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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]
The Itô integral of the process with respect to the Wiener process is denoted by (without the circle). For its definition, the same procedure is used as above in the definition of the Stratonovich integral, except for choosing the value of the process X {\displaystyle X} at the left-hand endpoint of each subinterval, i.e.,
Finding well-behaved subcalculi of a given process calculus. This is valuable because (1) most calculi are fairly wild in the sense that they are rather general and not much can be said about arbitrary processes; and (2) computational applications rarely exhaust the whole of a calculus. Rather they use only processes that are very constrained ...
To compute the integral, we set n to its value and use the reduction formula to express it in terms of the (n – 1) or (n – 2) integral. The lower index integral can be used to calculate the higher index ones; the process is continued repeatedly until we reach a point where the function to be integrated can be computed, usually when its index is 0 or 1.
More so than the other seminal process calculi (CCS and CSP), the development of ACP focused on the algebra of processes, and sought to create an abstract, generalized axiomatic system for processes, [2] and in fact the term process algebra was coined during the research that led to ACP.
where g 1 (x) = V x a g is the total variation of g in the interval [a, x], and g 2 (x) = g 1 (x) − g(x). Both g 1 and g 2 are monotone non-decreasing. Now, if f is bounded, the Lebesgue–Stieltjes integral of f with respect to g is defined by
The maximum process of a càdlàg process M is written as M* t = sup s ≤t | M s |. For any p ≥ 1 and bounded predictable integrand, the stochastic integral preserves the space of càdlàg martingales M such that E[(M* t) p] is finite for all t. If p > 1 then this is the same as the space of p-integrable martingales, by Doob's inequalities.
It is therefore a synthesis of stochastic analysis (the extension of calculus to stochastic processes) and of differential geometry. The connection between analysis and stochastic processes stems from the fundamental relation that the infinitesimal generator of a continuous strong Markov process is a second-order elliptic operator.