TITLE: Solving a SDE with quadratic drift
QUESTION [5 upvotes]: I am wondering whether the following SDE can be solved explicitly? 
$$
d X_t = X_t^2 d t + X_t d B_t
$$
where $B_t$ is a standard Brownian motion. If not, can we say some thing about the moments of the solution, i.e., $E(|X_t|^n)$?
Thank you very much for any hints!
Anand

REPLY [7 votes]: Actually your SDE may be solved explicitly. Look at the more general SDE,
\begin{align}
dX_{t} = (a X_{t}^{n} + b X_{t}) dt + c X_{t} dW_{t}
\end{align}
where $n > 1$ and $a,b,c \in \mathbb{R}$. It has a solution given by
\begin{align}
dX_{t} = \Theta_{t} \Bigl( X_{0}^{1-n} + a(1-n)\int_{0}^{t} \Theta^{n-1}_{s} ds \Bigr)^{\frac{1}{1-n}}
\end{align}
with
\begin{align}
\Theta_{t} = e^{ (b-\frac{1}{2}c^2) t + c W_{t}}
\end{align}
See p. 125 in "Numerical Solution of Stochastic Differential Equations", 1995, Peter E. Kloeden and Eckhard Platen.