<p>We investigate some generic properties of solutions of nonlinear McKean–Vlasov stochastic differential equations (MVSDEs), called also mean-field SDEs. These are SDEs whose coefficients are functions of the position and the marginal distribution of the solution in a non linear way. We prove that in the space of uniformly continuous function, the coefficients leading to a unique pathwise solution of the MVSDE is a residual set in the sense of Baire. Equivalently, we say that existence and uniqueness of solutions is a generic property. By using similar techniques, we also show that the convergence of Picard successive approximations and the convergence of the Euler numerical scheme are generic properties.</p>

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Generic Properties of McKean–Vlasov Stochastic Differential Equations

  • Mohamed Amine Mezerdi,
  • Brahim Mezerdi

摘要

We investigate some generic properties of solutions of nonlinear McKean–Vlasov stochastic differential equations (MVSDEs), called also mean-field SDEs. These are SDEs whose coefficients are functions of the position and the marginal distribution of the solution in a non linear way. We prove that in the space of uniformly continuous function, the coefficients leading to a unique pathwise solution of the MVSDE is a residual set in the sense of Baire. Equivalently, we say that existence and uniqueness of solutions is a generic property. By using similar techniques, we also show that the convergence of Picard successive approximations and the convergence of the Euler numerical scheme are generic properties.