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Register a new userSDEs with random and irregular coefficients
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Authors
Guohuan Zhao
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We prove the existence and uniqueness for SDEs with random and irregular coefficients through solving a backward stochastic Kolmogorov equation and using a modified Zvonkins type transformation.
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We obtain $T_2(C)$ for stochastic differential equations with Dini continuous drift and $T_1(C)$ stochastic differential equations with singular coefficients.
The paper investigates existence and uniqueness for a stochastic differential equation (SDE) with distributional drift depending on the law density of the solution. Those equations are known as McKean SDEs. The McKean SDE is interpreted in the sense of a suitable singular martingale problem. A key tool used in the investigation is the study of the corresponding Fokker-Planck equation.
Forward-backward stochastic differential equations (FBSDEs) have attracted significant attention since they were introduced almost 30 years ago, due to their wide range of applications, from solving non-linear PDEs to pricing American-type options. Here, we consider two new classes of multidimensional FBSDEs with distributional coefficients (elements of a Sobolev space with negative order). We introduce a suitable notion of a solution, show existence and uniqueness of a strong solution of the first FBSDE, and weak existence for the second. We establish a link with PDE theory via a nonlinear Feynman-Kac representation formula. The associated semi-linear second order parabolic PDE is the same for both FBSDEs, also involves distributional coefficients and has not previously been investigated; our analysis uses mild solutions, Sobolev spaces and semigroup theory.
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 analyze multi-dimensional mean-field stochastic differential equations where the drift depends on the law in form of a Lebesgue integral with respect to the pushforward measure of the solution. We show existence and uniqueness of Malliavin differentiable strong solutions for irregular drift coefficients, which in particular include the case where the drift depends on the cumulative distribution function of the solution. Moreover, we examine the solution as a function in its initial condition and introduce sufficient conditions on the drift to guarantee differentiability. Under these assumptions we then show that the Bismut-Elworthy-Li formula proposed in Bauer et al. (2018) holds in a strong sense, i.e. we give a probabilistic representation of the strong derivative with respect to the initial condition of expectation functionals of strong solutions to our type of mean-field equations in one-dimension.
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