Fractional smoothness of functionals of diffusion processes under a change of measure

Abstract

Let $v:[0,T]\times \R^d \to \R$ be the solution of the parabolic backward equation $ \partial_t v + (1/2) \sum_{i,l} [\sigma \sigma^\perp]_{il} \partial_{x_i \partial_{x_l} v + \sum_{i} b_i \partial_{x_i}v + kv =0$ with terminal condition $g$, where the coefficients are time- and state-dependent, and satisfy certain regularity assumptions. Let $X=(X_t)_{t\in [0,T]}$ be the associated $\R^d$-valued diffusion process on some appropriate $(\Omega,\cF,\Q)$. For $p\in [2,\infty)$ and a measure $d\P=\lambda_T d\Q$, where $\lambda_T$ satisfies the Muckenhoupt condition $A_\alpha$ for $\alpha \in (1,p)$, we relate the behavior of $\|g(X_T)-\ept g(X_T) \|_{L_p(\P)}$, $\|\nabla v(t,X_t) \|_{L_p(\P)}$ and $\|D^2 v(t,X_t) \|_{L_p(\P)}$ to each other, where $D^2v:=(\partial_{x_i \partial_{x_l}v)_{i,l}$ is the Hessian matrix.