91Ó°ÊÓ

The Ornstein-Uhlenbeck process Let \(\left\\{W_{t}\right\\}_{t \geq 0}\) denote standard Brownian motion under \(\mathbb{P} .\) The Ornstein-Uhlenbeck process, \(\left\\{X_{t}\right\\}_{t \geq 0}\), is the unique solution to Langevin's equation, $$ d X_{t}=-\alpha X_{t} d t+d W_{t}, \quad X_{0}=x $$ This equation was originally introduced as a simple idealised model for the velocity of a particle suspended in a liquid. In finance it is a special case of the Vasicek model of interest rates (see Exercise 19). Verify that $$ X_{t}=e^{-\alpha t} x+e^{-\alpha t} \int_{0}^{t} e^{\alpha s} d W_{s} $$ and use this expression to calculate the mean and variance of \(X_{t}\)

Use the Feynman-Kac stochastic representation formula to solve $$ \frac{\partial F}{\partial t}(t, x)+\frac{1}{2} \sigma^{2} \frac{\partial^{2} F}{\partial x^{2}}(t, x)=0 $$ subject to the terminal value condition $$ F(T, x)=x^{4} $$