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Register a new userNonlinear McKean-Vlasov diffusions under the weak Hormander condition with quantile-dependent coefficients
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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.
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The work concerns the stability for a type of multivalued McKean-Vlasov SDEs with non-Lipschitz coefficients. First, we prove the existence and uniqueness of strong solutions for multivalued McKean-Vlasov stochastic differential equations with non-Lipschitz coefficients. Then, we extend the classical It^{o}s formula from SDEs to multivalued McKean-Vlasov SDEs. Next, the exponential stability of second moments, the exponentially 2-ultimate boundedness and the almost surely asymptotic stability for their solutions in terms of a Lyapunov function are shown.
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.
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.
We present a simple uniqueness argument for a collection of McKean-Vlasov problems that have seen recent interest. Our first result shows that, in the weak feedback regime, there is global uniqueness for a very general class of random drivers. By weak feedback we mean the case where the contagion parameters are small enough to prevent blow-ups in solutions. Next, we specialise to a Brownian driver and show how the same techniques can be extended to give short-time uniqueness after blow-ups, regardless of the feedback strength. The heart of our approach is a surprisingly simple probabilistic comparison argument that is robust in the sense that it does not ask for any regularity of the solutions.
The work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters. First, we prove the existence and uniqueness of these equations under non-Lipschitz conditions. Second, we construct maximum likelihood estimators of these parameters and then discuss their strong consistency. Third, a numerical simulation method for the class of path-dependent McKean-Vlasov stochastic differential equations is offered. Moreover, we estimate the errors between solutions of these equations and that of their numerical equations. Finally, we give an example to explain our result.
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