1
$\begingroup$

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

Improve this question
$\endgroup$

2 Answers 2

Reset to default
1
$\begingroup$

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.

Improve this answer
$\endgroup$
0
$\begingroup$

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

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .