Primer año de posgradoProbability & StatisticsPlanificado
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
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Se basa en
Requisitos previos obligatorios, hasta el fondo: 48
Aprenderás a
- 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.
Definición
- 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.
Teorema
- 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).
Errores frecuentes
- Applying the ordinary chain rule to Itô integrals.
- Thinking Brownian paths are differentiable because they are continuous.
Descubrimientos opcionales
Lecturas complementarias
- Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus (1988)
- Bernt Øksendal, Stochastic Differential Equations