Martingales and Stochastic Integration
This domain tests filtrations, martingales, Ito integrals, stopping times and change of measure.
Core definitions
- A filtration represents information available over time. A process M_t is adapted if M_t is known at time t. It is a martingale if E[M_t | F_s] = M_s for s < t, with finite expectations.
- The Ito integral integral_0^t H_s dW_s is defined for adapted integrands H_s with finite squared expectation. It has the isometry E[(integral_0^t H_s dW_s)^2] = E[integral_0^t H_s^2 ds].
- An Ito process with zero drift is a local martingale; if integrability conditions hold, it is a true martingale. The exponential martingale is M_t = exp(theta W_t - 0.5 theta^2 t).
- Girsanov's theorem changes the drift of Brownian motion under an equivalent measure, which is the basis of risk-neutral pricing.
Worked example
Let M_t = exp(theta W_t - 0.5 theta^2 t). Apply Ito to f(t,x) = exp(theta x - 0.5 theta^2 t). Then f_t = -0.5 theta^2 M_t, f_x = theta M_t and f_xx = theta^2 M_t. With mu = 0 and sigma = 1, Ito gives dM_t = (-0.5 theta^2 + 0.5 theta^2)M_t dt + theta M_t dW_t = theta M_t dW_t. There is no dt term, so M_t is a martingale (subject to integrability).
Traps that cost marks
Do not use optional stopping with an unbounded stopping time. Do not treat a process with zero drift as a true martingale without checking integrability. The integrand in an Ito integral must be adapted. Remember that a martingale has constant expectation, but a submartingale has increasing expectation.