Application of stochastic differential equations on some medical and finance problems and their numerical solutions

In this thesis, the concept of stochastic differential equations and its application areas are investigated. After the existence and uniqueness of equations are mentioned and analytical and numerical solution methods are determined, stability analyzes of numerical methods are discussed. Stochastic differential equations is used for two important issues such as estimation of tumor growth and stock price which addressed in the application section. As a first application, we are approximated the YHOO stock prices between 01.01.2005-01.01.2015 with Black-Scholes model, which is a financial stochastic differential equation model. As a second application, we study on the behavior and growth of parathyroid cancer in the human body. Firstly deterministic Gompertz model is identified for this, then it is investigated deterministically change of parathyroid cancer respect to time, which is obtained through 41 patients in the literature. Then we describe the stochastic Gompertz model based on deterministic Gompertz's law with the diffusion coefficient in our stochastic model, using the data taken from the patients. Changing in tumor growth over the time is approximately calculated with this defined stochastic model. We solve both models numerically and analitically with defined methods. Obtained solutions are compared with the actual data and they are supported with graphs and error tables. Keywords: Stochastic differential equations, maximum likelihood parameter estimation method, non-parametric estimation method, Black-Scholes model, stochastic Gompertz model.