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
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. Google Scholar

[2]

N. Bouleau and D. Lépingle, Numerical Methods for Stochastic Processes, Wiley series in probability and mathematical statistics. Wiley & Sons, Inc, New York, 1994. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

M. B. Giles, Multilevel Monte Carlo path simulation, Operation Research, 56 (2008), 607-617. doi: 10.1287/opre.1070.0496. Google Scholar

[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. Google Scholar

[7]

A. M. GarsiaE. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, volume 24 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1997. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

D. Nualart, Malliavin calculus and related topics, Stochastic processes and related topics (Georgenthal, 1990), Math. Res., Akademie-Verlag, Berlin, 61 (1991), 103–127. Google Scholar

[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.Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[2]

N. Bouleau and D. Lépingle, Numerical Methods for Stochastic Processes, Wiley series in probability and mathematical statistics. Wiley & Sons, Inc, New York, 1994. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

M. B. Giles, Multilevel Monte Carlo path simulation, Operation Research, 56 (2008), 607-617. doi: 10.1287/opre.1070.0496. Google Scholar

[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. Google Scholar

[7]

A. M. GarsiaE. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, volume 24 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1997. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

D. Nualart, Malliavin calculus and related topics, Stochastic processes and related topics (Georgenthal, 1990), Math. Res., Akademie-Verlag, Berlin, 61 (1991), 103–127. Google Scholar

[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.Google Scholar

[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. Google Scholar

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