Stochastic Processes in Continuous Time

SVETLOZAR T. RACHEV, PhD, Dr Sci

Frey Family Foundation Chair-Professor, Department of Applied Mathematics and Statistics, Stony Brook University, and Chief Scientist, FinAnalytica

YOUNG SHIN KIM, PhD

Research Assistant Professor, School of Economics and Business Engineering, University of Karlsruhe and KIT

MICHELE LEONARDO BIANCHI, PhD

Research Analyst, Specialized Intermediaries Supervision Department, Bank of Italy

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

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.

In 1900, the father of modern option pricing theory, Louis Bachelier, proposed using Brownian motion for modeling stock market prices. There are several reasons why Brownian motion is a popular process. First, Brownian motion is the milestone ...

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