# 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  December 2018 Early access  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 and Continuous Dynamical Systems - B, 2018, 23 (10) : 4455-4476. doi: 10.3934/dcdsb.2018171
##### References:
 [1] H. Allouba and W. Zheng, Brownian-time processes: The PDE connection and the half-derivative generator, Annals of probability, 29 (2001), 1780-1795.  doi: 10.1214/aop/1015345772. [2] N. Bouleau and D. Lépingle, Numerical Methods for Stochastic Processes, Wiley series in probability and mathematical statistics. Wiley & Sons, Inc, New York, 1994. [3] M. T. Barlow and M. Yor, Semi-martingale inequalities via the Garsia-Rodemich-Rumsey lemma, and applications to local times, Journal of Functional Analysis, 49 (1982), 198-229.  doi: 10.1016/0022-1236(82)90080-5. [4] N. El Karoui and M. Mrad, An exact connection between two solvable SDEs and a non linear Utility Stochastic PDEs, SIAM Journal on Financial Mathematics, 4 (2013), 697-736.  doi: 10.1137/10081143X. [5] M. B. Giles, Multilevel Monte Carlo path simulation, Operation Research, 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496. [6] E. Gobet and M. Mrad, Strong approximation of stochastic processes at random times and application to their exact simulation, Stochastics, 89 (2017), 883-895.  doi: 10.1080/17442508.2016.1267179. [7] A. M. Garsia, E. Rodemich and H. Jr. Rumsey, A real variable lemma and the continuity of paths of some Gaussian processes, Indiana University Mathematics Journal, 20 (1970), 565-578.  doi: 10.1512/iumj.1971.20.20046. [8] S. Heinrich, Multilevel monte carlo methods, In LSSC '01 Proceedings of the Third International Conference on Large-Scale Scientific Computing, volume 2179 of Lecture Notes in Computer Science, pages 58–67. Springer-Verlag, 2001. doi: 10.1007/3-540-45346-6_5. [9] A. Kohatsu-Higa and M. Sanz-Solé, Existence and regularity of density for solutions to stochastic differential equations with boundary conditions, Stochastics Stochastics Rep., 60 (1997), 1-22.  doi: 10.1080/17442509708834096. [10] H. Kunita, Stochastic Flows and Stochastic Differential Equations, volume 24 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1997. [11] M. Musiela and T. Zariphopoulou, Investment and valuation under backward and forward dynamic exponential utilities in a stochastic factor model, In Advances in Mathematical Finance, pages 303–334. Birkhäuser Boston, 2007. doi: 10.1007/978-0-8176-4545-8_16. [12] M. Musiela and T. Zariphopoulou, Stochastic partial differential equations and portfolio choice, In Contemporary Quantitative Finance, pages 195–216. Springer, 2010. doi: 10.1007/978-3-642-03479-4_11. [13] D. Nualart, Malliavin calculus and related topics, Stochastic processes and related topics (Georgenthal, 1990), Math. Res., Akademie-Verlag, Berlin, 61 (1991), 103–127. [14] C. Rhee and P. W. Glynn, A new approach to unbiased estimation for SDEs, In C. Laroque, J. Himmelspach, R. Pasupathy, O. Rose and A. M. Uhrmacher, editors, Proceedings of the 2012 Winter Simulation Conference, (2012), 495–503. [15] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Comprehensive Studies in Mathematics. Berlin: Springer, third edition, 1999. doi: 10.1007/978-3-662-06400-9.

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##### References:
 [1] H. Allouba and W. Zheng, Brownian-time processes: The PDE connection and the half-derivative generator, Annals of probability, 29 (2001), 1780-1795.  doi: 10.1214/aop/1015345772. [2] N. Bouleau and D. Lépingle, Numerical Methods for Stochastic Processes, Wiley series in probability and mathematical statistics. Wiley & Sons, Inc, New York, 1994. [3] M. T. Barlow and M. Yor, Semi-martingale inequalities via the Garsia-Rodemich-Rumsey lemma, and applications to local times, Journal of Functional Analysis, 49 (1982), 198-229.  doi: 10.1016/0022-1236(82)90080-5. [4] N. El Karoui and M. Mrad, An exact connection between two solvable SDEs and a non linear Utility Stochastic PDEs, SIAM Journal on Financial Mathematics, 4 (2013), 697-736.  doi: 10.1137/10081143X. [5] M. B. Giles, Multilevel Monte Carlo path simulation, Operation Research, 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496. [6] E. Gobet and M. Mrad, Strong approximation of stochastic processes at random times and application to their exact simulation, Stochastics, 89 (2017), 883-895.  doi: 10.1080/17442508.2016.1267179. [7] A. M. Garsia, E. Rodemich and H. Jr. Rumsey, A real variable lemma and the continuity of paths of some Gaussian processes, Indiana University Mathematics Journal, 20 (1970), 565-578.  doi: 10.1512/iumj.1971.20.20046. [8] S. Heinrich, Multilevel monte carlo methods, In LSSC '01 Proceedings of the Third International Conference on Large-Scale Scientific Computing, volume 2179 of Lecture Notes in Computer Science, pages 58–67. Springer-Verlag, 2001. doi: 10.1007/3-540-45346-6_5. [9] A. Kohatsu-Higa and M. Sanz-Solé, Existence and regularity of density for solutions to stochastic differential equations with boundary conditions, Stochastics Stochastics Rep., 60 (1997), 1-22.  doi: 10.1080/17442509708834096. [10] H. Kunita, Stochastic Flows and Stochastic Differential Equations, volume 24 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1997. [11] M. Musiela and T. Zariphopoulou, Investment and valuation under backward and forward dynamic exponential utilities in a stochastic factor model, In Advances in Mathematical Finance, pages 303–334. Birkhäuser Boston, 2007. doi: 10.1007/978-0-8176-4545-8_16. [12] M. Musiela and T. Zariphopoulou, Stochastic partial differential equations and portfolio choice, In Contemporary Quantitative Finance, pages 195–216. Springer, 2010. doi: 10.1007/978-3-642-03479-4_11. [13] D. Nualart, Malliavin calculus and related topics, Stochastic processes and related topics (Georgenthal, 1990), Math. Res., Akademie-Verlag, Berlin, 61 (1991), 103–127. [14] C. Rhee and P. W. Glynn, A new approach to unbiased estimation for SDEs, In C. Laroque, J. Himmelspach, R. Pasupathy, O. Rose and A. M. Uhrmacher, editors, Proceedings of the 2012 Winter Simulation Conference, (2012), 495–503. [15] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Comprehensive Studies in Mathematics. Berlin: Springer, third edition, 1999. doi: 10.1007/978-3-662-06400-9.
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