Theorems regarding stochastic processes

Theorems regarding stochastic processes are mathematical results that describe the behavior, properties, and characteristics of stochastic processes, which are systems that evolve over time with randomness. Here are some important theorems and concepts in the study of stochastic processes: 1. **Markov Property**: A stochastic process satisfies the Markov property if the future state of the process depends only on the present state and not on the past states.

The Bruss–Duerinckx theorem is a result in the field of probability theory and mathematical finance, specifically related to the pricing and replication of contingent claims in incomplete markets. It presents conditions under which a contingent claim can be obtained as the limit of portfolios in a given financial market. The theorem states that if a financial market is incomplete, then under certain conditions, there exists an equivalent martingale measure (a probability measure that allows for the pricing of contingent claims).

The Bussgang theorem is a result in signal processing and statistics, named after Julian J. Bussgang, who introduced it in the context of nonlinear systems. The theorem states that if a Gaussian random process is passed through a nonlinear system, the cross-correlation of the output signal with the input signal can be expressed in terms of the correlation of the input signal alone.

The Clark–Ocone theorem is a fundamental result in the theory of stochastic calculus and financial mathematics, particularly in the context of stochastic processes. This theorem provides a way to express a certain class of random variables (specifically, adapted, or predictable functionals of a process) in terms of an integral with respect to a martingale and a stochastic integral.

Foster's theorem, often discussed in the context of stochastic processes and in particular for Markov chains and Markov decision processes, provides insights into the long-term behavior of certain types of random processes. One common application of Foster's theorem is in the study of Markov chains with continuous state spaces. In its simplest form, Foster's theorem relates to the existence of a stationary distribution for a Markov chain.

Ignatov's theorem refers to a result in the field of functional analysis, particularly concerning the properties of bounded linear operators on Banach spaces. Specifically, it deals with the existence of certain types of fixed points or invariant elements under the action of a non-expansive operator.

The Kolmogorov continuity theorem is a fundamental result in the theory of stochastic processes, particularly in the study of Brownian motion and other continuous-time processes. It provides conditions under which a collection of random variables (typically indexed by time) possesses a continuous version, which means that the sample paths of the process can be modified to be continuous with probability one.