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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.

wander very far from then the \mean-reversion" term (Yt )dt becomes larger, forcing Yt back toward . The coe cients of the stochastic di erential equation (11) satisfy the hypotheses of Theorem 2, and so for every possible initial state y0 2 R there is a unique solution Yt. In fact, it is possible to give an explicit representation of the solution.

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