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Weak solutions to Vlasov-McKean equations under Lyapunov-type conditions

Mehri, S ; Sharif University of Technology | 2019

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  1. Type of Document: Article
  2. DOI: 10.1142/S0219493719500424
  3. Publisher: World Scientific Publishing Co. Pte Ltd , 2019
  4. Abstract:
  5. We present a Lyapunov-type approach to the problem of existence and uniqueness of general law-dependent stochastic differential equations. In the existing literature, most results concerning existence and uniqueness are obtained under regularity assumptions of the coefficients with respect to the Wasserstein distance. Some existence and uniqueness results for irregular coefficients have been obtained by considering the total variation distance. Here, we extend this approach to the control of the solution in some weighted total variation distance, that allows us now to derive a rather general weak uniqueness result, merely assuming measurability and certain integrability on the drift coefficient and some non-degeneracy on the dispersion coefficient. We also present an abstract weak existence result for the solution of law-dependent stochastic differential equations with merely measurable coefficients, based on an approximation with law-dependent stochastic differential equations with regular coefficients under Lyapunov-type assumptions. © 2019 World Scientific Publishing Company
  6. Keywords:
  7. Existence and uniqueness of weak solution ; Girsanov theorem ; Lyapunov method ; Vlasov-McKean equations ; Weighted total variation
  8. Source: Stochastics and Dynamics ; Volume 19, Issue 6 , 2019 ; 02194937 (ISSN)
  9. URL: https://www-worldscientific-com.ezp2.semantak.com/doi/abs/10.1142/S0219493719500424