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

Abstract

Let $v:[0,T]\times {\mathbb{R}}^d\to \mathbb{R}$ be the solution of the parabolic backward equation $ \partial_t v + (1/2) \sum_{i,l} [σσ^\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 ${\mathbb{R}}^d$-valued diffusion process on some appropriate $(Ω,\cF,\Q)$. For $p\in [2,\infty )$ and a measure $d¶=λ_T d\Q$, where ${\lambda }_T$ satisfies the Muckenhoupt condition $A_{\alpha }$ for $\alpha \in (1,p)$, we relate the behavior of $\|g(X_T)-\ept g(X_T) \|_{L_p(¶)}$, $\Vert \nabla v(t,X_t){\Vert }_{L_p(\P )}$ and $\Vert D^{2}v(t,X_t){\Vert }_{L_p(\P )}$ to each other, where $D^2v:=(\partial_{x_i \partial_{x_l}v)_{i,l}$ is the Hessian matrix.