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

Stefan Geiss (University of Innsbruck)
Emmanuel Gobet (École Polytechnique)


Let $v:[0,T]\times {\mathbf R}^d \to {\mathbf R}$ be the solution of the parabolic backward equation $$\partial_t v + (1/2) \sum_{i,j} [\sigma \sigma^\top]_{i,j} \partial_{x_i}\partial_{x_j}v+ \sum_{i} b_i \partial_{x_i}v + kv =0$$ 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 ${\mathbf R}^d$-valued diffusion process on some appropriate $(\Omega,{\mathcal F},{\mathbb Q})$. For $p\in [2,\infty)$ and a measure $d{\mathbb P}=\lambda_T d{\mathbb Q}$, where $\lambda_T$ satisfies the Muckenhoupt condition $A_p$, we relate the behavior of \[  \|g(X_T)-{\mathbf E}_{\mathbb P}(g(X_T)|{\mathcal F}_t) \|_{L_p({\mathbb P})}, \quad     \|\nabla v(t,X_t)  \|_{L_p({\mathbb P})},           \quad    \|D^2 v(t,X_t)  \|_{L_p({\mathbb P})} \]to each other, where $D^2v:=(\partial_{x_i}\partial_{x_j}v)_{i,j}$ is the Hessian matrix.

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Pages: 1-14

Publication Date: June 13, 2014

DOI: 10.1214/ECP.v19-2786


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