Section 18.6

Mean-Reverting SDEs

Expectations and variances under Ornstein-Uhlenbeck, Vasicek and CIR dynamics.

Toolkit

Mean-reverting SDEs of the form $dX_t = \kappa(\theta - X_t)\,dt + \sigma(X_t)\,dW_t$ pull $X_t$ toward $\theta$ at rate $\kappa$ ; since $\mathbb{E}[\int \sigma\,dW]=0$ , moments are found by taking expectations and solving the resulting ODE, not by simulating the noise.

Key facts:

X_t=\theta+(X_0-\theta)e^{-\kappa t}+\sigma\int_0^t e^{-\kappa(t-s)}\,dW_s \quad\text{(OU/Vasicek, constant }\sigma\text{)},

\mathbb{E}[X_t]=\theta+(X_0-\theta)e^{-\kappa t},\qquad \operatorname{Var}(X_t)=\frac{\sigma^2}{2\kappa}\left(1-e^{-2\kappa t}\right).

As $t\to\infty$ : $X_t\to \mathcal N\!\big(\theta,\sigma^2/2\kappa\big)$ .

For CIR, $dX_t=\kappa(\theta-X_t)dt+\sigma\sqrt{X_t}\,dW_t$ : mean is identical to OU/Vasicek,

\operatorname{Var}(X_t)=X_0\frac{\sigma^2}{\kappa}\big(e^{-\kappa t}-e^{-2\kappa t}\big)+\theta\frac{\sigma^2}{2\kappa}\big(1-e^{-\kappa t}\big)^2.

Feller condition $2\kappa\theta\ge\sigma^2$ keeps $X_t\ge 0$ .

Problems

18.13 Mean of the Vasicek Rate+1 variation Medium Free