The Dobric-Ojeda Process with Applications to Option Pricing and the Stochastic Heat Equation

Replacing Black-Scholes\u27 driving process, Brownian motion, with fractional Brownian motion allows for incorporation of a past dependency of stock prices but faces a few major downfalls, including the occurrence of arbitrage when implemented in the financial market. We present the development, testing, and implementation of a simplified alternative to using fractional Brownian motion for pricing derivatives. By relaxing the assumption of past independence of Brownian motion but retaining the Markovian property, we are developing a competing model that retains the mathematical simplicity of the standard Black-Scholes model but also has the improved accuracy of allowing for past dependence. This is achieved by replacing Black-Scholes\u27 underlying process, Brownian motion, with the Dobric-Ojeda process.In the second half of the dissertation, we introduce a Dobric-Ojeda type stochastic noise. This noise is intended to serve as an approximation for fractional noise in a partial differential equation. We implement this Dobric-Ojeda noise in the stochastic heat equation and compare the solution to the analogue with fractional noise. As in option pricing, we aim to provide a more mathematically tractable alternative to fractional noise with similar properties