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Suppose further that the walk stops if it reaches 0 or m ≥ a; the time at which this first occurs is a stopping time. If it is known that the expected time at which the walk ends is finite (say, from Markov chain theory), the optional stopping theorem predicts that the expected stop position is equal to the initial position a.
Example of a stopping time: a hitting time of Brownian motion.The process starts at 0 and is stopped as soon as it hits 1. In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time [1]) is a specific type of “random time”: a random variable whose value is interpreted as the time at ...
The class of semimartingales is closed under optional stopping, localization, change of time and absolutely continuous change of probability measure (see Girsanov's Theorem). If X is an R m valued semimartingale and f is a twice continuously differentiable function from R m to R n, then f(X) is a semimartingale. This is a consequence of Itō's ...
Pandya theorem (nuclear physics) Pomeranchuk's theorem ; Reeh–Schlieder theorem (local quantum field theory) Spin–statistics theorem ; Stone–von Neumann theorem (functional analysis, representation theory of the Heisenberg group, quantum mechanics) Supersymmetry nonrenormalization theorems ; Vafa–Witten theorem
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Its original application in physics was as a model for the velocity of a massive Brownian particle under the influence of friction. It is named after Leonard Ornstein and George Eugene Uhlenbeck . The Ornstein–Uhlenbeck process is a stationary Gauss–Markov process , which means that it is a Gaussian process , a Markov process , and is ...
In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local martingale that is bounded from above is a submartingale; however, a local ...
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. [ citation needed ] Partition functions are functions of the thermodynamic state variables , such as the temperature and volume .