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2014 Fractional smoothness of functionals of diffusion processes under a change of measure
Stefan Geiss, Emmanuel Gobet
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Electron. Commun. Probab. 19: 1-14 (2014). DOI: 10.1214/ECP.v19-2786

Abstract

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.

Citation

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Stefan Geiss. Emmanuel Gobet. "Fractional smoothness of functionals of diffusion processes under a change of measure." Electron. Commun. Probab. 19 1 - 14, 2014. https://doi.org/10.1214/ECP.v19-2786

Information

Accepted: 13 June 2014; Published: 2014
First available in Project Euclid: 7 June 2016

zbMATH: 06349193
MathSciNet: MR3225866
Digital Object Identifier: 10.1214/ECP.v19-2786

Subjects:
Primary: 60H30
Secondary: 35Bxx, 35K10, 46B70

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