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. Notice that this system of equations may be written in vector form as (1), where now X t and (t, x) are N vectors with entries Xti and i (t, x), respectively; dWt is the d vector of increments dWtj of the component Brownian motions Wtj ; and (t, x) is the N d matrix with entries ij (t, x).
In nancial applications, the Ornstein-Uhlenbeck ... For a friendlier introduction, try Steele’s new book Stochastic Calculus with Financial Applications. where A(0) = 1. If A(t) is di erentiable and B(t) is continuous, then
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Introduction, Stochastic calculus, Stochastic Calculus, Filtering, and Stochastic Control, Applications, Stochastic Calculus: An Introduction with Applications, An introduction, Calculus, An Introduction To Stochastic Modeling, Stochastic, Stochastic difierential equations, Stochastic Differential Equations, SC505 STOCHASTIC PROCESSES Class Notes