# American Institute of Mathematical Sciences

December  2018, 23(10): 4455-4476. doi: 10.3934/dcdsb.2018171

## Convergence rate of strong approximations of compound random maps, application to SPDEs

 1 Centre de Mathématiques Appliquées, Ecole Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau Cedex, France 2 Laboratoire Analyse, Géométrie et Applications (UMR CNRS 7539), Institut Galile, Université Paris 13, France

* Corresponding author

Received  April 2017 Revised  January 2018 Published  June 2018

Fund Project: This work was funded jointly by Chaire Risques Financiers of the Risk Fondation and the Finance for Energy Market Research Centre

We consider a random map $x\mapsto F(ω,x)$ and a random variable $Θ(ω)$, and we denote by ${{F}^{N}}(ω,x)$ and ${{\Theta }^{N}}(ω)$ their approximations: We establish a strong convergence result, in ${\bf{L}}_p$-norms, of the compound approximation ${{F}^{N}}(ω,{{\Theta }^{N}}(ω) )$ to the compound variable $F(ω,Θ(ω))$, in terms of the approximations of $F$ and $Θ$. We then apply this result to the composition of two Stochastic Differential Equations (SDEs) through their initial conditions, which can give a way to solve some Stochastic Partial Differential Equations (SPDEs), in particular those from stochastic utilities.

Citation: Emmanuel Gobet, Mohamed Mrad. Convergence rate of strong approximations of compound random maps, application to SPDEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4455-4476. doi: 10.3934/dcdsb.2018171
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