Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose I have a diffusion $dX_t = a(X_t)dt + b(X_t)dW_t$. Is there a straightforward way of estimating the variance of $X_T$ for some time $T$, assuming that $T$ is large enough so that a simple Euler approximation isn't accurate?

Clearly, Monte-Carlo methods could be used here, but I'd like something more analytical.

Any ideas?

Many thanks.

share|cite|improve this question
What conditions over a and b ? are they deterministic, are they adapted ? how smooth are they ? – TheBridge Aug 11 '11 at 22:39

One of the methods is the forward Kolmogorov's equation: $$ \begin{cases} \frac{\partial m}{\partial t} &= a(x)\frac{\partial m}{\partial x} + b(x)\frac{\partial^2 m}{\partial x^2}, \\ m(0,x) &= f(x) \end{cases} $$ where $m(t,x) = \mathsf E_xf(X_t) = \mathsf E[f(X_t)|X_0=x]$.

In your case you should make calculations for $f_1 = x$ and $f_2 = x^2$. Then the variance will be given by $$ V[X_T] = m_2(T,x) - (m_1(T,x))^2 $$

share|cite|improve this answer
Does $\mathsf E_xf(X_t)$ stand for expectation with respect to distribution of $X_t$ with initial condition being $x$ ? – Sasha Aug 11 '11 at 15:42
@Sasha: Yes, it does. Thank you for the comment, I will put it in the answer. – Ilya Aug 11 '11 at 15:49
Thanks for your response. I'll most likely end up working in high dimensions, so the FKE isn't really an option. I'm looking for something a good deal cheaper, computationally speaking. On the other hand, it doesn't need to be as accurate either. A rough estimate would do. – Simon Aug 11 '11 at 16:07
@Simon: as far as I know, there are MC methods and FKE. I also would be happy to learn if there are others. The usual point is tha MC are working better in large dimensions. If you're looking for the steady behavior ($T\approx \infty$) or $T$ is just a large number (sorry for the informal statement)? – Ilya Aug 11 '11 at 16:20
I'm just assuming $T$ is a large-ish number. A steady state does not necessarily exist. I was wondering if I could get some useful information from stochastic Taylor expansions, maybe. – Simon Aug 11 '11 at 16:37

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.