Parameter sensitivity for stochastic differential equations in tumor growth, exit distributions, and biomathematical modeling

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2013-05

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Abstract

This dissertation covers three applications of stochastic differential equations. In each case a deterministic model is considered along with its corresponding results. An equivalent stochastic version is constructed to see how the model changes. Stochastic differential equations are important because each simulation will give different results that are the result of unknown, random phenomena. This is important because a deterministic model cannot include every variable that is inherit in a dynamical system. This randomness comes from Brownian motion which is described mathematically as a Wiener process.

The three models considered were chosen because of their deterministic behavior. The first of which is a model for tumor growth. Under certain parameter conditions the growth is easily modeled. However very different results are observed if the parameter values are changed. Since every person will have different health qualities and characteristics it is interesting to see how the stochastic model differs from the deterministic model. Parameter sensitivity under the stochastic model is of interest here.

The second model is for population dynamics in cannibalistic fish. This model is created under new assumptions and the results are compared with current research. The interesting part of this model is the stable limit cycles, or oscillations, in the solutions of the model. The stochastic model is then considered to see how the results would change with the added ``noise'' or randomness.

Lastly a stochastic model representing a collateralized debt obligation (CDO) is constructed to see how bundles of mortgages will default or be paid off. A deterministic model would give one solution but every borrower has a different probability of defaulting. Specifically the exit time of this process is desired which represents the fate of each borrower's mortgage. We are interested in the mean exit time as well as the corresponding probability distributions. Hypothesis testing is used in an attempt to determine adequate sample sizes that reflect the overall behavior.

For all of the above numerical simulations are performed in Matlab to answer these questions.

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Keywords

Stochastic differential equations, Exit distributions, Biomathematical modeling, Cannibalism, Solid tumor model, Jump-diffusion process, Brownian motion

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