Electronic Communications in Probability

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

Stefan Geiss and Emmanuel Gobet

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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.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 35, 14 pp.

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

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Zentralblatt MATH identifier

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 46B70: Interpolation between normed linear spaces [See also 46M35] 35K10: Second-order parabolic equations 35Bxx: Qualitative properties of solutions

Parabolic PDE Qualitative properties of solutions Diffusion Interpolation

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Geiss, Stefan; Gobet, Emmanuel. Fractional smoothness of functionals of diffusion processes under a change of measure. Electron. Commun. Probab. 19 (2014), paper no. 35, 14 pp. doi:10.1214/ECP.v19-2786. https://projecteuclid.org/euclid.ecp/1465316737

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