§ 06 / sdes

Stochastic Differential Equations

GBM, Ornstein–Uhlenbeck, Brownian bridge, existence/uniqueness, and numerical schemes.

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Section 1 / 11

Stochastic Differential Equations

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Many deterministic models in the natural and economic sciences can be expressed using differential equations. With the Itô integral and the Itô formula, we now have the foundations for stochastic differential equations, which allow many stochastic models to be formulated mathematically.

Given a probability space with Brownian motion and its standard Brownian filtration, and two continuous functions

μ:R>0×R→R,σ:R>0×R→R>0,\mu:\R_{>0}\times\R\to\R, \qquad \sigma:\R_{>0}\times\R\to\R_{>0},

we call

dXt=μ(t,Xt) dt+σ(t,Xt) dBt,X0=x0\boxed{ dX_t=\mu(t,X_t)\,dt+\sigma(t,X_t)\,dB_t, \qquad X_0=x_0 }

a stochastic differential equation (SDE).

In integral form,

Xt=x0+∫0tμ(s,Xs) ds+∫0tσ(s,Xs) dBs.\boxed{ X_t=x_0+\int_0^t\mu(s,X_s)\,ds +\int_0^t\sigma(s,X_s)\,dB_s. }

A continuous adapted stochastic process XX satisfying this equation is called a solution of the SDE.

Every solution is automatically an Itô process. We call μ(t,Xt)\mu(t,X_t) the drift coefficient and σ(t,Xt)\sigma(t,X_t) the diffusion coefficient. The two terms play different roles in describing the local trend and local variability of XX.