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Mean reflected stochastic differential equations with jumps

Part of: Probabilistic methods, simulation and stochastic differential equations Markov processes Stochastic analysis

Published online by Cambridge University Press:  15 July 2020

Phillippe Briand ,
Abir Ghannoum  and
Céline Labart
Phillippe Briand*
Affiliation:
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA
Abir Ghannoum*
Affiliation:
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA and Univ. Libanaise, LaMA-Liban
Céline Labart*
Affiliation:
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA
*
*Postal address: Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
**Postal address: 73000 Chambéry, France and P.O. Box 37, Tripoli, Liban
***Postal address: 73000Chambéry, France. Email address: [email protected]
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Abstract

In this paper, a reflected stochastic differential equation (SDE) with jumps is studied for the case where the constraint acts on the law of the solution rather than on its paths. These reflected SDEs have been approximated by Briand et al. (2016) using a numerical scheme based on particles systems, when no jumps occur. The main contribution of this paper is to prove the existence and the uniqueness of the solutions to this kind of reflected SDE with jumps and to generalize the results obtained by Briand et al. (2016) to this context.

Keywords

Reflected stochastic differential equationjumpsparticle systems

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 65C35: Stochastic particle methods 60J75: Jump processes
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 2 , June 2020 , pp. 523 - 562
Copyright
© Applied Probability Trust 2020

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References

Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9, 203–228.CrossRefGoogle Scholar
Briand, P., Chaudru de Raynal, P.-E., Guillin, A. and Labart, C. (2016). Particles systems and numerical schemes for mean reflected stochastic differential equations. Submitted.Google Scholar
Briand, P., Elie, R. and Hu, Y. (2018). BSDEs with mean reflexion. Ann. Appl. Prob. 28, 482–510.CrossRefGoogle Scholar
Carmona, R. and Delarue, F. (2018) Probabilistic Theory of Mean Field Games with Applications I. Springer, New York.Google Scholar
Carmona, R. and Delarue, F. (2018) Probabilistic Theory of Mean Field Games with Applications II. Springer, New York.CrossRefGoogle Scholar
Crépey, S. and Matoussi, A. (2008). Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison. Ann. Appl. Prob. 18, 2041–2069.CrossRefGoogle Scholar
Dumitrescu, R. and Labart, C. (2016). Numerical approximation of doubly reflected BSDEs with jumps and RCLL obstacles. J. Math. Anal. Appl. 442, 206–243.CrossRefGoogle Scholar
Dumitrescu, R. and Labart, C. (2016). Reflected scheme for doubly reflected BSDEs with jumps and RCLL obstacles. J. Comput. Appl. Math. 296, 827–839.CrossRefGoogle Scholar
Essaky, E. H., Harraj, N. and Ouknine, Y. (2005). Backward stochastic differential equation with two reflecting barriers and jumps. Stoch. Anal. Appl. 23, 921–938.CrossRefGoogle Scholar
Essaky, E. H. (2008). Reflected backward stochastic differential equation with jumps and RCLL obstacle. Bull. Sci. Math. 132, 690–710.CrossRefGoogle Scholar
Föllmer, H. and Schied, A. (2002). Convex measures of risk and trading constraints. Finance Stoch. 6, 429–447.CrossRefGoogle Scholar
Fournier, N. and Guillin, A. (2015). On the rate of convergence in Wasserstein distance of the empirical measure. Prob. Theory Relat. Fields 162, 707–738.CrossRefGoogle Scholar
Hamadène, S. and Hassani, M. (2006). BSDEs with two reacting barriers driven by a Brownian motion and an independent Poisson noise and related Dynkin game. Electron. J. Prob. 11, 121–145.CrossRefGoogle Scholar
Hamadène, S. and Ouknine, Y. (2003). Reflected backward stochastic differential equation with jumps and random obstacle. Electron. J. Prob. 8, 1–20.CrossRefGoogle Scholar
Kohatsu-Higa, A. (1992). Reflecting stochastic differential equations with jumps. Tech. Rep. 92--30, University of Puerto Rico.Google Scholar
Lasry, J.-M. and Lions, P.-L. (2006). Jeux à champ moyen. I. Le cas stationnaire. C. R. Acad. Sci. Paris 343, 619–625.CrossRefGoogle Scholar
Lasry, J.-M. and Lions, P.-L. (2006). Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Acad. Sci. Paris 343, 679–684.CrossRefGoogle Scholar
Lasry, J.-M. and Lions, P.-L. (2007). Large investor trading impacts on volatility. Ann. Inst. H. Poincaré, Anal. Non Linéaire 24, 311–323.Google Scholar
Lasry, J.-M. and Lions, P.-L. (2007). Mean field games. Japanese J. Math. 2, 229–260.CrossRefGoogle Scholar
Lepingle, D. (1995). Euler scheme for reflected stochastic differential equations. Math. Comput. Simul. 38, 119–126.CrossRefGoogle Scholar
Menaldi, J.-L. and Robin, M. (1985). Reflected diffusion processes with jumps. Ann. Prob. 13, 319–341.CrossRefGoogle Scholar
Pettersson, R. (1995). Approximations for stochastic differential equations with reflecting convex boundaries. Stoch. Process. Appl. 59, 295–308.CrossRefGoogle Scholar
Pettersson, R. (1997). Penalization schemes for reflecting stochastic differential equations. Bernoulli 3, 403–414.CrossRefGoogle Scholar
Quenez, M. C. and Sulem, A. (2014). Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps. Stoch. Process. Appl. 124, 3031–3054.CrossRefGoogle Scholar
Rüschendorf, L. and Rachev, S. T. (1998). Mass Transportation Problems. Vol. 1: Theory. Vol. 2: Applications. Springer, New York.Google Scholar
Skorokhod, A. V. (1961). Stochastic equations for diffusion processes in a bounded region. Theory Prob. Appl. 6, 264–274.CrossRefGoogle Scholar
Slominski, L. (1994). On approximation of solutions of multidimensional SDEs with reflecting boundary conditions. Stoch. Process. Appl. 50, 197–219.CrossRefGoogle Scholar
Slominski, L. (2001). Euler’s approximations of solutions of SDEs with reflecting boundary. Stoch. Process. Appl. 92, 317–337.CrossRefGoogle Scholar
Yu, H. and Song, M. (2012). Numerical solutions of stochastic differential equations driven by Poisson random measure with non-Lipschitz coefficients. J. Appl. Math. 2012.Google Scholar