Mwaka wa kwanza wa uzamiliProbability & StatisticsImepangwa
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
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 48 (hadi chini kabisa)
Utajifunza
- 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.
Fasili
- 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.
Nadharia
- 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).
Mawazo potofu ya kawaida
- Applying the ordinary chain rule to Itô integrals.
- Thinking Brownian paths are differentiable because they are continuous.
Ugunduzi wa hiari
Usomaji zaidi
- Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus (1988)
- Bernt Øksendal, Stochastic Differential Equations