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. P

    I A new interpretation of Quantum Mechanics

    These papers claim to present a realistic stochastic interpretation of quantum mechanics that obeys a stochastic form of local causality. (A lecture I recently attended mentioned these papers). It also claims the Born rule as a natural consequence rather than an assumption. This appears to me to...
  2. T

    I Evaluating a Stochastic Average

    Hi all, I am not familiar with stochastic processes, but I would like to know how to evaluate the following expectation value: $$\mathbb{E}[e^{\int_{0}^{t}d\tau(V_{i}(\tau)-V_{j}(\tau))}]$$ where ##\mathbb{E}[V_{i}(t)] = 0,\mathbb{E}[V_{i}(t),V_{j}(t')] = \gamma\delta_{ij}\delta(t-t')## for some...
  3. WMDhamnekar

    Stochastic mathematics in application to finance

    Get your friends and family to play this simple little game that illustrates a key aspect of stochastic mathematics in application to finance. Draw on a big sheet of paper a sequence of 30 squares and label them consecutively 0 (bankrupt), 1, 1, 2, 2, 3, 3, 4, . . . , 14, 14, 15...
  4. T

    A Brownian Motion (Langevin equation) correlation function

    So the Langevin equation of Brownian motion is a stochastic differential equation defined as $$m {d \textbf{v} \over{dt} } = - \lambda \textbf{v} + \eta(t)$$ where the noise function eta has correlation function $$\langle \eta_i(t) \eta_j(t') \rangle=2 \lambda k_B T \delta_{ij} \delta(t -...
  5. V

    Adding noise and solving stochastic ODEs in Python

    The Coupled ODE Model Below are my coupled differential equations, where the only variable I try to meddle with is the ITMblood. The motivation here is if I try to increase ITMblood (in the next section I will show how I do it), at some concentration of ITMblood (most likely a very huge one) ...
  6. lelouch_v1

    Probability Density of ##x## (Wiener Process)

    Suppose that W(t) is just a Wiener process (i.e. a Gaussian in general). I want to know what the probability density for x, P(x), is. I started off by just assuming that I want to measure the expectation value of an observable f(x), so ##<f(x)>=\int_{W=0}^{W=t}{P(W)f(g(W))dW} \ \ ,\ \ x=g(W) ##...
  7. thaiqi

    Quantum Nelson's Stochastic Interpretation of QM: Which Material is Best?

    Hello,everyone. Which material is best for Nelson's stochastic interpretation of quantum mechanics?
  8. MathematicalPhysicist

    A Is Stochastic String Theory the Key to Original Ideas in Physics?

    Well there was today some lecture on String theory which I got late since I was reading something else in Condensed matter physics and then I looked at a book that I have of stochastic Integration with jumps and by associativity of ideas I searched google for Stochastic String Theory and got to...
  9. F

    I Question about ergodic theorems

    Hello everyone. I am currently reading the book Probability statistics and random processes for electrical engineering by Alberto Leon Garcia. In page 540, one can find example 9.47, in which is shown how a stationary random process doesn't have to be ergodic by defining a random variable A of...
  10. fresh_42

    I QM through stochastic optimization on spacetimes

    I have a simple question as a layman in the field: Is this worth reading, and even more, is it a contribution to possibly shorten the endless discussions in this subforum? https://www.nature.com/articles/s41598-019-56357-3.pdf
  11. T

    Coupled linear stochastic differential equations

    In order to solve for ##x##, I need to re-write the equation for ##dx## so it is independent of ##y## and ##dy##. However, I am having some issues with this. Can someone give me a push in the right direction?
  12. T

    Auto-correlation of a stochastic equation

    Here is my attempt: $$ \begin{align} \langle x(t)x(t+\tau) \rangle &= \langle (e^{-\beta W(t)}W(t))(e^{-\beta W(t+\tau)}W(t+\tau))\rangle \\ &= \langle (e^{-\beta W(t)}e^{-\beta W(t+\tau)})(W(t)W(t+\tau))\rangle \\ &= \langle (e^{-\beta W(t)}e^{-\beta W(t+\tau)})(\int_0^t...
  13. JorgeM

    A Are Chaotic and Stochastic processes related?

    Hello everyone. I have read on the web some people that mention something about "stochastic chaos" but I am not that sure about what it really means or if that actually exists. Two months ago , I started to study some chaotic systems but in stochastic systems I am not that familiarized in...
  14. A

    I Is throwing dice a stochastic or a deterministic process?

    As far as I understand it a stochastic process is a mathematically defined concept as a collection of random variables which describe outcomes of repeated events while a deterministic process is something which can be described by a set of deterministic laws. Is then playing (classical, not...
  15. A

    A Why is this Pilot-wave model on a discrete spacetime stochastic?

    Look at the paper in the link below: https://link.springer.com/content/pdf/10.1007%2Fs10701-016-0026-7.pdf It introduces a pilot-wave model on a discrete spacetime lattice. However, the pilot-wave model is not deterministic; the motion of quantum particles is described by a |Ψ|^2-distributed...
  16. V

    MHB How to modify Adaline Stochastic gradient descent

    Dear May I know how to modify my own Python programming so that I will get the same picture as refer to the attached file - Adaline Stochastic gradient descent (I am using the Anaconda Python 3.7) Prayerfully Tron
  17. R

    Drawing the line between the Pilotwave and Stochastic Interpretations

    Greetings… I requested some books that deals with the different interpretations of QM almost a year ago and I read lots of them. My favourite was Laloe's (so whoever recommended it, extra thanks for you). I noticed something interesting in the Bohmian mechanism, a common misconception if I may...
  18. F

    I How can I represent a stochastic process in 2D?

    Hello everyone. I have recently started working with a model whose output are two stochastic process which evolve trough time. Now, I have two 9*500 matrices, being 9 the number of times for which the model offers a value and 500 the number of realizations. I was wondering if someone could...
  19. F

    A Apliying PCA to two correlated stochastic processes

    Hello everyone, I have two matrices of size 9*51, meaning that I have 51 measurements of a stochastic process measured at 9 times, being precise, it is wind speed in the direction X, I have the same data for the direction Y. I am aware that both stochastic processes are not independent, so I...
  20. binbagsss

    Other Operations research, stochastic dynamic programming, sources

    My lecture notes and recommended textbook Hillier and Liberman are not enough for me. My methodology and formulation of problems still seems like too much guess-work. Can anyone recommend any good resources, lecture notes or textbooks, for stochastic DP? Many thanks
  21. J

    Expectation Value of a Stochastic Quantity

    Homework Statement I'm working on a process similar to geometric brownian motion (a process with multiplicative noise), and I need to calculate the following expectation/mean; \langle y \rangle=\langle e^{\int_{0}^{x}\xi(t)dt}\rangle Where \xi(t) is delta-correlated so that...
  22. Peter Morgan

    A Stochastic signal analysis in QM

    [Moderator's note: This discussion has been spun off from another thread since it was getting into a more technical area only tangentially related to the original thread's topic. Some posts have been edited slightly for the new context.] They're too technical for the question here, and only...
  23. T

    Stochastic calculus:Laplace transformation of a Wiener process

    Homework Statement I am asked to show that E[\exp(a*W_t)]=\exp(\frac{a^2t}{2}) Let's define: Z_t = \exp(a*W_t) W_t is a wiener process Homework Equations W_t \sim N(0,\sqrt{t}) The Attempt at a Solution I want to use the following formula. if Y has density f_Y and there's a ral function g...
  24. Aslet

    I Probability of a Stochastic Markov process

    Hi everyone! I'm approaching the physics of stochastic processes. In particular I am studying from "Handbook of stochastic processes - Gardiner". This book defines a stationary process like: $$ p(x_1, t_1; x_2, t_2; ...; x_n, t_n) = p(x_1, t_1 + \epsilon; x_2, t_2 + \epsilon; ...; x_n, t_n +...
  25. PHstud

    I Autocorellation of a stochastic process

    Hello ! I am trying an exercice to get a better grip of what is the autocorellation meaning. I know the mathematical formula, but let's consider a case. If in the case above, the probabilty of the red curve to happen (so w2) is Pr, the blue one Pb and the green on Pg, what would be the...
  26. N

    I Stochastic Minimax partition problem card game

    Hey all, ran into a game theory problem I can't solve. A and B have a set of 10 random numbers from 1-10, players can make so called "piles", a pile has a goal number from 1 to 10, if 6,3 are on the table, a pile of nine may be started, the pile is added to by adding sets of numbers that sum to...
  27. A

    A Stochastic differential equations with time uncertainty....

    Hi all, I'm wondering if anyone is able to point me in a direction regarding an aspect of stochastic differential equations. I have a situation in which I need to propagate a stochastic DE through time using measurement updates - however, the exact time at which each measurement arrives is...
  28. S

    A What is stochastic heating of very small grains?

    I recently came across a paper where there was a term ''stochastic heating of very small grains". I went through some papers on this area but I am still confused about it. What is the vibrational temperature associated with very small grains? Does temperature has got a meaning if it's a highly...
  29. D

    Double stochastic matrix positive recurrent?

    Homework Statement When is a Markov chain with double stochastic matrix positive recurrent? Homework Equations Double stochastic matrix is when the sum of the column vectors, and not just the row vectors, is 1. The Attempt at a Solution I know I have to show that the expected value of the...
  30. T

    I About stochastic differential equation and probability density

    I have two questions about the use of stochastic differential equation and probability density function in physics, especially in statistical mechanics. a) I wonder if stochastic differential equation and PDF is an approximation to the actual random process or is it a law like Newton's second...
  31. nomadreid

    I Odd use of terms (“stationary stochastic process”., etc.)

    I am trying to make sense of a Russian author’s use of terms (I have to translate his article). I have three questions, but please don't think you need to answer all three before answering. Thanks for any insights! [1] He uses the term “probability density distribution” ρ(ξ) of a stationary...
  32. Fyj

    Studying Take module on Quantum Computing or Stochastic Mechanics?

    Hi. I was wondering if you could help me out a bit here. I have a choice to take either a module on stochastic dynamics in statistical mechanics or quantum computing for my masters. Both seem really interesting so it’s a close tie between which one has more current/future applications. The...
  33. F

    Construct a Markov Chain: How to Generate Xn's Using the Sequence U0, U1, ...?

    Homework Statement Describe the construction of a Markov chain X0, X1, ... on Ω ∈ (0, 1) with state space S = {1, 2, ..., s} and S X S PTM P and initial state X0 ~ ν (probabilities distributed like vector ν). Use the sequence U0, U1, ... to generate the Xn's Homework Equations U0, U1 is a...
  34. E

    A Measuring the degree of convergence of a stochastic process

    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...
  35. Avi Nandi

    A Stochastic Differential equation: Dichotomous Process

    I am studying a dichotomous markov process. The master equation is given in this link https://en.wikipedia.org/wiki/Telegraph_process. I want to calculate the mean and correlation function given also in the link. But actually I can't make any progress. How from this master equation governing the...
  36. kev931210

    Stochastic dynamics and Angular velocity of a molecular motor

    Homework Statement Homework Equations Langevin equation (I included all the equations in the next section.[/B]The Attempt at a Solution I do not know how I can proceed from this point. I'm stuck since I have no information on the drag coefficient. Maybe my approach is wrong, and there may...
  37. Alex_Doge

    Proof that an interval is a confidence interval for Geom(q)

    Hello Physicsforum Homework Statement I have a problem proving this: Given C(x)=[0, 3/x] for all x\in\chi, with \chi=\Omega being the sample space and P_q=Geom(q) being the geometric distribution. I have to show that C(x) is a confidence Interval for q but I don't know how to get started...
  38. C

    MHB Stochastic differential equations

    Hello to all, I am a new member, but I've been reading and getting help from this forum for a year! I recently started to study about stochastic calculus because I am considering risk management/ actuarial/ finance job. I would appreciate your help. If we have Poisson $(\lambda)$ and $W$...
  39. T

    Stochastic Matrices in Cosmology

    Good afternoon all, I'm taking a linear algebra course this semester, and upon entering the topic of 'Applications of Matrix Operations', my professor has given our class the opportunity to earn some extra credit points by writing a paragraph or two on the application of stochastic matrices in...
  40. M

    Is Discarding the Higher Random Number in a Stochastic Queue Simulation Correct?

    Hello guys! I am learning a bit on queue theory in one of my courses, and decided to try and do some simulations even though it is not mandatory. (Curriculum only covers the steady state, where it can be treated mathematically with ease). Im looking at a birth-death process, where the time til...
  41. E

    Integral in Gardiner's book on stochastic methods

    Dear all, I have troubles in one proof of the book Handbook of stochastic methods by Gardiner. In the paragraph 3.7.3 he writes this integral \sum_i\int d\vec x \frac{\partial}{\partial x_i}[-A_ip_1\log(p_1/p_2)] where p_1 and p_2 are two solutions of the Chapman-Kolmogorov equation and \vec A...
  42. H

    True/False : Stationary process In stochastic process

    Stochastic process problem! 1. If Xn and Yn are independent stationary process, then Vn= Xn / Yn is wide-sense stationary. (T/F) 2. If Xn and Yn are independent wide sense stationary process, then Wn = Xn / Yn is wide sense stationary (T/F) I solve this problem like this: 1...
  43. H

    About stochastic process....Help please

    Given a Gaussian process X(t), identify which of the following , if any, are gaussian processes. (a)X(2t) solution said that X(2t) is not gaussian process, since and similarly Given Poisson process X(t) (a) X(2t) soultion said that X(2t) is not poisson process, since same reason above...
  44. FrancescoMi

    MATLAB [Matlab] Simulation of Stochastic Process

    Hi all, I have this dynamic: is a Mean Reverting process. I want to simulate the sde with MATLAB but I am a beginner and I have some problems. I show you the code that I have created: %% Simulazione prezzo Geometric Ornstein-Ulenbeck clear all clc %Parameters mu = 0.5; sigma = 0.12; eta =...
  45. NATURE.M

    Logistic regression: Stochastic Gradient Ascent (in python)

    So I've been following through a online course in machine learning offered by Stanford university. I have been recently reading up on logistic regression and stochastic gradient ascent. Here is a link to the original notes: http://cs229.stanford.edu/notes/cs229-notes1.pdf (pages 16-19). Here...
  46. G

    How do I solve this stochastic differential equation?

    Homework Statement I am trying to solve this \begin{align} d X_t = - b^2 X_t (1 - X_t)^2 dt + b \sqrt{1 - X_t^2} dW_t \end{align} where $b$ is a constant. Note that I have the answer here and can provide it if necessary. But I want to know how one would come up with it. Homework EquationsThe...
  47. laramman2

    Applied What Are the Best Textbooks for Self-Studying Nonlinear Dynamics?

    What textbooks would you recommend for self studying Nonlinear Dynamics? I am a undergraduate junior who will be doing research on nonlinearity of spiking neurons. I have taken courses on ODE, vector calculus, probability, statistics, and linear algebra.