What is Stochastic processes: Definition and 44 Discussions

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space. The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.Based on their mathematical properties, stochastic processes can be grouped into various categories, which include random walks, martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes. The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications.

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  1. cianfa72

    A Karhunen–Loève theorem expansion random variables

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  2. E

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  3. WMDhamnekar

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  4. S

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  5. S

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  6. JorgeM

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  7. S

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  8. F

    A Apliying PCA to two correlated stochastic processes

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  9. J

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  10. F

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  11. E

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  12. E

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    Consider a sample consisting of {y1,y2,...,yk} realisations of a random variable Y, and let S(k) denote the variance of the sample as a function of its size; that is S(k)=1/k( ∑ki=1(yi−y¯)2) for y¯=1/k( ∑ki=1 yi) I do not know the distribution of Y, but I do know that S(k) tends to zero as k...
  13. Raptor112

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    By reading Heinz-Peter Breuer: A Piece Wise Deterministic Process (where you have a deterministic time-evolution + a jump process and which is just a particular type of stochastic process) may be defined in terms of a Liouville master equation for its probability density : Where the first...
  14. A

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  15. J

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    Let x(a) be the extinction probability of a branching process whose offspring is Poisson distributed with parameter a. I need to find the limit as a approaches infinity x(a)e^a. I tried computing x(a) directly using generating functions, and I found that it's the solution to e^(a(s-1))=s, but...
  16. R

    Stochastic processes for a physicist?

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  17. B

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    I'm a grad student studying electrical/computer engineering. Since I have a month of winter break coming up soon, I want to use it to study some more about probability theory and stochastic processes. Has anyone previously done a self study or partner study over a break like this? If so, how did...
  18. R

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  19. G

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  20. C

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  21. L

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  22. R

    MHB Applied Stochastic processes: difference of uniform distributions

    Two independent random variables X and Y has the same uniform distributions in the range [-1..1]. Find the distribution function of Z=X-Y, its mean and variance. =Using change of variables technique seems to be easiest. fX(x) = 1/2 fY(y) =1/2 f = 1/4 ( -1<X<1 , -1<Y<1) Using u =x -y...
  23. R

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  24. R

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  25. X

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    Hello there. The stochastic calc book I'm going through ( and others I've seen ) uses the phrase "\mathscr{F}-measurable" random variable Y in the section on measure theory. What does this mean? I'm aware that \mathscr{F} is a \sigma-field over all possible values for the possible values of...
  26. mnb96

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    Hello, when we have a deterministic signal f:ℝ→ℝ that is square integrable we can typically write f \in L^2(\mathbb{R}). However, what if \{ f(t): \; t\in \mathbb{R} \} are random variables, i.e. f is a continuous-time stochastic process? What is the notation to denote the space of "square...
  27. A

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  28. D

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  29. sunrah

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  30. C

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    Hey! Just as the title suggests I am looking for a good book on stochastic processes which isn't just praised because it is used everywhere, but because the students actually find it thorough, crystal-clear and attentive to detail. Hopefully with solved exercises and problems too! Anyone...
  31. S

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  32. C

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  33. M

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  34. C

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  35. E

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  36. D

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  37. R

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  38. F

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  39. S

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  40. T

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    I had this problem on my last midterm and received no credit for these parts. 1. Express trains arrive at Hiawatha station according to a Poisson process at rate 4 per hour, and independent of this, Downtown local buses arrive according to a Poisson process at rate 8 per hour. a. Given that 10...
  41. D

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    Homework Statement http://img411.imageshack.us/img411/4274/50122514bc3.png Homework Equations http://img133.imageshack.us/img133/4624/68596500xm4.png The Attempt at a Solution I don't know how to start I've found this: Let X be the the winnings per bet and let the total profit...
  42. D

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  43. W

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    Homework Statement I need someone to reassure me (or correct me) on this problem: The process X(t) = e^{At} is a family of exponentials depending on the random variable A. Express the mean \eta(t) , the autocorrelation R(t_1,t_2) , and the first order density f(x,t) of X(t) in terms of...
  44. S

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