Meanfield stochastic differential equations and associated PDEs
Abstract
In this paper we consider a meanfield stochastic differential equation, also called Mc KeanVlasov 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 meanfield 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 meanfield stochastic differential equation with respect to the probability law and a corresponding Itô 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2014
 DOI:
 10.48550/arXiv.1407.1215
 arXiv:
 arXiv:1407.1215
 Bibcode:
 2014arXiv1407.1215B
 Keywords:

 Mathematics  Probability
 EPrint:
 37 pages. The results were presented by Rainer Buckdahn at the "Workshop in Probability and its Applications (1720 March 2014)" in Mathematical Institute of University of Oxford on March 18, 2014