First-year graduateProbability & StatisticsPlanned
Stochastic processes
Continuous-time processes: Poisson processes, continuous-time Markov chains, Brownian motion and its path properties, continuous-time martingales, and an introduction to Itô calculus and stochastic differential equations.
math.prob.stochastic-processes
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 48
You will learn to
- Construct the Poisson process and derive its basic properties.
- State the defining properties of Brownian motion and use its basic path properties.
- Apply Itô's formula and solve simple stochastic differential equations such as that of geometric Brownian motion.
Definition
- Standard Brownian motion is a process B with B₀ = 0, continuous paths, and independent increments with B_t − B_s normally distributed with mean 0 and variance t − s for s < t.
- The Itô integral ∫ H dB is defined as a limit in L² of sums in which the integrand is evaluated at the left endpoint of each interval.
Theorem
- Brownian paths are almost surely continuous and nowhere differentiable.
- Itô's formula: for twice continuously differentiable f, df(B_t) = f′(B_t) dB_t + ½ f″(B_t) dt.
- The equation dS = μS dt + σS dB has the solution S_t = S₀ exp((μ − σ²/2)t + σB_t).
Common misconceptions
- Applying the ordinary chain rule to Itô integrals.
- Thinking Brownian paths are differentiable because they are continuous.
Optional discoveries
Further reading
- Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus (1988)
- Bernt Øksendal, Stochastic Differential Equations