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Stochastic dominance relations are a family of stochastic orderings used in decision theory: [1] Zeroth-order stochastic dominance: A ≺ ( 0 ) B {\displaystyle A\prec _{(0)}B} if and only if A ≤ B {\displaystyle A\leq B} for all realizations of these random variables and A < B {\displaystyle A<B} for at least one realization.
Let (Ω, Σ, P) be a probability space.Let X : I × Ω → S be a stochastic process, where the index set I and state space S are both topological spaces.Then the process X is called sample-continuous (or almost surely continuous, or simply continuous) if the map X(ω) : I → S is continuous as a function of topological spaces for P-almost all ω in Ω.
In the theory of probability for stochastic processes, the reflection principle for a Wiener process states that if the path of a Wiener process f(t) reaches a value f(s) = a at time t = s, then the subsequent path after time s has the same distribution as the reflection of the subsequent path about the value a. [1]
Stochastic chemical kinetics provide yet another example of the use of the master equation. A master equation may be used to model a set of chemical reactions when the number of molecules of one or more species is small (of the order of 100 or 1000 molecules). [4]
Consider the autonomous Itō stochastic differential equation: = + with initial condition =, where denotes the Wiener process, and suppose that we wish to solve this SDE on some interval of time [,]. Then the Milstein approximation to the true solution X {\displaystyle X} is the Markov chain Y {\displaystyle Y} defined as follows:
For n ≥ 2, the n-dimensional Wiener process started at the origin is transient from its starting point: with probability one, i.e., X t > 0 for all t > 0. It is, however, neighbourhood-recurrent for n = 2, meaning that with probability 1, for any r > 0, there are arbitrarily large t with X t < r; on the other hand, it is truly transient for n > 2, meaning that X t ≥ r for all t ...
A common example of a first-hitting-time model is a ruin problem, such as Gambler's ruin. In this example, an entity (often described as a gambler or an insurance company) has an amount of money which varies randomly with time, possibly with some drift. The model considers the event that the amount of money reaches 0, representing bankruptcy.
A σ-algebra defines the set of events that can be measured, which in a probability context is equivalent to events that can be discriminated, or "questions that can be answered at time ". Therefore, a filtration is often used to represent the change in the set of events that can be measured, through gain or loss of information .