Stochastic delay differential equations

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2017
Aladağlı, E. Ezgi
In many areas of science like physics, ecology, biology, economics, engineering, financial mathematics etc. phenomenas do not show their effect immediately at the moment of their occurrence. Generally, they influence the future states. In order to understand the structure and quantitative behavior of such systems, stochastic delay differential equations (SDDEs) are constructed while inserting the information that are obtained from the past phenomena into the stochastic differential equations (SDEs). SDDEs become a new interest area due to the their potential to capture reality better. It can be said that SDDEs are in the infancy stage when we consider the SDEs. Some numerical approaches to SDDEs are constructed because obtaining closed form solutions by the help of stochastic calculus is very difficult most of the time and for some equations it is impossible. In recent years, scientist who are interest in economy and finance study option pricing formulation for systems that include time delay which can be stochastic or deterministic. The aim of this thesis is to understand general forms of SDDEs and their solution process for the deterministic time delay. Some examples are provided to see the exact solution process. Moreover, we examine numerical techniques to obtain approximate solution processes. In order to understand effect of delay term, these techniques are used to simulate the solution process for different choices of delay terms and coefficients. In the application part of the thesis, we investigate the stock returns and European call option price when the system is modeled with SDDEs. 
Citation Formats
E. E. Aladağlı, “Stochastic delay differential equations,” M.S. - Master of Science, Middle East Technical University, 2017.