Brownian Motion and Stochastic Calculus
This domain tests the construction and properties of Brownian motion, stochastic differential equations and Ito's lemma.
Brownian motion and SDEs
- Standard Brownian motion W_t starts at 0, has independent normal increments W_t - W_s ~ N(0, t - s), and has continuous but nowhere differentiable paths. Its quadratic variation over [0,t] is t.
- Arithmetic Brownian motion has dX_t = mu dt + sigma dW_t. Geometric Brownian motion has dS_t = mu S_t dt + sigma S_t dW_t, with solution S_t = S_0 exp((mu - 0.5 sigma^2)t + sigma W_t).
- Ito's lemma for f(t,x) with dX_t = mu dt + sigma dW_t is df = (f_t + mu f_x + 0.5 sigma^2 f_xx) dt + sigma f_x dW_t.
Worked example
Let f(W_t) = W_t^2. Here f_t = 0, f_x = 2W_t and f_xx = 2, with mu = 0 and sigma = 1. Ito's lemma gives df = (0 + 0 + 0.5 x 1 x 2) dt + 1 x 2W_t dW_t = dt + 2W_t dW_t. Hence W_t^2 = 2 integral_0^t W_s dW_s + t. The extra dt term is the Ito correction; ordinary calculus would wrongly give only 2W_t dW_t. This also shows that W_t^2 - t is a martingale.
Traps that cost marks
The commonest error is applying the ordinary chain rule and omitting the 0.5 sigma^2 f_xx term. Do not confuse the drift mu of S_t with the drift mu - 0.5 sigma^2 of log S_t. Brownian paths are continuous but not differentiable, so dW_t is not an ordinary differential. Check whether a model is stated in arithmetic or geometric form before solving it.