What is Stochastic: Definition and 165 Discussions

Stochastic (from Greek στόχος (stókhos) 'aim, guess') refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselves, these two terms are often used synonymously. Furthermore, in probability theory, the formal concept of a stochastic process is also referred to as a random process.Stochasticity is used in many different fields, including the natural sciences such as biology, chemistry, ecology, neuroscience, and physics, as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, cryptography, and telecommunications. It is also used in finance, due to seemingly random changes in financial markets as well as in medicine, linguistics, music, media, colour theory, botany, manufacturing, and geomorphology. Stochastic modeling is also used in social science.

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  1. R

    Stochastic processes for a physicist?

    I was wondering how useful a course in basic stochastic processes is if you want to pursue a career in physics? And especially for a theoretical physicist or astronomer. Im going to have to choose two courses next semester and I think I'm going to choose Special relativity and Mathematical...
  2. I

    MHB Find the expectation and covariance of a stochastic process

    The problem is:Let $W(t)$, $t ≥ 0$, be a standard Wiener process. Define a new stochastic process $Z(t)$ as $Z(t)=e^{W(t)-(1/2)\cdot t}$, $t≥ 0$. Show that $\mathbb{E}[Z(t)] = 1$ and use this result to compute the covariance function of $Z(t)$. I wonder how to compute and start with the...
  3. S

    Example of Stochastic DE for elementary physics?

    Are there any tutorials that apply stochastic differential equations to the settings of elementary physics problems ? - for example, an object sliding down a not-frictionless ramp. The ramps of everyday life don't have a constant coefficient of friction. A better model for them would be...
  4. A

    MHB Determine dynamics of stochastic differential equations (SDE)

    Hi guys, I´ve just started with SDE and Itô´s lemma but don't really know where and how to begin. I´ve realized that both a Reimann integral and Itô integral is needed both cannot figure out how to solve these questions. Would be much appreciated if someone would help me.
  5. A

    'Mean Aversion' in Stochastic Differential Equations

    I had a brief question regarding SDEs. Typically, I've seen models like the Ornstein-Uhlenbeck process that generally revert back to the mean over time. However, I've been trying to find a stochastic differential equation/process that avoids the mean, such as a sharp increase followed by a sharp...
  6. C

    MHB Stochastic Differential Equation using Ito's Lemma

    I am new to SDE, and especially Ito's Lemma. I have a question that I simply cannot answer. It attached. Can someone please help?
  7. Ravi Mohan

    Probability distribution of a stochastic variable

    I am studying an article which involves stochastic variables http://www.rmki.kfki.hu/~diosi/prints/1985pla112p288.pdf. The author defines a probability distribution of a stochastic potential V by a generator functional G[h] = \left<exp\left(i\int...
  8. R

    Stochastic differential equations question. An over view

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  9. B

    Stationary Increments & Markov Property - True or False?

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  10. R

    MHB Applied Stochastic Processes

    The first image contains questions And the second image shows my answer which i manage to do so far, but some of them i could not do it.
  11. Z

    Optimizing Plane Fitting Using Stochastic Gradient Descent

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  12. G

    Looking for text on stochastic processes

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  13. G

    MHB Stochastic Taylor's Expansion (Ito Rule)

    Good day. For $k\geq0$ a continuous function on $\mathbb{R}_+$ and $\{W_t\}$ a standard Brownian motion, could you help me find the Taylor's expansion of the following exponential: $e^{-\int_0^t k(s) dW_s}.$ For the case where $e^{-k W_t}$ where $k>0$ is a constant, I was able to recompute...
  14. C

    Stochastic processes with memory

    Can anyone provide references for stochastic processes where future steps do depend on the past state of the system? Most of the material I'm finding deals purely with memoryless processes. Thanks!
  15. L

    MHB Applied Stochastic Processes

    I really need your help for a solution to these exercises. I will be so grateful.1/ passengers arrive at a train station according to a poisson process of rate lambda per minute and trains depart station according to a renewal process with inter-departure times uniformly distributed between a...
  16. W

    What is the Role of Epsilon in Stochastic Continuity?

  17. 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...
  18. R

    Applied Stochastic Processes - 2?

    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...
  19. R

    Applied Stochastic Processess?

    Random variable X is distributed according to Gaussian distribution P(x)= (1/sqrt(2πσ^2) * exp(- (x-μ)^2 / (2σ^2) ) 1. Calculate its first three moments 2. What are the distributions of Y= X+b; Z=σX; W=X^2 P(x)= (1/sqrt(2πσ^2) * exp(- (x-μ)^2 / (2σ^2) ) so 1.1. the moment of first...
  20. R

    MHB Applied Stochastic Processes: characteristic functions

    Find characteristic functions of 1. The random variable X uniformly distributed on[-1..1] 2. The random variable Y distributed exponentially (with exponent λ) 3. The random variable Z=X+Y
  21. X

    Measure Theory Q's wrt Stochastic Processes

    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...
  22. mnb96

    Question on notation of stochastic processes

    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...
  23. S

    Sampling from a stochastic process

    Homework Statement Given X(t) = cos(2\pi50t + ω), where the stochastic variable ω is uniformly distributed between 0 and 2\pi. Suppose the sampling frequency fs is 30 Hz. What frequency interval is covered after the sampling? Homework Equations Normalized frequency when sampling can be...
  24. M

    What is Stochastic Overflow? | Carl Sagan

    In his book "The Demon-Haunted World" Carl Sagan talks about this, I wonder what kind of physics phenom is that. t.i.a.
  25. J

    Maximal Entropy Random Walk - quantum corrections to stochastic models

    Imagine there is a complex system and we are interested in its basic statistical properties, like the stationary probability distribution. For example for a single electron wandering in defected lattice of semiconductor. Physics offers two basic ways of answering such question: - from one side...
  26. D

    Stochastic Process prerequsites and difficulty?

    My university is offering a course called "Stochastic Process". The only prerequisites to this course according to my university is a course in Probability which uses the book by Rosen. I've read elsewhere that the course actually requires more of analysis (functional analysis and measure...
  27. S

    Basic Stochastic Calculus Question, why does dB^2 = dt?

    As the title says, while using Stochastic Calculus, can someone explain some of the properties of differentials? Why does dB_t dB_t=dt Also, why does dt dt=0 and dB_t dt=0 I don't really get why these work?
  28. B

    Uniform pdf from difference of two stochastic variables?

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  29. W

    Stochastic approximation applied to fixed source problem

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  30. A

    Does anyone have some expertise in stochastic processes?

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  31. O

    Expected Value and First Order Stochastic Dominance

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  32. H

    Text book suggestion for stochastic process

    Hello. I plan on doing independent study on the Stochastic Process and time series models. I have already learned two semesters worth of statistics (Mathematical Statistics and Applications by Wackerly, Mendenhall and Scheaffer). And I have taken a semester of multiple regression models. I...
  33. R

    Infinitesimal generators of bridged stochastic process

    I hope someone can put me on the right track here. I need to derive the infinitesimal generator for a bridged gamma process and have come a bit stuck (its for a curve following stochastic control problem - don't ask). Any tips, papers, books that could guide me out of my hole would be greatly...
  34. J

    Example of a non-Gaussian stochastic process?

    Consider stochastic process ##X(t)## with properties $$ \langle X(t) \rangle = 0, $$ $$ \langle X(t) X(t-\tau) \rangle = C_0e^{-|\tau|/\tau_c}. $$ For example, the position of a Brownian particle in harmonic potential can be described by ##X##. In that case, the probability...
  35. Mandelbroth

    Regarding continuous stochastic variables and probability

    One of my math teachers discussed stochastic ("random") variables today. In an example, he discussed the probability of picking a random number n, such that n\inℝ, in the interval [0,10]. He proceeded to say that the probability of picking the integer 4 (n = 4) is 0, supporting his claim with...
  36. O

    Stochastic Process Intg: Why & How?

    Why does: \int_0^t d(e^{-us} X(s)) = \sigma \int_0^t e^{-us} dB(s) for stochastic process X(t) and Wiener process B(t)? Also, why is the following true: \int_0^t d(e^{-us} X(s)) = e^{-ut}X(t) - X(0)
  37. O

    Strange right-arrow symbol ([itex]\mapsto[/itex]) in stochastic calculus.

    What does the right arrow mean in this context: " ...Then the process t \mapsto \int_{0}^t \phi_s dM_s are well-defined continuous local martingales, whose quadratic variations are given by ... " Is this supposed to mean "the process X that is the mapping X: t \mapsto \int_{0}^t \phi_s dM_s"
  38. D

    Know of a field that combines EM, fluid mech and stochastic processes?

    Touch of background, about half way through a mechatronics engineering degree, I found that I love fluid mechanics and from previous studies I know I enjoy EM. To throw a curve ball in I also am fascinated by the concept of randomness and the associated mathematics. Now for the question, could...
  39. N

    Understanding the Factor 2 in the Langevin Stochastic Differential Equation

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  40. J

    Solving stochastic differentials for time series forecasting

    I am trying to reproduce results of a paper. The model is: dS = (v-y-\lambda_1)Sdt + \sigma_1Sdz_1 \\ dy = (-\kappa y - \lambda_2)dt + \sigma_2 dz_2 \\ dv = a((\bar{v}-v)-\lambda_3)dt + \sigma_3 dz_3 \\ dz_1dz_2 = \rho_{12}dt \\ dz_1dz_3 = \rho_{13}dt \\ dz_2dz_3 = \rho_{23}dt \\...
  41. S

    Applying Ito's Lemma: Solving a Stochastic Differential Equation

    Homework Statement I'm trying to figure out how to use Ito's Lemma, but all I have are notes and proofs. It would help if someone could go through one actual example with me: Use Ito's Lemma to solve the stochastic differential equation: X_t=2+\int_{0}^{t}(15-9X_s)ds+7\int_{0}^{t}dB_s and...
  42. P

    Issue in understanding stochastic ordering

    This is not actually a problem but I need clarification for stochastic ordering From Wikipedia, there is stated: (http://en.wikipedia.org/wiki/Stochastic_ordering) Usual stochastic order: a real random variable A is less than a random variable B in the "usual stochastic order" if Pr(A>x)...
  43. sunrah

    Stochastic Processes: Maximising profits $$

    Homework Statement A company incurs manufacturing costs of $q per item. The product is sold at a retail price of $p per item with p > q. The customer demand K (e.g. number of items that will be sold when the number of items offered is large enough) is a discrete random variable in N. The...
  44. S

    Stochastic modelling, poisson process

    Homework Statement Suppose a book of 600 pages contains a total of 240 typographical errors. Develop a poisson approximation for the probability that three partiular successive pages are error-free. The Attempt at a Solution I say that the number of errors is poissondistributed...
  45. S

    Comparing Stochastic Optimization Problems: Z_t vs. Y_t

    Hi everyone, I am comparing the following optimization problems: Prob Space =(Omega, F_T, (F_t)_{t=1, ..T}, P). Let X be an adapted process. I denote E[|F_t] as the conditional expectation given F_t. 1.Z_t(omega)= max{ E[X_s | F_t](omega) : s=t, ..,T} 2. Y_t(omega)=max{E[X_tau |...
  46. J

    MHB Stationary distribution for a doubly stochastic matrix.

    I can find the stationary distribution vector $\boldsymbol\pi$ for a stochastic matrix $P$ using: $\boldsymbol\pi P=\boldsymbol\pi$, where $\pi_1+\pi_2+\ldots+\pi_k=1$ However, I can't find a textbook that explains how to do this for a doubly stochastic (bistochastic) matrix. Could somebody...
  47. S

    Stationary distribution for a doubly stochastic matrix.

    Homework Statement I can find the stationary distribution vector \boldsymbol\pi for a stochastic matrix P using: \boldsymbol\pi P=\boldsymbol\pi, where \pi_1+\pi_2+\ldots+\pi_k=1 However, I can't find a textbook that explains how to do this for a doubly stochastic (bistochastic) matrix...
  48. L

    Idea of adapted stochastic process doesn't make sense to me

    The technical definition of an adapted stochastic process can be found here https://en.wikipedia.org/wiki/Adapted_process. I understand the following chain of consequences from this definition: {X_i} is adapted \Rightarrow Each random variable X_i is measurable with respect to the...
  49. C

    Good books on stochastic processes?

    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...
  50. K

    Proving Covariance for Stationary Stochastic Processes

    If a stoch. process Xt has independent and weak stationary increments. var(Xt) = σ^2 for all t, prove that Cov(xt,xs) = min(t,s)σ^2 I'm not sure how to do this. I tried using the definition of covariance but that doesn't really lead me anywhere. If it's stationary that means the distribution...
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