Suppose I have a stochastic differential equation of the form $$dX_t=b(t,X_t)dt + \sigma(t,X_t)dW_t \quad X_0=x_0 \in \mathbb R^n.$$ My professor said that in some cases one wants to compute the derivative of $X_t$ with respect to the initial datum $\frac{\partial X_t}{ \partial x_0 }.$
Where can I find a reference for this?
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
The proof goes through creating new systems satisfied by the derivative processes and then showing existence-uniqueness for the entire system.