MathOverflow is a question and answer site for professional mathematicians. 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

Having read this link on math stackexchange, I would like to submit to your wisdom the following questions.

Is it possible, mutatis mutandis, to repeat the same reasoning for a fractional Brownian motion?

More specifically: a real valued Gaussian process $B^H:=\{B^H(t)\}_{t\geq 0}$ in a probability space $(\Omega,\mathscr{F},\mathbb{P})$ is a fractional Brownian motion (fBm) with Hurst parameter $H \in(0,1)$ if for all $s,t\in \mathbb{R}_+$

  1. $B^H(0)=0$,

  2. $\mathbb{E}B^H(t)=0$,

  3. $\operatorname{Cov}[B^H(t),B^H(s)]=\frac{1}{2} \left(t^{2 H}+s^{2 H}-|t-s|^{2 H}\right)$.

In addition, the It\^o formula for fBm is written as: $$ f(B^H(t))= \displaystyle\int_0^t f'\left(B^H(s)\right)\, d B^H(s) + H \displaystyle\int_0^t f''\left(B^H(s)\right) s^{2H-1}\, ds. $$ Taking the expectation in both sides of the above equality, we obtain: $$ \mathbb{E}\left[f(B^H(t))\right]= H \displaystyle\int_0^t \mathbb{E}\left[f''\left(B^H(s)\right)\right] s^{2H-1}\, ds. $$ I might continue as; changing the expectation by the conditional expectation $\mathbb{E}_x$ with respect to the event $\{X_0=x\}$ where $X(t)=B^H(t)+ x $, it follows: $$ \mathbb{E}_x\left[f(X^H(t))\right]= H \displaystyle\int_0^t \mathbb{E}_x\left[f''\left(X^H(s)\right)\right] s^{2H-1}\, ds. $$ And if we put $$ m(x,t; H)= \mathbb{E}_x\left[f\left(X^H(t)\right)\right]. $$ We get: $$ \displaystyle\frac{\partial}{\partial t} m(x,t; H)= H \, t^{2H-1}\displaystyle\frac{\partial^2}{\partial x^2 }m(x,t; H) $$ Knowledge that fBm is not a semimartingale nor a Markov process except for cases $H=\frac{1}{2}$. I have some doubts about the last deduction.1.

share|cite|improve this question
up vote 4 down vote accepted

Your construction has been carried out for much more general situations. See

Baudoin, F., Coutin, L. Operators associated with a stochastic differential equation driven by fractional Brownian motions, Stoch. Proc. Appl. 117, 5, 550–574, 2007. ArXiv: math/0509511.

share|cite|improve this answer

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.