TITLE: Reversing a diffusion bridge.
QUESTION [6 upvotes]: Suppose I have an $n$-dimensional Itô SDE
$$dX_t = \sigma(X_t) dW_t + \lambda(X_t)dt$$
and I'm interested in diffusion bridges from $X_0=a\in\mathbb R^n$ to $X_T=b\in\mathbb R^n$.
Now let $Y_t$ be a diffusion bridge of the SDE 
$$dY_t=\sigma(Y_t)dW_t - \lambda(Y_t) dt$$
from $Y_0=b$ to $Y_T=a$.
Based on a (very) non-rigorous approximation argument I would conjecture that the laws of $X_t$ and $Y_{T-t}$ agree. 
To make my proof rigorous would involve $\varepsilon$'s and $\delta$'s that I think would quite long.  Is this a known result? Is there a proof using a theorem about diffusions. Or is it wrong?
Pointers etc are very much appreciated.

REPLY [3 votes]: The laws of the processes do not agree. Time-reversal for stochastic processes is more complicated than just to swich signs in the drift compnent. Think of an Ornstein-Uhlenbeck
process started from an equilibrium for example. 
Just google for time-reversed diffusions for more help.