Implicit function theorem for stochastic differential equation

Question

For each $\theta\in \mathbb{R}$, we consider a stochastic differential equation (SDE): $$ d X_t =b(t,X_t,\theta)dt+\sigma dW_t,\; t\in [0,T];\quad X_0=x_0\in \mathbb{R}, $$ where $\sigma\ge 0$ and the function $b$ is sufficiently smooth such that the above SDE admits a square integrable adapted solution $X^\theta$. To study the regularity of $\theta\to X^\theta$, for the ODE case with $\sigma=0$, one approach is to apply the implicit function theorem (IFT). However, I didn't find any literature for applying the same approach to study the sensitivity of SDEs.

Is it because it is difficult to view the SDE (which is in fact an integral equation) as an operator on proper function spaces, and to study the invertibility of the Frechet derivative? Could you point me to some references on applying IFT in the context of SDE?

Answer 1

See here Differentiable dependence on the initial condition of the solution of a SDE

about continuous dependence on parameters

As mentioned in the comments, Kunita's lectures cover this eg. Lectures on Stochastic Flows And Applications, i.e. depending on the regularity of the coefficients we have analogous differentiability in initial conditions

Another source is in Varadhan's notes too Stochastic1

As you mentioned, indeed the proof goes through creating new systems satisfied by the derivative processes and then showing existence-uniqueness for the entire system.