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


 
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1. Title Title of document Fractional smoothness of functionals of diffusion processes under a change of measure
 
2. Creator Author's name, affiliation, country Stefan Geiss; University of Innsbruck; Austria
 
2. Creator Author's name, affiliation, country Emmanuel Gobet; École Polytechnique; France
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Parabolic PDE; Qualitative properties of solutions; Diffusion; Interpolation
 
3. Subject Subject classification 60H30; 46B70; 35K10; 35Bxx
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Academy of Finland
 
7. Date (YYYY-MM-DD) 2014-06-13
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/2786
 
10. Identifier Digital Object Identifier 10.1214/ECP.v19-2786
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 19
 
12. Language English=en en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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