Questions tagged [stochastic-processes]

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.

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

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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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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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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…
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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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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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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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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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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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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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What is the Kolmogorov Extension Theorem good for?

The Kolmogorov Extension Theorem says, essentially, that one can get a process on $\mathbb{R}^T$ for $T$ being an arbitrary, non-empty index set, by specifying all finite dimensional distributions in a "consistent" way. My favorite formulation of…
Michael Greinecker
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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…
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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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Probability of completing a self-avoiding chessboard tour

Someone asked a question about self-avoiding random walks, and it made me think of the following: Consider a piece that starts at a corner of an ordinary $8 \times 8$ chessboard. At each turn, it moves one step, either up, down, left, or right,…
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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…
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Can I apply the Girsanov theorem to an Ornstein-Uhlenbeck process?

Let $W_t$ be a standard Brownian motion, and $X_t$ a measurable adapted process. Girsanov's theorem says that under certain conditions, the Brownian motion with drift $Y_t = W_t - \int_0^t X_s\,ds$ can be a Brownian motion under a certain…
Nate Eldredge
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