Probability Theory III: Stochastic Calculus
IUrii Vasil'evich Prokhorov, I︠U︡riĭ Vasilʹevich Prokhorov, Ūrij Vasil'evič Prohorov, ︠I︡Uriĭ Vasilʹevich Prokhorov, I͡ Uri ĭ Vasilʹevich Prokhorov
Springer Science & Business Media, 1998 - Business & Economics - 253 pages
Preface In the axioms of probability theory proposed by Kolmogorov the basic "probabilistic" object is the concept of a probability model or probability space. This is a triple (n, F, P), where n is the space of elementary events or outcomes, F is a a-algebra of subsets of n announced by the events and P is a probability measure or a probability on the measure space (n, F). This generally accepted system of axioms of probability theory proved to be so successful that, apart from its simplicity, it enabled one to embrace the classical branches of probability theory and, at the same time, it paved the way for the development of new chapters in it, in particular, the theory of random (or stochastic) processes. In the theory of random processes, various classes of processes have been studied in depth. Theories of processes with independent increments, Markov processes, stationary processes, among others, have been constructed. In the formation and development of the theory of random processes, a significant event was the realization that the construction of a "general theory of ran dom processes" requires the introduction of a flow of a-algebras (a filtration) F = (Ftk::o supplementing the triple (n, F, P), where F is interpreted as t the collection of events from F observable up to time t.
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absolute continuity applications assume Banach space boundary bounded calculus called Chap coefficients complete concept condition consider constant convergence corresponding defined Definition denote depend derivative diffusion distribution domain English transl equality equivalent established example exists fact formula function give given Hence holds important independent increments initial Itō Jacod jumps Krylov limit locally Markov martingale Math means measure method natural o-algebra obtain operator particular predictable present probability problem process with independent proof proved question random variables reflection representation respect result satisfied semimartingale sense sequence Skorokhod smooth solution space square-integrable stationary stochastic basis stochastic differential equations stochastic integral stochastic processes strong solution sufficient Suppose Theorem theory tion trajectories triple turns uniformly unique values variation Watanabe weak Wiener process