This thesis consists of three subsequent parts addressing the applications of stochastic
processes to the analysis and solutions of parabolic equations with discontinuous coefficients
that are of mathematical interest.
The first two parts consist of three manuscripts, in which we analyze solutions
of Fickian convection dispersion equations with discontinuous coefficients...
This thesis considers one of the classical problems in the actuarial mathematics literature, the decay of the probability of ruin in the collective risk model. The
claim number process N(t) is assumed to be a renewal process, the resulting model
being referred as the Sparre Andersen risk model. The inter-claim...
In this dissertation, we study two risk models. First, we consider the dual risk process which models the surplus of a company that incurs expenses at a constant rate and earns random positive gains at random times. When the surplus is invested in a risky asset following a geometric Brownian...
The flow of incompressible, viscous fluids in R³ is governed by the non-linear Navier-Stokes equations. Two common linearizations of the Navier-Stokes equations, the Stokes equations and the Oseen equations, are studied in this thesis using probabilistic methods.
The incompressibility condition presents new challenges for the well known theory relating partial...
This thesis contains three manuscripts addressing the application of stochastic processes to the analysis and solution of partial differential equations (PDEs) in mathematical physics.
In the first manuscript, one dimensional diffusion and Burgers equation are considered. The Fourier transform of the solution to each PDE is represented as the expected...
This dissertation consists of three papers studying optimization, conservation
and valuation of contingent claims in economic resource nianagement under
uncertainty.
In the first paper the Markovian optinial policies are studied for resource
management in a finite time horizon. Under sonic conditions, in particular, when
the prices are stochastic and there...