Controlled diffusion Mean Field Games with common noise and McKean-Vlasov second order backward SDEs

Рассматривается игра среднего поля с одним внешним шумом, коэффициент диффузии которого включает управление. Доказывается существование слабого решения с релаксацией при некоторых условиях на коэффициент диффузии. Далее, показывается, что при отсутствии этого внешнего шума игра среднего поля описывается обратным стохастическим дифференциальным уравнением типа Маккина-Власова второго порядка.

Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application

<p style='text-indent:20px;'>In this paper we solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in Peng [<xref ref-type="bibr" rid="b12">12</xref>] from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues <inline-formula><tex-math id="M1">\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> and construct corresponding eigenfunctions. Moreover, the order of growth for these <inline-formula><tex-math id="M2">\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> are obtained: <inline-formula><tex-math id="M3">\begin{document}$ \lambda_m\sim m^2 $\end{document}</tex-math></inline-formula>, as <inline-formula><tex-math id="M4">\begin{document}$ m\rightarrow +\infty $\end{document}</tex-math></inline-formula>. As applications, we give an explicit estimation formula about the statistic period of solutions of Forward-Backward SDEs. Besides, by a meticulous example we show the subtle situation in time-dependent case that some eigenvalues appear when the solution of the associated Riccati equation does not blow-up, which does not happen in time-invariant case.</p>

Mean square rate of convergence for random walk approximation of forward-backward SDEs

Let (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk $B^n$ from the underlying Brownian motion B by Skorokhod embedding, one can show $L_2$-convergence of the corresponding solutions $(Y^n,Z^n)$ to $(Y, Z).$ We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in $C^{2,\alpha}$. The proof relies on an approximative representation of $Z^n$ and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.