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

TL;DR

This paper investigates the fractional smoothness of functionals of diffusion processes under a change of measure, establishing relationships between the regularity of solutions to backward equations and the behavior of functionals.

Contribution

It introduces new connections between the fractional smoothness of functionals and the derivatives of solutions to backward parabolic equations under measure changes.

Findings

Established relations between the Lp norms of functionals and derivatives of the PDE solution.

Extended analysis to measures satisfying the Muckenhoupt condition.

Provided insights into the regularity properties of diffusion process functionals.

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.