§ 03 / Practice

Worked examples.

Progressive hints — reveal one step at a time, only when you are ready.

Brownian motion · Ch.1

Variance of a Brownian increment

Let BB be a standard Brownian motion. Compute Var(B3−B1)\mathrm{Var}(B_3-B_1).
Martingales · Ch.2

Is $B_t^2-t$ a martingale?

For Brownian motion BB, show that Mt=Bt2−tM_t=B_t^2-t is a martingale (sketch).
Itô integral · Ch.3

Mean of an Itô integral

For a suitable integrand HH, what is E[∫0THs dBs]\mathbb{E}\big[\int_0^T H_s\,dB_s\big]?
SDEs · Ch.6

Solve geometric Brownian motion

Solve dSt=μSt dt+σSt dBtdS_t=\mu S_t\,dt+\sigma S_t\,dB_t with S0>0S_0>0.
Girsanov · Ch.7

Drift removal by change of measure

In one sentence, what does Girsanov achieve for Brownian motion with drift?
Feynman–Kac · Ch.10

PDE meets expectation

Why do expectations of diffusion functionals solve parabolic PDEs?