solve for the optimal liquidation boundaries. The numerical solution of the system of variational inequalities can be obtained by applying a ﬁnite-diﬀerence scheme with the use of the Projected-Successive-Over-Relaxation (PSOR) method. The solution of the resulting equations for value functions are solved by the Successive Over Re- laxation (SOR) method. We refer to ?, Chap.9 for a detailed discussion on the projected SOR method. In each SOR iterative step in ﬁnding the numerical approximation of the value functions, we simply take the maximum value between the approximated function value and asset payoﬀ. Then the variational inequality (??) admits the linear complimentarity form:      L ≤ ≥ ∈ × R V (t, x) 0, V (t, x) f(t, x), (t, x) [0, T¯) , +     (L V (t, x))(f(t, x) − V (t, x)) = 0, (t, x) ∈ [0, T¯) × R , (4.1) + ∈ R V (T¯, x) = f(T¯, x), x . + L We now consider the discretization of the partial diﬀerential equation V (t, x) = 0, over a − bounded uniform grid with discretizations in time (δt = T¯/N), and space (δx = (X X )/M), max min where X and X are the upper and lower bounds on the values of x on the grid. Applying the max min Crank-Nicolson method for x-derivatives and backward diﬀerence for t-derivatives on the resulting equation leads to the grid equation: − − − α − V − − + (1 β)V − γ − V − = α V − + (1 + β)V + γ V (4.2) i,j 1 i 1,j 1 i,j 1 i,j 1 i+1,j 1 i,j i 1,j i,j i,j i+1,j where the coeﬃcients are given by 21

text: Proof.

text: The

text: delayed

text: liquidation

text: premium

text: associated

text: with

text: the

text: long-call-short-put

text: position

text: is

text: given

text: by

text: E

text: τ

text: L

text: (t,

text: x)

text: sup

text: e

text: r(u

text: t)

text: G

text: (u,

text: X

text: G

text: (u,

text: X

text: du

text: X

text: x

text: call

text: put

text: ∈T

text: call

text: u

text: put

text: u

text: t

text: τ

text: t,T¯

text: t

text: E

text: τ

text: sup

text: e

text: r(u

text: t)G

text: (u,

text: X

text: du

text: X

text: x

text: ∈T

text: stock

text: u

text: t

text: τ

text: t,T¯

text: t

text: The

text: last

text: term

text: is

text: delayed

text: liquidation

text: premium

text: associated

text: with

text: selling

text: the

text: stock

text: S.

text: As

text: an

text: interesting

text: consequence

text: of

text: Proposition

text: consider

text: a

text: pair

text: of

text: call

text: and

text: put

text: with

text: strike

text: K

text: with

text: maturity

text: T

text: and

text: another

text: pair

text: with

text: strike

text: K

text: and

text: maturity

text: T

text: with

text: T¯

text: min

text: T

text: T

text: The

text: respective

text: long-call-short-put

text: positions

text: will

text: yield

text: the

text: same

text: optimal

text: timing

text: strategy.

text: This

text: strategy

text: is

text: identical

text: to

text: that

text: of

text: selling

text: the

text: underlying

text: stock

text: S,

text: which

text: is

text: independent

text: of

text: strike

text: and

text: maturity.

text: Numerical

text: Implementation

text: We

text: now

text: summarize

text: the

text: numerical

text: scheme

text: used

text: to

text: solve

text: the

text: variational

text: inequality

text: satisﬁed

text: by

text: the

text: value

text: function

text: for

text: the

text: optimal

text: trading

text: problem

text: A

text: ﬁnite

text: diﬀerence

text: scheme

text: is

text: used

text: to

text: solve

text: for

text: the

text: optimal

text: liquidation

text: boundaries.

text: The

text: numerical

text: solution

text: of

text: the

text: system

text: of

text: variational

text: inequalities

text: can

text: be

text: obtained

text: by

text: applying

text: a

text: ﬁnite-diﬀerence

text: scheme

text: with

text: the

text: use

text: of

text: the

text: Projected-Successive-Over-Relaxation

text: (PSOR)

text: method.

text: The

text: solution

text: of

text: the

text: resulting

text: equations

text: for

text: value

text: functions

text: are

text: solved

text: by

text: the

text: Successive

text: Over

text: Re-

text: laxation

text: (SOR)

text: method.

text: We

text: refer

text: to

text: for

text: a

text: detailed

text: discussion

text: on

text: the

text: projected

text: SOR

text: method.

text: In

text: each

text: SOR

text: iterative

text: step

text: in

text: ﬁnding

text: the

text: numerical

text: approximation

text: of

text: the

text: value

text: functions,

text: we

text: simply

text: take

text: the

text: maximum

text: value

text: between

text: the

text: approximated

text: function

text: value

text: and

text: asset

text: payoﬀ.

text: Then

text: the

text: variational

text: inequality

text: admits

text: the

text: linear

text: complimentarity

text: form:

text: L

text: R

text: V

text: (t,

text: x)

text: V

text: (t,

text: x)

text: f(t,

text: x),

text: (t,

text: x)

text: T¯)

text: (L

text: V

text: (t,

text: x))(f(t,

text: x)

text: V

text: (t,

text: x))

text: (t,

text: x)

text: T¯)

text: R

text: R

text: V

text: (T¯,

text: x)

text: f(T¯,

text: x),

text: x

text: L

text: We

text: now

text: consider

text: the

text: discretization

text: of

text: the

text: partial

text: diﬀerential

text: equation

text: V

text: (t,

text: x)

text: over

text: a

text: bounded

text: uniform

text: grid

text: with

text: discretizations

text: in

text: time

text: (δt

text: T¯/N),

text: and

text: space

text: (δx

text: (X

text: X

text: )/M),

text: max

text: min

text: where

text: X

text: and

text: X

text: are

text: the

text: upper

text: and

text: lower

text: bounds

text: on

text: the

text: values

text: of

text: x

text: on

text: the

text: grid.

text: Applying

text: the

text: max

text: min

text: Crank-Nicolson

text: method

text: for

text: x-derivatives

text: and

text: backward

text: diﬀerence

text: for

text: t-derivatives

text: on

text: the

text: resulting

text: equation

text: leads

text: to

text: the

text: grid

text: equation:

text: α

text: V

text: β)V

text: γ

text: V

text: α

text: V

text: β)V

text: γ

text: V

text: i,j

text: i

text: i,j

text: i,j

text: i,j

text: i

text: i,j

text: i,j

text: where

text: the

text: coeﬃcients

text: are

text: given

text: by


text:  (cid:18) (cid:19)        αi,j = δt 4(cid:18)(σδx2)2 − A((cid:19)tj, xi−1)8+δxA(tj, xi+1) ,     −δt σ2  β = (cid:18) r + , (cid:19) (4.3)  2 (δx)2   σ2 A(t , x − ) + A(t , x ) j i 1 j i+1 γ = δt + , i,j 4(δx)2 8δx − − for i = 1, 2, ..., M 1 and j = 1, 2, ..., N 1. The system to be solved backward in time is M1− V − = r , (4.4) j 1 j 1 j where the right-hand side is            α − V − + α V   0   1,j 1 0,j 1 1,j 0,j   0 .. rj = M2j Vj + .. + . , (4.5) . 0 0 γ − − V − + γ − V M 1,j 1 M,j 1 M 1,j M,j and      − −   1 β γ   − −1,j −   α 1 β γ   2,j − −2,j −   α 1 β γ  M1j =  3,j ... ..3.,j ...  , (4.6) − − − α − 1 β γ − M 2,j − M− 2,j  αM−1,j 1 β      1 + β γ   1,j   α 1 + β γ   2,j 2,j   α 1 + β γ  M2j =  3,j ... .3.,.j ...  , (4.7) α − 1 + β γ − M 2,j M 2,j α − 1 + β (cid:2) (cid:3) M 1,j V = V , V , . . . , V − T . (4.8) j 1,j 2,j M 1,j This leads to a sequence of stationary complementarity problems. Since the trader can close her position anytime prior to expiry, the value function V (t, x) must satisfy the constraint 22 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: αi,j

text: δt

text: −δt

text: σ2

text: β

text: r

text: σ2

text: A(t

text: x

text: A(t

text: x

text: j

text: i

text: j

text: γ

text: δt

text: i,j

text: for

text: i

text: M

text: and

text: j

text: N

text: The

text: system

text: to

text: be

text: solved

text: backward

text: in

text: time

text: is

text: V

text: r

text: j

text: j

text: j

text: where

text: the

text: right-hand

text: side

text: is

text: α

text: V

text: α

text: V

text: rj

text: Vj

text: γ

text: V

text: γ

text: V

text: M

text: M,j

text: M

text: M,j

text: and

text: β

text: γ

text: α

text: β

text: γ

text: α

text: β

text: γ

text: α

text: β

text: γ

text: M

text: M−

text: β

text: β

text: γ

text: α

text: β

text: γ

text: α

text: β

text: γ

text: α

text: β

text: γ

text: M

text: M

text: α

text: β

text: M

text: V

text: V

text: V

text: V

text: T

text: j

text: M

text: This

text: leads

text: to

text: a

text: sequence

text: of

text: stationary

text: complementarity

text: problems.

text: Since

text: the

text: trader

text: can

text: close

text: her

text: position

text: anytime

text: prior

text: to

text: expiry,

text: the

text: value

text: function

text: V

text: (t,

text: x)

text: must

text: satisfy

text: the

text: constraint

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text:    where     ≡ t  ζ ζ(t) = (cid:18) − , (cid:19)   2T σ2(T t)   − ≡ − x X  b1 b1(t, x) = σ2(T −0t) + λ1 ,      c1 = λ1θ, (cid:112) (cid:18) (cid:19)  ≡ b (t, x) 1 d d (t, x) = 2ζ(t) θ + . 1 1 2ζ(t) √ (cid:90) (cid:0) (cid:1) Similarly, by applying y = 2ζ(z + b2 ), we obtain 2ζ ∞ − (cid:90) (cid:18) (cid:19) H (t, x) = p λ exp (ζz2 + b z + c ) dz 2 √2 2 2 2 θ − ∞ √2π b2 4ζc √1 −y2 = p λ exp( 2 2 ) √ exp dy 2 2 √2ζ 4ζ 2π 2ζ(θ+b2 ) 2 − 2ζ √π b2 4ζc − = p λ exp( 2 2 )Φ( d ), 2 2 2  ζ 4ζ (cid:18) (cid:19)   where   −   b ≡ b (t, x) = − x X−0 − λ , 2 2 σ2(T t) 2     − (cid:18) (cid:19)  c2 = λ2θ, (cid:112)  ≡ b (t, x) 2 d d (t, x) = 2ζ(t) θ + . 2 2 2ζ(t) Next, we compute N and N in the numerator with changes of variables y = ζ(z + b1 )2 and (cid:90) 1 2 2ζ y = ζ(z + b2 )2 respectively. (cid:0) (cid:1) 2ζ (cid:90)θ − N (t, x) = p λ z exp (ζz2 + b z + c ) dz 1 −∞ 1 1 1 1 ∞ − = − p (cid:18)λ 1 exp(b21 (cid:19)4ζc1 ) exp(−y)dy − b1 H (t, x) 1 1 1 ζ(θ+b1 )2 2ζ 4ζ 2ζ √ 2ζ − − − 1 − b b2 4ζc − b √π b2 4ζc = p λ exp ζ(θ + 1 )2 exp( 1 1 ) p λ 1 exp( 1 1 )Φ(d ), 1 1 1 1 1 (cid:90) 2ζ 2ζ 4ζ 2ζ ζ 4ζ (cid:0) (cid:1) and ∞ (cid:90) − N (t, x) = p λ z exp (ζz2 + b z + c ) dz 2 2 2 2 2 θ∞ − = p λ(cid:18) 1 exp(b22 (cid:19)4ζc2 ) exp(−y)dy − b2 H (t, x) 2 2 2 ζ(θ+b2 )2 2ζ 4ζ 2ζ √ 2ζ − − 1 − b b2 4ζc − b √π b2 4ζc − = p λ exp ζ(θ + 2 )2 exp( 2 2 ) p λ 2 exp( 2 2 )Φ( d ). 2 2 2 2 2 2ζ 2ζ 4ζ 2ζ ζ 4ζ 26 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: t

text: ζ

text: ζ(t)

text: t)

text: x

text: X

text: x)

text: λ1

text: λ1θ,

text: b

text: (t,

text: x)

text: d

text: d

text: (t,

text: x)

text: θ

text: Similarly,

text: by

text: applying

text: y

text: we

text: obtain

text: 2ζ

text: H

text: (t,

text: x)

text: p

text: λ

text: exp

text: b

text: z

text: c

text: dz

text: √2

text: θ

text: √2π

text: √1

text: p

text: λ

text: exp(

text: exp

text: dy

text: √2ζ

text: 4ζ

text: 2π

text: 2ζ

text: √π

text: p

text: λ

text: exp(

text: )Φ(

text: d

text: ζ

text: 4ζ

text: where

text: b

text: b

text: (t,

text: x)

text: x

text: λ

text: t)

text: λ2θ,

text: b

text: (t,

text: x)

text: d

text: d

text: (t,

text: x)

text: θ

text: Next,

text: we

text: compute

text: N

text: and

text: N

text: in

text: the

text: numerator

text: with

text: changes

text: of

text: variables

text: y

text: ζ(z

text: and

text: 2ζ

text: y

text: ζ(z

text: respectively.

text: 2ζ

text: N

text: (t,

text: x)

text: p

text: λ

text: z

text: exp

text: b

text: z

text: c

text: dz

text: p

text: exp(−y)dy

text: H

text: (t,

text: x)

text: 2ζ

text: 4ζ

text: 2ζ

text: 2ζ

text: b

text: b

text: √π

text: p

text: λ

text: exp

text: ζ(θ

text: exp(

text: p

text: λ

text: exp(

text: )Φ(d

text: 2ζ

text: 2ζ

text: 4ζ

text: 2ζ

text: ζ

text: 4ζ

text: and

text: N

text: (t,

text: x)

text: p

text: λ

text: z

text: exp

text: b

text: z

text: c

text: dz

text: θ∞

text: p

text: exp(−y)dy

text: H

text: (t,

text: x)

text: 2ζ

text: 4ζ

text: 2ζ

text: 2ζ

text: b

text: b

text: √π

text: p

text: λ

text: exp

text: ζ(θ

text: exp(

text: p

text: λ

text: exp(

text: )Φ(

text: d

text: 2ζ

text: 2ζ

text: 4ζ

text: 2ζ

text: ζ

text: 4ζ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: t ∈ X = X + σβ + D, t [0, T ], (2.3) t 0 t T where σ > 0 is a constant parameter. The trader observes the price evolution of the asset, and F ≡ F the corresponding ﬁltration ( ) ≤ ≤ is generated by the log-price process X. In other words, t 0 t T the trader does not directly observe the standard Brownian bridge β over time. Mathematically, this F means that β is not adapted to . The ﬁnancial interpretation is that β reﬂects the uninformative t noise in the markets such as market sentiments and rumors. Without direct observation of β , the t trader is unable to decompose X into β and D at any time t < T . At time T , the trader observes t t the realization of D and thus X . The parameter σ allows us to control eﬀect of β on the log-price T ﬂuctuation. By inspecting equation (2.3), the ratio t can be viewed as the rate at which the value T of D is revealed, going linearly from fully hidden at time 0 to fully revealed at time T . By Proposition 1 of ?, the log-price process satisﬁes the stochastic diﬀerential equation (SDE) dX = A(t, X )dt + σdW , (2.4) t t t ∈ F for t [0, T ], where (W ) ≤ ≤ is an -adapted standard Brownian motion under the probability t 0 t T P measure . The drift term is − − a(t, X ) (X X ) A(t, X ) = t − t 0 , (2.5) t T t where (cid:16) (cid:17) (cid:82) (cid:16) (cid:17) (cid:82)∞ − − E | −∞ zexp z σx2(TX−0t) z2 2Tσ2t(T−t) dF (z) a(t, x) := [D Xt = x] = ∞ − − , (2.6) −∞ exp z x X−0 z2 t − dF (z) σ2(T t) 2Tσ2(T t) · and F ( ) is the cumulative distribution function of the random variable D. We refer to Ap- F pendix A of ? for the derivation of (2.4). The -Brownian motion W appearing in (2.4) can be t considered as the market information accessible by the trader. Unlike β , the value of W contains t t real information relevant to the risky asset price. (2.4)-(2.6) represent how the price innovations reﬂect the probability distribution of D and market information ﬂow. The incomplete information in (2.3) is reformulated as a complete information by projecting the log-price innovations onto the observable ﬁltration. In that sense, the information-based approach is more ﬂexible to add addi- tional interpretation and intuition based on observable price process, as we present in the following sessions of the paper. To understand the conditional expectation in ((cid:90)2.6), we start with the deﬁnition ∞ E | [D X = x] = zπ (z)dz, (2.7) t −∞ t 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: t

text: X

text: X

text: σβ

text: D,

text: t

text: T

text: t

text: t

text: T

text: where

text: σ

text: is

text: a

text: constant

text: parameter.

text: The

text: trader

text: observes

text: the

text: price

text: evolution

text: of

text: the

text: asset,

text: and

text: F

text: F

text: the

text: corresponding

text: ﬁltration

text: is

text: generated

text: by

text: the

text: log-price

text: process

text: X.

text: In

text: other

text: words,

text: t

text: t

text: T

text: the

text: trader

text: does

text: not

text: directly

text: observe

text: the

text: standard

text: Brownian

text: bridge

text: β

text: over

text: time.

text: Mathematically,

text: this

text: F

text: means

text: that

text: β

text: is

text: not

text: adapted

text: to

text: The

text: ﬁnancial

text: interpretation

text: is

text: that

text: β

text: reﬂects

text: the

text: uninformative

text: t

text: noise

text: in

text: the

text: markets

text: such

text: as

text: market

text: sentiments

text: and

text: rumors.

text: Without

text: direct

text: observation

text: of

text: β

text: the

text: t

text: trader

text: is

text: unable

text: to

text: decompose

text: X

text: into

text: β

text: and

text: D

text: at

text: any

text: time

text: t

text: T

text: At

text: time

text: T

text: the

text: trader

text: observes

text: t

text: t

text: the

text: realization

text: of

text: D

text: and

text: thus

text: X

text: The

text: parameter

text: σ

text: allows

text: us

text: to

text: control

text: eﬀect

text: of

text: β

text: on

text: the

text: log-price

text: T

text: ﬂuctuation.

text: By

text: inspecting

text: equation

text: the

text: ratio

text: t

text: can

text: be

text: viewed

text: as

text: the

text: rate

text: at

text: which

text: the

text: value

text: T

text: of

text: D

text: is

text: revealed,

text: going

text: linearly

text: from

text: fully

text: hidden

text: at

text: time

text: to

text: fully

text: revealed

text: at

text: time

text: T

text: By

text: Proposition

text: of

text: the

text: log-price

text: process

text: satisﬁes

text: the

text: stochastic

text: diﬀerential

text: equation

text: (SDE)

text: dX

text: A(t,

text: X

text: )dt

text: σdW

text: t

text: t

text: t

text: F

text: for

text: t

text: T

text: where

text: (W

text: is

text: an

text: -adapted

text: standard

text: Brownian

text: motion

text: under

text: the

text: probability

text: t

text: t

text: T

text: P

text: measure

text: The

text: drift

text: term

text: is

text: a(t,

text: X

text: (X

text: X

text: A(t,

text: X

text: t

text: t

text: t

text: T

text: t

text: where

text: E

text: zexp

text: z

text: dF

text: (z)

text: a(t,

text: x)

text: [D

text: Xt

text: x]

text: exp

text: z

text: x

text: t

text: dF

text: (z)

text: t)

text: t)

text: and

text: F

text: is

text: the

text: cumulative

text: distribution

text: function

text: of

text: the

text: random

text: variable

text: D.

text: We

text: refer

text: to

text: Ap-

text: F

text: pendix

text: A

text: of

text: for

text: the

text: derivation

text: of

text: The

text: -Brownian

text: motion

text: W

text: appearing

text: in

text: can

text: be

text: t

text: considered

text: as

text: the

text: market

text: information

text: accessible

text: by

text: the

text: trader.

text: Unlike

text: β

text: the

text: value

text: of

text: W

text: contains

text: t

text: t

text: real

text: information

text: relevant

text: to

text: the

text: risky

text: asset

text: price.

text: represent

text: how

text: the

text: price

text: innovations

text: reﬂect

text: the

text: probability

text: distribution

text: of

text: D

text: and

text: market

text: information

text: ﬂow.

text: The

text: incomplete

text: information

text: in

text: is

text: reformulated

text: as

text: a

text: complete

text: information

text: by

text: projecting

text: the

text: log-price

text: innovations

text: onto

text: the

text: observable

text: ﬁltration.

text: In

text: that

text: sense,

text: the

text: information-based

text: approach

text: is

text: more

text: ﬂexible

text: to

text: add

text: addi-

text: tional

text: interpretation

text: and

text: intuition

text: based

text: on

text: observable

text: price

text: process,

text: as

text: we

text: present

text: in

text: the

text: following

text: sessions

text: of

text: the

text: paper.

text: To

text: understand

text: the

text: conditional

text: expectation

text: in

text: we

text: start

text: with

text: the

text: deﬁnition

text: E

text: [D

text: X

text: x]

text: zπ

text: (z)dz,

text: t

text: t

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text: 1.8 2 1.6 1.8 1.6 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 t t (a) (b) Figure 1: Path simulation of (2.4) with the prior belief on the future log-price following (a) two-point − discrete distribution and (b) normal distribution. Parameters: (a) δ = 0.3, δ = 0.3, p = 0.5, p = 0.5; u d u d (b) µ = 0, σ = 0.3. Common parameters: X = 1, T = 1, σ = 0.4. D 0 9 X X ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: t

text: t

text: (a)

text: (b)

text: Figure

text: Path

text: simulation

text: of

text: with

text: the

text: prior

text: belief

text: on

text: the

text: future

text: log-price

text: following

text: (a)

text: two-point

text: discrete

text: distribution

text: and

text: (b)

text: normal

text: distribution.

text: Parameters:

text: (a)

text: δ

text: δ

text: p

text: p

text: u

text: d

text: u

text: d

text: (b)

text: µ

text: σ

text: Common

text: parameters:

text: X

text: T

text: σ

text: D

text: X

text: X

text: ##LTFigure##

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text: 12 FRANC¸OIS GREER Theorem 27. [8] A primitive embedding of an even lattice of signature (m , m−) + into an even unimodular lattice of signature (n , n−) exists if + ≤ m + m− min(n , n−). + + When this inequality is strict, the embedding is unique up to isomorphism. ⊂ \ ∈ N The locus NL [O(Λ) D] decomposes into components indexed by m such | 2n ∈ that m2 n. Let v Λ be a(cid:91)lattice vector of self-intersection 2n/m2 so that mv m m has self-intersection 2n. Then we can write \{ ∈ ⊂ ⊥} NL = O(Λ) ψ D : H2,0(ψ) v . 2n m | m2 n 4. Simultaneous Resolution In this section, we study ﬂat families of surfaces with rational double points, focus- ing on the A case. ρ → Theorem 28. [6] Let π : X B be a ﬂat family of surfaces over a smooth variety B, such that each ﬁber X has at worst ADE singularities. Then af(cid:101)ter a ﬁnite base (cid:48) → b (cid:48) (cid:48) → (cid:48) change B B in the category of analytic spaces, the new family π : X B (cid:48) → (cid:48) admits a simultaneous resolution, a proper birational morphism X X which restricts to a minimal resolution on each ﬁber of π. E´tale locally it suﬃces to consider a versal family. In the case of the A singularity ρ (x2 + y2 = zρ+1), this family is given by − − · · · (3) x2 + y2 = zρ+1 + s zρ 1 + s zρ 2 + + s 1 2 ρ ∈ C in the deformation coordinate (cid:126)s S = ρ. The base change required is given by the elementary symmetric polynomials (cid:80) s = σ ((cid:126)t), i i+1 ∈ C (cid:39) C where (cid:126)t T = Spec [t , t , . . . , t ]/ t ρ. The cover is Galois with deck 0 1 ρ i group Sρ+1, branched over the discrimin(cid:89)ant hypersurface. After base change, the equation can be factored ρ x2 + y2 = (z + t ). i i=0 Singularities in the ﬁbers occur over the big diagonal in T, which is the hyperplane (cid:39) Z ⊂ C → arrangement dual to the root system A ρ n. The S covering T S in ρ 3 the case of A is pictured below. 2 ##LTFigure##

text: FRANC¸OIS

text: GREER

text: Theorem

text: A

text: primitive

text: embedding

text: of

text: an

text: even

text: lattice

text: of

text: signature

text: (m

text: m−)

text: into

text: an

text: even

text: unimodular

text: lattice

text: of

text: signature

text: (n

text: n−)

text: exists

text: if

text: m

text: m−

text: min(n

text: n−).

text: When

text: this

text: inequality

text: is

text: strict,

text: the

text: embedding

text: is

text: unique

text: up

text: to

text: isomorphism.

text: N

text: The

text: locus

text: NL

text: [O(Λ)

text: D]

text: decomposes

text: into

text: components

text: indexed

text: by

text: m

text: such

text: that

text: n.

text: Let

text: v

text: Λ

text: be

text: vector

text: of

text: self-intersection

text: so

text: that

text: mv

text: m

text: m

text: has

text: self-intersection

text: Then

text: we

text: can

text: write

text: NL

text: O(Λ)

text: ψ

text: D

text: v

text: m

text: n

text: Simultaneous

text: Resolution

text: In

text: this

text: section,

text: we

text: study

text: ﬂat

text: families

text: of

text: surfaces

text: with

text: rational

text: double

text: points,

text: focus-

text: ing

text: on

text: the

text: A

text: case.

text: ρ

text: Theorem

text: Let

text: π

text: X

text: B

text: be

text: a

text: ﬂat

text: family

text: of

text: surfaces

text: over

text: a

text: smooth

text: variety

text: B,

text: such

text: that

text: each

text: ﬁber

text: X

text: has

text: at

text: worst

text: ADE

text: singularities.

text: Then

text: a

text: ﬁnite

text: base

text: b

text: change

text: B

text: B

text: in

text: the

text: category

text: of

text: analytic

text: spaces,

text: the

text: new

text: family

text: π

text: X

text: B

text: admits

text: a

text: simultaneous

text: resolution,

text: a

text: proper

text: birational

text: morphism

text: X

text: X

text: which

text: restricts

text: to

text: a

text: minimal

text: resolution

text: on

text: each

text: ﬁber

text: of

text: π.

text: E´tale

text: locally

text: it

text: suﬃces

text: to

text: consider

text: a

text: versal

text: family.

text: In

text: the

text: case

text: of

text: the

text: A

text: singularity

text: ρ

text: this

text: family

text: is

text: given

text: by

text: s

text: zρ

text: s

text: zρ

text: s

text: ρ

text: C

text: in

text: the

text: deformation

text: coordinate

text: S

text: ρ.

text: The

text: base

text: change

text: required

text: is

text: given

text: by

text: the

text: elementary

text: symmetric

text: polynomials

text: s

text: σ

text: i

text: C

text: C

text: where

text: T

text: Spec

text: [t

text: t

text: t

text: t

text: ρ.

text: The

text: cover

text: is

text: Galois

text: with

text: deck

text: ρ

text: i

text: group

text: branched

text: over

text: the

text: hypersurface.

text: After

text: base

text: change,

text: the

text: equation

text: can

text: be

text: factored

text: ρ

text: (z

text: t

text: i

text: Singularities

text: in

text: the

text: ﬁbers

text: occur

text: over

text: the

text: big

text: diagonal

text: in

text: T,

text: which

text: is

text: the

text: hyperplane

text: Z

text: C

text: arrangement

text: dual

text: to

text: the

text: root

text: system

text: A

text: ρ

text: n.

text: The

text: S

text: covering

text: T

text: S

text: in

text: ρ

text: the

text: case

text: of

text: A

text: is

text: pictured

text: below.

text: ##LTFigure##