How do I compute the derivative $\frac{\partial X_t}{\partial \sigma}$, where $dX_t=\theta (\mu-X_t)dt+\sigma \sqrt{X_t}dZ_t$?

## 2 Answers

You may want to see Kunita's "Stochastic Flows and Stochastic Differential Equations". Among other things, he develops a calculus of semimartingales with spatial parameters there, and discusses smoothness and differentiation with respect to parameters.

By differentiating the SDE wrt. $\sigma$, we obtain for $Y_t:=\frac{\partial X_t}{\partial \sigma}$, $$dY_t = -\theta Y_t dt +\frac{\sigma Y_t}{2\sqrt{X_t}}dZ_t + \sqrt{X_t}dZ_t. $$