Week 6

Itô integral, Itô formula (7.1,7.2)

We covered most of these sections. I didn’t prove the Itô formula.

Examples (7.3)

We looked the examples of the OU process and geometric Brownian Motion.

Notes: Stochastic intergrals and SDEs

Exercises

• 7.1
• 7.3
• 7.5
• 7.6
• 7.7

• Let
  \(X_t = e^{-\gamma t}W_{\frac{\sigma^2}{2\gamma}(e^{2\gamma t}-1)},\) where $W_t$ is a Wiener process.
  Deduce that it satisfies the Ornstein–Uhlenbeck SDE
  \(dX_t = -\gamma X_t\,dt + \sigma\, dW_t.\)

• Define a stochastic integral using Riemann sums with partition $P={0=t_0<t_1<\dots<t_n=t}$ and evaluation point
  \(t_* = \lambda t_{j+1} + (1-\lambda)t_j,\qquad \lambda\in[0,1].\) Using this definition, derive the formula for
  \(\int_0^t W_s\, dW_s.\)
• Consider the Gaussian process \(Q = \int_0^te^{-\gamma(t-s)}dW_s\) which appeared in the solution of the OU process. Derive its covariance function.