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LECTURE 12: STOCHASTIC DIFFERENTIAL EQUATIONS, …

LECTURE 12: STOCHASTIC DIFFERENTIAL EQUATIONS, DIFFUSION. PROCESSES, AND THE FEYNMAN-KAC FORMULA. 1. Existence and Uniqueness of Solutions to SDEs It is frequently the case that economic or financial considerations will suggest that a stock price, exchange rate, interest rate, or other economic variable evolves in time according to a STOCHASTIC DIFFERENTIAL equation of the form (1) dXt = (t, Xt ) dt + (t, Xt ) dWt where Wt is a standard Brownian motion and and are given functions of time t and the current state x. More generally, when several related economic variables X 1 , X 2 , .. , X N are considered, the vector Xt = (Xt1 , Xt2 , .. , XtN )T may evolve in time according to a system of STOCHASTIC DIFFERENTIAL equations of the form d ij (t, Xt ) dWtj , X.

LECTURE 12: STOCHASTIC DIFFERENTIAL EQUATIONS, DIFFUSION PROCESSES, AND THE FEYNMAN-KAC FORMULA 1. Existence and Uniqueness of Solutions to SDEs It is frequently the case that economic or nancial considerations will suggest that a stock price, exchange rate, interest rate, or other economic variable evolves in time according to a …

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