Transcription of LECTURE 12: STOCHASTIC DIFFERENTIAL EQUATIONS, …
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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. (2) dXti = i (t, Xt ) dt +. j=1. where Wt = (Wt1 , Wt2 , .. , Wtd ) is a d dimensional Brownian motion.
LECTURE 12: STOCHASTIC DIFFERENTIAL EQUATIONS, DIFFUSION PROCESSES, AND THE FEYNMAN-KAC FORMULA 1. Existence and Uniqueness of Solutions to SDEs ... then its absolute value is de ned to be jxj = pP i x 2 i. 1. In practice, one rarely needs to verify this condition, because the following is true, by the ...
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