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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

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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.

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