Stochastic Processes in Continuous Time

Abstract

The dynamic of a financial asset's returns and prices can be expressed using a deterministic process if there is no uncertainty about its future behavior, or with a stochastic process in the more likely case when the value is uncertain. Stochastic processes in continuous time are the most used tool to explain the dynamics of a financial asset's returns and prices. They are the building blocks with which to construct financial models for portfolio optimization, derivatives pricing, and risk management. Continuous-time processes allow for more elegant theoretical modeling compared to discrete-time models and many results proven in probability theory can be applied to obtain a simple evaluation method. Keywords: Brownian motion; Poisson process; jump processes; Levy processes; stochastic process X =; gamma process; exponential process; variance gamma; process; stable process; standard CTS process; generalized tempered stable; process; modified tempered stable; process; standard MTS process; normal tempered stable; process; standard NTS process; normal inverse Gaussian; standard NIG process; Kim-Rachev tempered stable; standard KRTS process; rapidly decreasing tempered stable; standard RDTS process; Brownian motion; Independent increments; Stationary increments; time-changed Brownian motion; The Modified Tempered Stable Processes with Application to Finance; conditional expectation; change of measure; Esscher transform; Brownian Motion and Stochastic Calculus; Risk Assessment Decisions in Banking and Finance