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Abstract
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In this expository talk, I will present material from Sections 15.3 and 15.4 of "Introduction to Stochastic Processes," volume 2, by Karlin and Taylor. I will start with a simple definition of a diffusion process and formally show how to use that definition to obtain hitting probabilities and expected hitting times. Specifically, these quantities are solutions of differential equations associated with the diffusion process. Hitting in this context refers to a diffusion process reaching, or hitting, a particular value; therefore, a hitting probability is the probability that a diffusion process reaches one given value before another value. In insurance, one can interpret a hitting probability as the probability that an insurance company goes bankrupt. This talk will lay some of the groundwork for further expository talks in optimal stochastic control by presenting the connection between expectations involving diffusion processes and differential equations--without the added complication of optimal control. |
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