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Stochastic Differential Equations

Stochastic Differential Equations Steven P. Lalley December 2, 2016. 1 SDEs: Definitions Stochastic Differential Equations Many important continuous-time Markov processes for instance, the Ornstein-Uhlenbeck pro- cess and the Bessel processes can be defined as solutions to Stochastic Differential Equations with drift and diffusion coefficients that depend only on the current value of the process. The general form of such an equation (for a one-dimensional process with a one-dimensional driving Brownian motion) is dXt = (Xt ) dt + (Xt ) dWt , (1).

This parallels the main existence/uniqueness result for ordinary differential equations, or more generally finite systems of ordinary differential equations x0(t) = F(x(t)); (7) which asserts that unique solutions exist for each initial value x(0) provided the function F is uniformly Lipschitz. Without the hypothesis that the function Fis ...

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