Related papers: Wellposedness of Second Order Backward SDEs
A class of backward doubly stochastic differential equations (BDSDEs in short) with continuous coefficients is studied. We give the comparison theorems, the existence of the maximal solution and the structure of solutions for BDSDEs with…
The results of the author and Gess [27] develop a robust well-posedness theory for a broad class of conservative stochastic PDEs, with both probabilistically stationary and non-stationary Stratonovich noise, and with irregular noise…
Although having been developed for more than two decades, the theory of forward backward stochastic differential equations is still far from complete. In this paper, we take one step back and investigate the formulation of FBSDEs. Motivated…
It is known that Markovian forward-backward stochastic differential equations provide nonlinear Feynman-Kac representation formulae for semilinear parabolic PDEs. We show that non-Markovian forward-backward stochastic differential equations…
We extend Peng's maximum principle for semilinear stochastic partial differential equations (SPDEs) in one space-dimension with non-convex control domains and control-dependent diffusion coefficients to the case of general cost functionals…
The classical Feynman-Kac identity builds a bridge between stochastic analysis and partial differential equations (PDEs) by providing stochastic representations for classical solutions of linear Kolmogorov PDEs. This opens the door for the…
We give a proof of the equivalence of the electric-magnetic duality on one side and helicity conservation of the tree level amplitudes on the other side within general models of nonlinear electrodynamics. Using modified Feynman rules…
In this paper a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for…
The existence and uniqueness of measure-valued solutions to stochastic nonlinear, non-local Fokker-Planck equations is proven. This type of stochastic PDE is shown to arise in the mean field limit of weakly interacting diffusions with…
In this book we establish under suitable assumptions the uniqueness and existence of viscosity solutions of Kolmogorov backward equations for stochastic partial differential equations (SPDEs). In addition, we show that this solution is the…
We obtain an existence and uniqueness theorem for fully coupled forward-backward SDEs (FBSDEs) with jumps via the classical solution to the associated quasilinear parabolic partial integro-differential equation (PIDE), and provide the…
The paper is devoted to the construction of a probabilistic particle algorithm. This is related to nonlin-ear forward Feynman-Kac type equation, which represents the solution of a nonconservative semilinear parabolic Partial Differential…
This paper, is an attempt to extend the notion of stochastic viscosity solution to reflected semi-linear stochastic partial differential equations (RSPDEs, in short) with non-Lipschitz condition on the coefficients. Our method is fully…
We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-*…
We adapt and generalise results of Loganathan on the cohomology of inverse semigroups to the cohomology of ordered groupoids. We then derive a five-term exact sequence in cohomology from an extension of ordered groupoids, and show that this…
In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with…
We present a deep recurrent neural network architecture to solve a class of stochastic optimal control problems described by fully nonlinear Hamilton Jacobi Bellmanpartial differential equations. Such PDEs arise when one considers…
We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…
We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential…
We consider stochastic PDEs \[dY_t = L(Y_t)\, dt + A(Y_t).\, dB_t, t > 0\] and associated PDEs \[du_t = L u_t\, dt, t > 0\] with regular initial conditions. Here, $L$ and $A$ are certain partial differential operators involving…