We provide an existence and uniqueness theory for an extension of backward SDEs to the second order. While standard Backward SDEs are naturally connected to semilinear PDEs, our second order extension is connected to fully nonlinear PDEs, as suggested in [4]. In particular, we provide a fully nonlinear extension of the Feynman-Kac formula. Unlike [4], the alternative formulation of this paper i...

We prove a version of the Feynman-Kac formula for Lévy processes and integro-differential operators, with application to the momentum representation of suitable quantum (Euclidean) systems whose Hamiltonians involve Lévytype potentials. Large deviation techniques are used to obtain the limiting behavior of the systems as the Planck constant approaches zero. It turns out that the limiting behavi...

Time evolution of a passive scalar is considered in a turbulent homogeneous incompressible Gaussian flow. The turbulent nature of the flow results in non-smooth coefficients in the corresponding evolution equation. A strong, in the probabilistic sense, solution of the equation is constructed using Wiener Chaos expansion, and the properties of the solution are studied. Among the results obtained...

This is a brief review on Brownian functionals in one dimension and their various applications, a contribution to the special issue “The Legacy of Albert Einstein” of Current Science. After a brief description of Einstein’s original derivation of the diffusion equation, this article provides a pedagogical introduction to the path integral methods leading to the derivation of the celebrated Feyn...

A simulation based method for the numerical solution of PDE with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to...

This article investigates an optimal allocation problem of assets under Gaussian state variables consisting of the risk-free rate and the multi dimensional market price of risk. The optimal portfolio consisting of a bond and several stocks is derived. The idea is that the indirect utility as a solution of a partial differential equation is expressed as a zero-coupon bond price in a different ec...

On the base of a Feynman-Kac–type formula involving Poisson stochastic processes, recently a Monte Carlo algorithm has been introduced, which describes exactly the realor imaginary-time evolution of many-body lattice quantum systems. We extend this algorithm to the exact simulation of time-dependent correlation functions. The techniques generally employed in Monte Carlo simulations to control f...

We design a theoretic tree-based functional representation of a class of Feynman-Kac particle distributions, including an extension of the Wick product formula to interacting particle systems. These weak expansions rely on an original combinatorial, and permutation group analysis of a special class of forests. They provide refined non asymptotic propagation of chaos type properties, as well as ...

This paper establishes an anticipating stochastic differential equation of parabolic type for the expectation of the solution of a stochastic differential equation conditioned on complete knowledge of the path of one of its components. Conversely, it is shown that any appropriately regular solution of this stochastic p.d.e. must be given by the conditional expectation. These results generalize ...

The goal of this paper is to give a new and short proof of the local Atiyah-Singer index theorem by using approximations of heat semigroups. The heat equation approach to index theorems is not new: It was suggested by Atiyah-Bott [1] and McKean-Singer [18], and first carried out by Patodi [21] and Gilkey [13]. Bismut in [8] introduces stochastic methods based on Feynman-Kac formula. For probabi...