Particles systems and numerical schemes for mean reflected stochastic differential equations

Mean reflected stochastic differential equations with jumps

“…For all , we have Under Assumption 4 and thanks to Itô’s formula we get where is given by (2.13). Thus, we obtain As a conclusion, using (2.15), Lemma 3, and the proof of Proposition 2.7 in [BCdRGL16], we deduce that the measure dK has the following density: …”

“…When h is smooth, we get an approximation error proportional to . By the way, we improve the speed of convergence obtained in [BCdRGL16]. Finally, we illustrate these results numerically in Section 5.…”

“…Finally, we deduce that K is Lipschitz continuous and so has a bounded density on [0, T ] for each (see Proposition 2.7 in [BCdRGL16] for more details).…”

“…As we consider a uniform convergence in time, getting the usual rate of convergence is not straightforward. If we only suppose that Assumption 2 holds, we obtain that According to the additional Assumption 3, and in view of the proof of Part (i) of [BCdRGL16, Theorem 3.2], we have…”

“…Stochastic differential equations with mean reflection have been introduced by Briand, Elie, and Hu in their backward forms in [BEH18]. In that work, the authors show that mean reflected stochastic processes exist and are uniquely defined by the associated system of equations of the following form: Because the reflection process K depends on the law of the position, the authors of [BCdRGL16], inspired by mean field games, study the convergence of a numerical scheme based on particle systems to compute solutions to (1.1) numerically.…”