# Time reversibility of stochastic differential equation

For Ito SDE $$dx = f(x,t)dt + g(x,t)dw$$

It transforms initial distribution $$p(x_0)$$ to distribution $$p(x_T)$$ at time $$T$$. My question is whether or not there exists a "reverse" sde $$dx = f'(x,t)dt + g'(x,t)dw$$ such that if the initial distribution is $$q(x_0)=p(x_T)$$, the transformed distribution at time $$T$$ under the sde is $$q(x_T)=p(x_0)$$. Under which assumption, such "reverse" sde exists and what are $$f',g'$$?

My naive guess is $$dx = [f(x,T-t) + g(x,T-t)g^T(x,T-t)\nabla log q(x_T|x_0)]dt + g(x,T-t)dw$$ But I am not sure about correctness and under what assumptions such "reverse" sde exists.

Update: My previous question only requires end marginals. It should be $$q(x_{T-t})=p(x_t)$$.

• That would mean to reverse entropy. Both processes are dissipative, both widen the distribution. Commented Mar 7, 2021 at 18:58
• Notation is crappy. What you likely mean is it transforms an initial distribution, say $p_{0}$, to some distribution $p_{T}$ at time $T$. Now, regarding you question I suggest you dig into literaturem focussing on key word: Doob's transform, h-transform, bridges. Commented Mar 7, 2021 at 19:39