# stochastic process problems

[230] There are different interpretations of a point process, such a random counting measure or a random set. for all {\displaystyle T} A stochastic process is defined as a collection of random variables X={Xt:t∈T} defined on a common probability space, taking values in a common set S (the state space), and indexed by a set T, often either N or [0, ∞) and thought of as time … A stochastic process may involve several related random variables. Σ , the difference [91][280] For example, the problem known as the Gambler's ruin is based on a simple random walk,[196][281] and is an example of a random walk with absorbing barriers. [169] For example, the supremum of a stochastic process or random field is not necessarily a well-defined random variable. { -dimensional Euclidean space or some manifold. [80] If the index set is : had the meaning of time, so [70][71] For example, a stochastic process can be interpreted or defined as a If the state space is the real line, then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. [162][163][165] The theorem can also be generalized to random fields so the index set is X , ]�!5�6� 3cP�ǫ�r58A�ݸ�+sۺ���,9E�'�:G�3���a}d��� ��v��0�Nyd�7�Y�ɊLɢ�18\\��8��Ͱ�� , or decreases by one with probability is a discrete-time martingale. ∈ {\displaystyle T} The name Khinchin is also written in (or transliterated into) English as Khintchine. , then for any two non-negative numbers Damiano Brigo, Bernard Hanzon and François LeGland, A Differential Geometric approach to nonlinear filtering: the Projection Filter, I.E.E.E. S The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. = n {\displaystyle p} ∞ {\displaystyle T} Although stochastic process theory and its applications have made great progress in recent years, there are still a lot of new and challenging problems existing in the areas of theory, analysis, and application, which cover the fields of stochastic control, Markov chains, renewal process, actuarial science, and so on. [56][57] If the index set is some interval of the real line, then time is said to be continuous. ( [1][5][30][53][58], A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. G differ from each other at most on a subset of X 1 n [180][184][185] The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, … 1Shandong University of Finance and Economics, Jinan, China, 2University of Melbourne, Melbourne, Australia, 3Central University of Finance and Economics, Beijing, China. {\displaystyle t\in T} [51][107] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled,[109][110] which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem. and for every choice of epochs {\displaystyle (\Omega ,{\cal {F}},P)} stream -valued random variables, which can be written as:[82], Historically, in many problems from the natural sciences a point } {\displaystyle R^{2}} This state space can be, for example, the integers, the real line or X t ) {\displaystyle t\in T} , {\displaystyle (\Omega ,{\mathcal {F}},P)} and a measurable space T E8>�;E�]v�Q����~o]m�Ј��Doe��I�w��A��A��n���YT�c�I�Rp7al�,2�\$���N�t�U�������M�v7u?J�e��C�H���+�. {\displaystyle t\in T} {\displaystyle \{X_{t}\in F{\text{ for all }}t\in G\cap U\}} [225] In a 1932 paper Kolmogorov derived a characteristic function for random variables associated with Lévy processes. 2 {\displaystyle S} X {\displaystyle T=[0,\infty )} t {\displaystyle X_{t_{2}}-X_{t_{1}}} ( ) X { {\displaystyle D[0,1]} : It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. -dimensional Euclidean space. {\displaystyle \left\{X_{t}\right\}} Y S C ) /ExtGState 19 0 R n [62] For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term stochastischer Prozeß was used in German by Aleksandr Khinchin,[65][66] though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931. ∈ X The year 1654 is often considered the birth of probability theory when French mathematicians Pierre Fermat and Blaise Pascal had a written correspondence on probability, motivated by a gambling problem. X T t , so the law of a stochastic process is a probability measure. The general idea is to establish a "best estimate" for the true value of some system from an incomplete, potentially noisy set of observations on that system. {\displaystyle X} [151][169], More precisely, a real-valued continuous-time stochastic process {\displaystyle t_{2}\in [0,\infty )} /Group << − [127][128], Defined on the real line, the Poisson process can be interpreted as a stochastic process,[51][129] among other random objects. has the same distribution, which means that for any set of X ( Deﬁnition: {X(t) : t ∈ T} is a discrete-time process if the set T is ﬁnite or countable. denotes function composition and Zakai, Moshe (1969), On the optimal filtering of diffusion processes. {\displaystyle n} Deﬁnition: {X(t) : t ∈ T} is a discrete-time process if the set T is ﬁnite or countable. T t ∞ {\displaystyle \{X(t,\omega ):t\in T\}} 43, 2 (1998), pp 247--252. https://en.wikipedia.org/w/index.php?title=Filtering_problem_(stochastic_processes)&oldid=840761007, Creative Commons Attribution-ShareAlike License, This page was last edited on 12 May 2018, at 00:25. [59][311] This theorem, which is an existence theorem for measures on infinite product spaces,[315] says that if any finite-dimensional distributions satisfy two conditions, known as consistency conditions, then there exists a stochastic process with those finite-dimensional distributions.

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