# Skorokhod's contribution to probability theory

I always thought of Skorokhod associated with the topology for convergence in the space of càdlàg (right continuous with left limits) functions in probability.

But it seems that he made several other contributions. He studied the reflected Brownian motion, proved the first existence theorem for stochastic differential equations.

What else did he do in the field?

## 1 Answer

Perhaps the best known contributions are the method of a single probability space, and the Skorokhod integral, which extends the Itō integral to non-adapted processes and is adjoint to the Malliavin derivative. He also did some foundational work on stochastic processes on manifolds with boundary. The Institute of Mathematics in Kiev maintains a Skorokhod webpage with a "short sketch of life and research", which is much longer than the Wikipedia entry. "His absence at international scientific conferences gave birth to the opinion among foreign scientists that "Skorokhod" was the collective name of a group of Soviet scientists, just as the group of French mathematicians united under the name "Bourbaki"".

Some details on his research from the webpage:"The first series of Skorokhod's works that gained him wide recognition was devoted to the limit theorems for random processes constructed on the basis of sums of independent random variables. These works accomplished the series of attempts of numerous mathematicians aimed at the generalization of the famous Donsker invariance principle to the case where the limit process is an arbitrary, not necessarily continuous, process with independent increments... Skorokhod proposed the method of a single probability space (mentioned earlier) and introduced several topologies in the space of functions that do not have discontinuities of the second kind, one of which is now widely known as the Skorokhod topology... One should mention his proof of the theorem on existence of solutions of stochastic differential equations by the method of a single probability space under the assumption that the coefficients of these equations are continuous functions (i.e., they may not satisfy the Lipschitz condition)...

In 1966, he proved that a sufficiently broad class of continuous Markov processes can be reduced to quasidiffusion processes by a random change of the time variable. In the monograph "Stochastic Equations for Complex Systems", he constructed stochastic differential equations for quasidiffusion processes taking values in spaces of complex structure (e.g., manifolds with boundary, manifolds with variable dimensionality, etc.)... In the 1970s, Skorokhod introduced several notions, which are now widely used not only by mathematicians, but also by physicists. Among them, one should mention the notions of extended stochastic integral (the Skorokhod integral), strong (weak) random linear operator, and stochastic semigroup."

• Skorokhod introduced the notion of stochastic semigroup? Can you say more about that? – Conrado Costa Sep 7 '15 at 22:30
• @Conrado Costa Here is his survey paper turpion.org/php/paper.phtml?journal_id=rm&paper_id=4021 From MR:"The author, in earlier papers, has investigated martingales and stochastic semigroups; and in his monograph Random Linear Operators he investigated the relationship between random differential and integral equations whose solutions are operator-valued random functions and stochastic semigroups... the author presents a number of new results... on the relationship between stochastic semigroups and random differential equations whose solutions are operator-valued". – Conifold Sep 7 '15 at 23:21