LECTURE 12: STOCHASTIC DIFFERENTIAL EQUATIONS, …

LECTURE12:STOCHASTICDIFFERENTIALEQUATION S,DIFFUSIONPROCESSES,ANDTHEFEYNMAN-KAC is frequentlythecasethateconomicor nancialconsiderationswillsuggestthata stock price,exchangerate,interestrate,or othereconomicvariableevolves in timeaccordingto a stochasticdi erentialequationof theform(1)dXt= (t; Xt)dt+ (t; Xt)dWtwhereWtis a standardBrownianmotionand and aregiven functionsof timetandthecurrentstatex. Moregenerally, whenseveralrelatedeconomicvariablesX1; X2; : : : ; XNareconsidered,thevectorXt= (X1t; X2t; : : : ; XNt)Tmay evolve in timeaccordingto asystemof stochasticdi erentialequationsof theform(2)dXit= i(t; Xt)dt+dXj=1 ij(t; Xt)dWjt;whereWt= (W1t; W2t; : : : ; Wdt) is ad be writtenin vectorformas (1),wherenowXtand (t; x) areN vectorswithentriesXitand i(t; x), respectively;dWtis thed vectorof incrementsdWjtof thecomponentBrownianmotio

LECTURE 12: STOCHASTIC DIFFERENTIAL EQUATIONS, DIFFUSION PROCESSES, AND THE FEYNMAN-KAC FORMULA 1. Existence and Uniqueness of Solutions to SDEs

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