| Description |
Random elements in functional spaces (cylindrical sigma-algebra, finite dimensional distributions of a stochastic process; Kolmogorov's theorem on finite dimensional distributions (with proof); mean and variance function of a stochastic process; stationarity properties of stochastic processes; Gaussian processes and their finite dimensional distributions, finite dimensional distribution of the Wiener process; equivalent processes and modifications (versions), stochastic continuity; separability (existence of a separable version - formulation without proof); continuity of sample paths, Kolmogorov's continuity criterion (proof), its application to the Wiener process (proof); definitions of continuous time filtration, stopping times and progressive measurability, progressive measurability of right-continuous processes, definition of continuous time martingales.)
Markov processes (transition kernels and the Chapman--Kolmogorov equation; operators acting on functions and measures, strong Markov property; 0-1 law of Blumenthal (proof); definition of the generating operator.)
Brownian motion (the Wiener process) (definition and its equivalent variants (with proofs of equivalence); covariance function and calculation of distributions, in particular integrals of the Wiener process; key properties (scale invariance, time shift, symmetry, time inversion) with proofs; Markov and strong Markov properties (proof); Brownian bridge and its basic properties; Brownian motion in higher dimensions; quadratic and linear variation of the Wiener process (proofs), non-differentiability (without proof); reflection principle (proof); derivation of the distribution of supremum of the Wiener process, and of the first hitting time; level sets, zero set and the arcsine law (proof); martingale properties of the Brownian motion, Wald's identities (without proofs).
Stochastic integration (problems with the classical definition of the integral with respect to the Wiener process; It^o's stochastic integral for step functions, proof of It^o's isometry for step functions; extension for progressively measurable functions, It\^o's isometry, It\^o integrable functions (on the half-line); martingale properties of the stochastic integral, existence of a continuous modification (with proofs), Wald's identities (without proof); It\^o processes, It\^o's formula (proof only for $df(W_t)$), its variants and applications to calculation of stochastic integrals, product rule for stochastic differentials; multidimensional It\^o's formula (proof using calculus rule for stochastic differentials); basic ideas of stochastic differential equations; basic examples (geometric Brownian motion, Ornstein-Uhlenbeck).)
L'evy processes (definition of infinitely divisible random variables with examples; L'evy-Khintchine formula (be able to show the integrability in the jump part), condition on the L'evy measure; definition of L'evy processes and examples; Poisson and compound Poisson processes; idea of a construction of the L'evy process from three parts (Gaussian, compound Poisson and small jumps), relation to the Poisson process; L'evy processes with finite linear variation (idea of the proof); operators P_t associated with the L'evy process and the resolvent U^q (definitions and formulations of basic properties); definition of the potential measure and recurrence/transience properties of the L'evy process, be able to apply the Chung-Fuchs criterium (proof for dimension one); s |
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