ON SWITCHING DIFFUSIONS: THEFEYNMAN-KAC FORMULA ANDNEAR-OPTIMAL CONTROLSbyNICHOLAS BARANAugust 2014Advisor: Dr. George YinMajor: MathematicsDegree: Doctor of Philosophy We consider diffusions in two different contexts. First, we consider the so-calledFeynman-Kac formula(s) for switching diffusions. These formulas provide stochasticrepresentations for ...

Suppose that a real valued process X is given as a solution to a stochastic differential equation. Then, for any twice continuously differentiable function f , the Feynman-Kac formula gives a condition for f(t,X) to be a local martingale. We generalize the Feynman-Kac formula in two main ways. First, it is extended to nondifferentiable functions. Second, the process X is not required to satisfy...

We consider linear parabolic stochastic integro-PDE’s of FeynmanKac type associated to Lévy-Itô diffusions. The solution of such equations can be represented as certain Feynman-Kac functionals of the associated diffusion such that taking expectation yields the deterministic Feynamn-Kac formula. We interpret the problem in the framework of white noise analysis and consider differentiation in the...

We develop a probabilistic theory for conversion of A→ B processes for a model Temporal Analysis of Products (TAP)-like experiments based on the general theory of Brownian motion with killing and the Feynman-Kac formula. Initial experimental testing for systems with a single catalyst particle is given, as well as numerical experiments for two-particle catalyst systems in simple geometric config...

Adapting ideas of Daubechies and Klauder we derive a continuum path-integral formula for the time evolution generated by a spin Hamiltonian. For this purpose we identify the finite-dimensional spin Hilbert space with the ground-state eigenspace of a suitable Schödinger operator on L(R), the Hilbert space of square-integrable functions on the Euclidean plane R, and employ the Feynman-Kac-Itô for...

In [4], the authors proved the uniqueness among the solutions which admit every exponential moments. In this paper, we prove that uniqueness holds among solutions which admit some given exponential moments. These exponential moments are natural as they are given by the existence theorem. Thanks to this uniqueness result we can strengthen the nonlinear Feynman-Kac formula proved in [4].

In [4], the authors proved the uniqueness among the solutions which admit every exponential moments. In this paper, we prove that uniqueness holds among solutions which admit some given exponential moments. These exponential moments are natural as they are given by the existence theorem. Thanks to this uniqueness result we can strengthen the nonlinear Feynman-Kac formula proved in [4].

In this paper we establish a version of the Feynman-Kac formula for the stochastic heat equation with a multiplicative fractional Brownian sheet. We prove the smoothness of the density of the solution, and the Hölder regularity in the space and time variables.