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Register a new userA stochastic McKean--Vlasov equation for absorbing diffusions on the half-line
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We study a finite system of diffusions on the half-line, absorbed when they hit zero, with a correlation effect that is controlled by the proportion of the processes that have been absorbed. As the number of processes in the system becomes large, the empirical measure of the population converges to the solution of a non-linear stochastic heat equation with Dirichlet boundary condition. The diffusion coefficients are allowed to have finitely many discontinuities (piecewise Lipschitz) and we prove pathwise uniqueness of solutions to the limiting stochastic PDE. As a corollary we obtain a representation of the limit as the unique solution to a stochastic McKean--Vlasov problem. Our techniques involve energy estimation in the dual of the first Sobolev space, which connects the regularity of solutions to their boundary behaviour, and tightness calculations in the Skorokhod M1 topology defined for distribution-valued processes, which exploits the monotonicity of the loss process $L$. The motivation for this model comes from the analysis of large portfolio credit problems in finance.
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In this paper we consider a class of {it conditional McKean-Vlasov SDEs} (CMVSDE for short). Such an SDE can be considered as an extended version of McKean-Vlasov SDEs with common noises, as well as the general version of the so-called {it conditional mean-field SDEs} (CMFSDE) studied previously by the authors [1, 14], but with some fundamental differences. In particular, due to the lack of compactness of the iterated conditional laws, the existing arguments of Schauders fixed point theorem do not seem to apply in this situation, and the heavy nonlinearity on the conditional laws caused by change of probability measure adds more technical subtleties. Under some structure assumptions on the coefficients of the observation equation, we prove the well-posedness of solution in the weak sense along a more direct approach. Our result is the first that deals with McKean-Vlasov type SDEs involving state-dependent conditional laws.
We consider $mathbb{R}^d$-valued diffusion processes of type begin{align*} dX_t = b(X_t)dt, +, dB_t. end{align*} Assuming a geometric drift condition, we establish contractions of the transitions kernels in Kantorovich ($L^1$ Wasserstein) distances with explicit constants. Our results are in the spirit of Hairer and Mattinglys extension of Harris Theorem. In particular, they do not rely on a small set condition. Instead we combine Lyapunov functions with reflection coupling and concave distance functions. We retrieve constants that are explicit in parameters which can be computed with little effort from one-sided Lipschitz conditions for the drift coefficient and the growth of a chosen Lyapunov function. Consequences include exponential convergence in weighted total variation norms, gradient bounds, bounds for ergodic averages, and Kantorovich contractions for nonlinear McKean-Vlasov diffusions in the case of sufficiently weak but not necessarily bounded nonlinearities. We also establish quantitative bounds for sub-geometric ergodicity assuming a sub-geometric drift condition.
In this paper, the strong existence and uniqueness for a degenerate finite system of quantile-dependent McKean-Vlasov stochastic differential equations are obtained under a weak H{o}rmander condition. The approach relies on the apriori bounds for the density of the solution to time inhomogeneous diffusions. The time inhomogeneous Feynman-Fac formula is used to construct a contraction map for this degenerate system.
This paper is devoted to investigating the Freidlin-Wentzells large deviation principle for a class of McKean-Vlasov quasilinear SPDEs perturbed by small multiplicative noise. We adopt the variational framework and the modified weak convergence criteria to prove the Laplace principle for McKean-Vlasov type SPDEs, which is equivalent to the large deviation principle. Moreover, we do not assume any compactness condition of embedding in the Gelfand triple to handle both the cases of bounded and unbounded domains in applications. The main results can be applied to various McKean-Vlasov type SPDEs such as distribution dependent stochastic porous media type equations and stochastic p-Laplace type equations.
We study a class of non linear integro-differential equations on the Wasserstein space related to the optimal control of McKean--Vlasov jump-diffusions. We develop an intrinsic notion of viscosity solutions that does not rely on the lifting to an Hilbert space and prove a comparison theorem for these solutions. We also show that the value function is the unique viscosity solution.
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