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Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

A stochastic process is a collection of random variables representing the evolution of a system of random variables over time. A typical example is a .

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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What's the difference between stochastic and random?

What's the difference between stochastic and random?
Red Banana
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What is the importance of the infinitesimal generator of Brownian motion?

I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have found lack any motivation or intuition. What is…
Potato
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Why did my friend lose all his money?

Not sure if this is a question for math.se or stats.se, but here we go: Our MUD (Multi-User-Dungeon, a sort of textbased world of warcraft) has a casino where players can play a simple roulette. My friend has devised this algorithm, which he himself…
Konerak
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An explanation of the Kalman filter

In the past 3 months, I've been trying to understand the Kalman filter. I have tried to implement it, I have watched YouTube tutorials, and I have read some papers about it and its operation (update, predicate, etc.). However, I still am unable to…
xsari3x
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Numerical phenomenon. Who can explain?

I was doing some software engineering and wanted to have a thread do something in the background to basically just waste CPU time for a certain test. While I could have done something really boring like for(i < 10000000) { j = 2 * i }, I ended up…
Jake Mirra
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Proving that $1$- and $2D$ simple symmetric random walks return to the origin with probability $1$

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the origin with probability $1$? Edit: note that while…
Isaac
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When can we interchange the derivative with an expectation?

Let $ (X_t) $ be a stochastic process, and define a new stochastic process by $ Y_t = \int_0^t f(X_s) ds $. Is it true in general that $ \frac{d} {dt} \mathbb{E}(Y_t) = \mathbb{E}(f(X_t)) $? If not, under what conditions would we be allowed to…
Jonas
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If a coin toss is observed to come up as heads many times, does that affect the probability of the next toss?

A two-sided coin has just been minted with two different sides (heads and tails). It has never been flipped before. Basic understanding of probability suggests that the probability of flipping heads is .5 and tails is .5. Unexpectedly, you flip the…
skurwa
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What is meant by a continuous-time white noise process?

What is meant by a continuous-time white noise process? In a discussion following a question a few months ago, I stated that as an engineer, I am used to thinking of a continuous-time wide-sense-stationary white noise process $\{X(t) \colon…
Dilip Sarwate
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Intuition for random variable being $\sigma$-algebra measurable?

Is there some sort of intuition or a good ilustrative example for random variables being $\sigma$-algebra measurable? I understand the definition, but when looking at martingales, the meaning of random variables being measurable eludes me. So my…
BoZenKhaa
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Convergence of $np(n)$ where $p(n)=\sum_{j=\lceil n/2\rceil}^{n-1} {p(j)\over j}$

Some years ago I was interested in the following Markov chain whose state space is the positive integers. The chain begins at state "1", and from state "n" the chain next jumps to a state uniformly selected from {n+1,n+2,...,2n}. As time goes on,…
user940
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Difference between weak ( or martingale ) and strong solutions to SDEs

Hi Im fairly new to SDE theory and am struggling with the difference between a weak ( or martingale ) solution and a strong solution to an SDE : $$ d(X_{t})=b(t,X_{t})dt + \sigma(t,X_{t})dW_{t} $$ Are these two differences and what do they really…
Monty
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What is "white noise" and how is it related to the Brownian motion?

In the Chapter 1.2 of Stochastic Partial Differential Equations: An Introduction by Wei Liu and Michael Röckner, the authors introduce stochastic partial differential equations by considering equations of the form $$\frac{{\rm d}X_t}{{\rm…
0xbadf00d
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How the Ornstein–Uhlenbeck process can be considered as the continuous-time analogue of the discrete-time AR(1) process?

Wikipedia says The Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of the discrete-time AR(1) process. I was wondering how the Ornstein–Uhlenbeck process can be considered as the continuous-time analogue of the…
Tim
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Difference between Modification and Indistinguishable

Would someone be able to offer a layman's explanation of what is means when two stochastic processes are a Modification of each other and when they are Indistinguishable? My Stochastic Analysis notes define the following: a) $\textit{The stochastic…
John Smith
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