# Time reversibility of Stratonovich Diffusion: Reference Request

Please consider the Stratonovich stochastic differential equation (SDE) $$dX = b(X)\circ dB$$ where $B$ is standard Brownian motion and $X(0)=X_0$. This corresponds to the Ito (SDE) $$dX = \frac{1}{2} b(X) b'(X) dt + b(X) dB.$$

I would like a reference showing (or even just stating) that trajectories of this equation are time-reversible in the following sense: that for all $m\geq 1$ and $t_m > t_{m-1} > \ldots > t_1 >0$, the joint distribution of $$(X(t_1), \ldots, X(t_m) )$$ is identical to the joint distribution of $$(X(-t_1), \ldots, X(-t_m) ).$$

Also, is there a particular term for this kind of time-reversibility? People also use time-reversibility to mean detailed balance for systems in equilibrium, which is different from this.

Motivation In a paper I am listing advantages of expressing diffusions in terms of the Stratonovich convention. I want to be able to briefly state that if the drift coefficient in a Stratonovich SDE is 0, then the equation is time-reversible in the sense I state above.

Edit: Further Explanation Here is a clarification of what I mean above, as well as a justification of my claim.

Let $B(t)$ for $t \in \mathbb{R}$ be two-sided Brownian motion with $B(0)=0$. Let $X(t)$ solve the above Stratonovich SDE. Let $Y(t)=X(-t)$. Then $$dY(t) = dX(-t) = -b(X(-t)) \circ dB(-t) = b(Y(t)) \circ d\tilde{B}(t)$$ where $\tilde{B}(t) = -B(-t)$ is also a Brownian motion. So $Y$ solves the same equation as $X$ with a different Brownian motion. These formal manipulations can be justified by letting $B$ be approximated by smooth stochastic processes and then taking the limit using the Wong-Zakai result.

Thanks for any help!

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