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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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brownian motion and stochastic calculus - Karatzas& Shreve : 1.3 definition, page 2.

My question concerns the definition 1.3 page 2 of Brownian motion and stochastic calculus of Karatzas & Shreve. Recall that a stochastic process is a collection of random variables (r.v.) $$ X = \{ X_t, t \in [0,\infty) \}. $$ Each of these r.v.…
megaproba
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Expected value for a Poisson Process

A machine works for an exponentially distributed time with rate μ and then fails. A repair crew checks the machine at times distributed according to a Poisson process with rate λ; if the machine is found to have failed then it is immediately…
user7990
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A type of stochastic jump process

Let $X \geq 1$ be an integer r.v. with $E[X]=\mu$. Let $X_i$ be a sequence of iid rvs with the distribution of $X$. On the integer line, we start at $0$, and want to know the expected position after we first cross $K$, which is some fixed integer.…
iMath
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Why do we need progressive measurability in the approximation of process by simple functions?

I am following Karatzas and Shreve- Brownian Motion and Stochastic Calculus. In the context of Stochastic integration one defines the martingale transform for simple functions: Now we would like to extend the this definition for more general…
Conrado Costa
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Markov chain transition matrix

Consider the Markov chain with the following transition matrix: $$P = \pmatrix{0& 0.5 &0 &0 &0 &0.5\\ 0.25 &0 &0.25 &0.25 &0 &0.25\\ 0 &0.5 &0 &0.5 &0 &0\\ 0 &0.25 &0.25 &0 &0.25 &0.25\\ 0 &0 &0 &0.5 &0 &0.5\\ 0.25 &0.25 &0 &0.25 &0.25 &0}$$ I'm…
mary
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Uniqueness of a local martingale problem

First, some notation: let $X = (X_{t})_{t\geq 0}$ be some strong Markov process in $E = \mathbb R^n$ with cadlag paths. Let us denote by $P_t$ the transition semigroup of $X$ and by $\mathbb B$ the Banach space of all bounded measurable real-valued…
SBF
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White Noise Space and Local Time

This question follows from the answer I gave to the question "Wiener Meets Sobolev". I was wondering in the context of White Noise Space if the Local Time at $x$ of a pre-Brownian motion is a notion that can be properly defined. Best regards. Edit:…
TheBridge
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Independent increments M-W

Let $Y_t$ be a stochastic process defined by $Y_t := M_t - W_t$, where $W_t$ is the Wiener process and $M_t := \max_{0 \leq u \leq t} W_u$. Does $Y_t$ have independent increments?
user19611
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Existence of increasing modification

I have the following, seemingly simple question: Consider a stochastic process $(X_t)$ satisfying $X_s\le X_t$ a.s. for all $s\le t.$ My question is: Does there exist a modification $\tilde{X}$ of $X$, which almost surely has increasing sample…
stochPro
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Lower semicontinuity of the indicator function: stochastic processes

Let $X$ be a Markov process given on a metric space $\mathcal X$ by a transition semigroup $P_t$ acting on $\mathbb B(\mathcal X)$ - the set of all bounded and Borel measurable functions. Such a function is said to be $\mathcal C$-lower…
SBF
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Diffusion property

Consider a continuous time real-valued Markov process $X_t$ given by an SDE: $$ dX_t = \mu(X_t)dt+\sigma (X_t)dW_t. $$ Let $\mu,\sigma\in C^1(\mathbb R)$ and $\sigma\ge0$. Moreover let us assume that $\mu,\sigma$ are such that there exists a…
SBF
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Martingale representation theorem for Levy processes

Is there an equivalent of martingale representation theorem for Levy processes in some form? I believe there is no such theorem in generality, but maybe there are some specific versions?
Grzenio
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stochastic process: determining whether process is cadlag

I had asked a similar problem before and Didier Piau was very kind to help me with the answer. I have another question on the same problem setting. Let $T$ be a random exponentially distributed time. $P(T>t)=e^{−t}$. Define $M$ via $M_t=1$ if…
babu
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$E [Z(u) \overline{Z(v)}]=\lim _{N \rightarrow \infty} E [Z_N(u) \overline{Z_N(v)}]=\int\sum_n f_n(t) \overline{f_n(s)} u(t) \overline{v(s)} d u d s$?

A stochastic process $\{Z(t), t \in \mathbb{R}\}$ is called a generalized stationary Gaussian process with mean $0$ and covariance $\{\gamma(s), s \in \mathbb{R}\}$ if for every test function $u(t)$, the random variable $Z(u)$ is normal with mean…
Lely
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diagonalization procedure for stochastic processes

This is a question about a proof I saw in a script about stochastic processes. First I state a theorem which is needed in the proof. After that there are two questions, which are highlighted. Between the questions I provide my thoughts / work so…
user20869
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