Mean-field stochastic differential equations and associated PDEs
| Autor: | Buckdahn, Rainer, Li, Juan, Peng, Shige, Rainer, Catherine |
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| Rok vydání: | 2014 |
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| Druh dokumentu: | Working Paper |
| Popis: | In this paper we consider a mean-field stochastic differential equation, also called Mc Kean-Vlasov equation, with initial data $(t,x)\in[0,T]\times R^d,$ which coefficients depend on both the solution $X^{t,x}_s$ but also its law. By considering square integrable random variables $\xi$ as initial condition for this equation, we can easily show the flow property of the solution $X^{t,\xi}_s$ of this new equation. Associating it with a process $X^{t,x,P_\xi}_s$ which coincides with $X^{t,\xi}_s$, when one substitutes $\xi$ for $x$, but which has the advantage to depend only on the law $P_\xi$ of $\xi$, we characterise the function $V(t,x,P_\xi)=E[\Phi(X^{t,x,P_\xi}_T,P_{X^{t,\xi}_T})]$ under appropriate regularity conditions on the coefficients of the stochastic differential equation as the unique classical solution of a non local PDE of mean-field type, involving the first and second order derivatives of $V$ with respect to its space variable and the probability law. The proof bases heavily on a preliminary study of the first and second order derivatives of the solution of the mean-field stochastic differential equation with respect to the probability law and a corresponding It\^{o} formula. In our approach we use the notion of derivative with respect to a square integrable probability measure introduced in \cite{PL} and we extend it in a direct way to second order derivatives. Comment: 37 pages. The results were presented by Rainer Buckdahn at the "Workshop in Probability and its Applications (17-20 March 2014)" in Mathematical Institute of University of Oxford on March 18, 2014; at the triangle seminar in UBO (Brest, France) on April 1, 2014; at the "7th International Symposium on BSDEs ( 22-27 June 2014)" in Shandong University, Weihai (China) on June 24, 2014 |
| Databáze: | arXiv |
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