# 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. ### A Karhunen–Loève theorem expansion random variables

Hi, in the Karhunen–Loève theorem's statement the random variables in the expansion are given by $$Z_k = \int_a^b X_te_k(t) \: dt$$ ##X_t## is a zero-mean square-integrable stochastic process defined over a probability space ##(\Omega, F, P)## and indexed over a closed and bounded interval ##[a...
2. ### Ensemble vs. time averages and Ashcroft and Mermin Problem 1.1

The question is as seen below: My attempt (note that my questions are in bold below) is below. Please note that I am self-studying AM: (a) By the independence of any interval ##dt## of time and time symmetry, we expect these two answers are the same (Is there any way to make this rigorous?)...
3. ### Using a Logarithmic Transformation for a Simpler Random Walk Model

Answer to 1. Answer to 2. How would you answer rest of the questions 4 and 5 ?
4. ### Understanding Recurrence in Probability: Solving for hN(1) and cNcN

I set hN(1)hN(1) equal to cNcN, but I'm confused on how I'd be able to solve it and because of that I was not able to conclude that 0 is recurrent when qx/px = infinity
5. ### A When should we use the Langevin equation and when should use Fokker-Planck

As everyone knows that we can go from Langevin equation to Fokker-Planck equation which gives the evolution of probability density function. But what I don't understand is what is exactly the main difference between them as long as they are both give the variance (which then we can for example...
6. ### 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...
7. ### Schools Who are the Top Non-Equilibrium Research Groups in North America and Europe?

I realize the question is quite broad but what research groups working on statistical physics, stochastic processes, and complex systems are generally considered the best? Would like to know about Europe and America alike.
8. ### 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...
9. ### 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...
10. ### 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...
11. ### A Stopping rule for quality control problem

Problem: Suppose I have a production process that yields output in batches of n items. For each batch, I can test whether they are of good or bad quality. Let q_i ∈ {1,0} be the quality of tested item i. If more than half of the items are ‘bad’, the batch should be discarded. In other words...
12. ### 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...
13. ### A Liouville Master Equation for an Open Quantum System

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...

15. ### Limit involving extinction probability of branching process

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. ### 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...
17. ### Partner study for grad students on winter break: Success stories?

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. ### 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.
19. ### Looking for text on stochastic processes

An introductory text is preferable. Topics relevant (not a deal-breaker if not covered): Poisson process, Markov chains, renewal theory, models for queuing, and reliability. Also, in the future I'd like to dabble in stochastic calculus, but my background in measure theory is non-existent. I've...
20. ### 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!
21. ### 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...
22. ### 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. ### 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...
24. ### 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
25. ### 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...
26. ### 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...
27. ### Does anyone have some expertise in stochastic processes?

Would anyone be able to help me with a problem involving martingales and another problem dealing with Poisson processes perhaps? I'm completely stuck on my last homework assignment in the class this term. Thanks for anyone who can help.
28. ### 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...
29. ### 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...
30. ### 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...
31. ### Solving Stochastic Processes Homework for PP (9/hora)

Homework Statement If people entering the engineering building following a PP (9/hora) and you know that between 10:00 am and 11:00 am came exactly 100 persons, what is the probability that between 10:00 am and 10:20 am entered less than 20 people If people entering the engineering...
32. ### Complex Variables or Stochastic Processes?

Hi, I am a math and physics major planning on going into biophysics for grad school, and i want to do computational/mathematical modelling/theoretical work in the field. I have one more math course to take and I am not sure which would be more useful. Here are their very brief course...
33. ### Stochastic processes: infinite server queue with batch poisson arrivals

Hi everyone, I am trying to solve this problem but I am stuck with doubts. Here are my ideas. Homework Statement Busloads of customers arrive at an infinite server queue at a Poisson rate λ Let G denote the service distribution. A bus contains j customers with probability aj = 1...
34. ### Why is waiting time memoryless? (in Stochastic Processes)

I am learning Stochastic Processes right now. Can someone some explain why waiting time is memoryless? Say, if a light bulb has been on for 10 hours, the probability that it will be on for another 5 is the same as the 1st 5 hours. It doesn't make sense to me, because the longer you use it...
35. ### Markov Processes & Diffusion: Textbook Reference

This semester I have a course on mathematical methods in physics. It's in three parts and the first professor is talking about Markov processes (discrete and continuous time) and diffusion. The problem is he doesn't have any notes or a reference textbook. Do you know any textbook on these topics?
36. ### Stochastic Processes, Poisson Process | Expected value of a sum of functions.

Homework Statement Suppose that passengers arrive at a train terminal according to a poisson process with rate "$". The train dispatches at a time t. Find the expected sum of the waiting times of all those that enter the train. Homework Equations F[X(t+s)-X(s)=n]=((($t)^n)/n!)e^(-\$t))...
37. ### Betting & Stochastic Processes

Dear Community, I am faced with a challenge. I can't quite grasp the implications of this. I schould've listened better in statistics lectures! :blushing: I would really appreciate your help. :) I have a not normal stochastic process P on which I can bet. Obviously, I don't know the...
38. ### Help with stochastic processes

From my extremely small and inadequate knowledge of stochastic processes (and Wikipedia): A stochastic process is a process in which some later state is determined by predictable actions and by a random element. Now the question: this "random element" is this meant to compensate for...
39. ### Markov Chain of Stochastic Processes

I would like to construct a model using a markov chain that has different stochastic processes for each state in the chain. Is there a term for such a thing, or anything similar to it? Thanks
40. ### Stochastic Processes - Poisson Process question

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. ### Stochastic processes: martingales

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. ### Stochastic Processes Homework: Rewriting Expectation

Homework Statement I know that per definition E(N)= \sum P(N=k) \cdot k . But how can I rewrite the above expectation towards the 'usual definition'?
43. ### Solving Stochastic Processes Homework Problem

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. ### Stochastic Processes: Introduction and Tips

Hi all, Im going to be researching into Stochastic processes don't know anything about it except the title, Thought I might get on here to get an introduction, see what other people know about it and tips that would be helpful in understanding the concepts? so if anybody knows anything about...