Pathwise differentiability with respect to a parameter of solutions of stochastic differential equations

Abstract

We consider a stochastic differential equation

$$x^u (t) = v^u (t) + \int_o^t {\sigma (u,s,x_{s^ - }^u )ds_s + } \int_o^t {f(u,s,x_{s^ - }^u ,x)q(ds,dx)}$$

where S is a semimartingale and q a random measure and where the “coefficients” depend on a parameter u. We prove under suitable differentia-bility-conditions that the solution X ^u(t, ω) can be choosen for each u in such a way that the mapping u↷X ^u(t, ω) is continuously differentiable for every (t, ω).