Solutions of Stratonovich SDEs

Definition

First, consider the ODE

A solution of it could be defined as any function $f(x)$ such that

Here, ${\int }_0^{x}F(t,f(t))dt$$ is the Riemann--Stieltjes integral or the Lebesgue integral fo the one-variable function $h(t)=F(t,f(t))$. This may seem a notational truism, but it isn't. It is a way to effectively check if a function $f$ is a solution, without using differentiation.

Analogously, a solution of a Stratonovich SDE

is a stochastic process $X_t$ such that almost surely

Here we have:

• ${\int }_0^t{\sigma }^i(s,X_s)\circ d{W}_s^i$ stands for the corresponding Stratonovich integral.
• What about ${\int }_0^{t}b(s,X_s)ds$? We have that $X_s$ is a curve-valued random variable. For each result of the draw, you have a function. So $b(s,X_s)$ is another function. And ${\int }_0^{t}b(s,X_s)ds$ is, too. So ${\int }_0^{t}b(s,X_s)ds$ is a new stochastic process obtained by composition and Riemann-integration of the individual curves $X_t$.

Examples

• Pure noise:

  $$d{X}_t=\circ d{W}_t,X_0=0\Rightarrow X_t=W_t.$$

• Drift + noise:

  $$d{X}_t=dt+\circ d{W}_t,X_0=0\Rightarrow X_t=t+W_t.$$

• Geometric (multiplicative noise):

  $$d{X}_t=\alpha X_{t}dt+\beta X_t\circ d{W}_t\Rightarrow X_t=X_0\exp (\alpha t+\beta W_t).$$

• OU with additive noise:

  $$d{X}_t=-\theta X_{t}dt+\sigma \circ d{W}_t\Rightarrow X_t=e^{-\theta t}(X_0+\sigma {\int }_0^t{e}^{\theta s}\circ d{W}_s).$$

• Vector field flow:
  If $d{X}_t=V(X_t)\circ d{W}_t$, then

  $$X_t={\varphi }_{W_t}(X_0)$$

  where $\varphi$ is the deterministic flow of $\dot{y}=V(y)$. I have to see/prove it.

When does $X_t=F(t,W_t)$ work?

Suppose $X_t=F(t,W_t)$. Stratonovich chain rule:

For $d{X}_t=b(X_t)dt+\sigma (X_t)\circ d{W}_t$, we need

Compatibility requires

so drift and diffusion must be proportional.
In $n$ dimensions this is generalized as $[b,\sigma ]=0$, where $[-,-]$ is the Lie bracket. That is, the vector fields defining the drift and the noise must commute!!