Definition of stochastic process. Cylinder #x03C3;-algebra,
finite-dimensional distributions, the Kolmogorov theorem.-
Characteristics of a stochastic process. Mean and covariance functions.
Characteristic functions.- Trajectories. Modifications. Filtrations.-
Continuity. Differentiability. Integrability.- Stochastic processes with
independent increments. Wiener and Poisson processes. Poisson point
measures.- Gaussian processes.- Martingales and related processes in
discrete and continuous time. Stopping times.- Stationary discrete- and
continuous-time processes. Stochastic integral over measure with
orthogonal values.- Prediction and interpolation.- Markov chains:
Discrete and continuous time.- Renewal theory. Queueing theory.- Markov
and diffusion processes.- It#x00F4; stochastic integral. It#x00F4;
formula. Tanaka formula.- Stochastic differential equations.- Optimal
stopping of random sequences and processes.- Measures in a functional
spaces. Weak convergence, probability metrics. Functional limit
theorems.- Statistics of stochastic processes.- Stochastic processes in
financial mathematics (discrete time).- Stochastic processes in
financial mathematics (continuous time).- Basic functionals of the risk
theory.