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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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Stochastic Processes Question on Proving a Random Walk is a Martingale

Given a random walk for which 0 is an absorbing state and such that from any positive state, the process is equally likely to go up or down one unit, $p_i = q_i = 1/2$. Also note, $R_0 = 1$ (for the absorbing state to be 0). a) Show that the…
user380668
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Integral with respect to a martingale

In my survival analysis course we use integrals that integrate with respect to some finite variation process. One of which is the counting process, which is straight forward to understand since you just sum over jump times. But I have no clue what…
ChuckP
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Expected value of a simple binary stochastic model

Consider a model with two binary stochastic nodes $x_0,x_1$ with probability of being on of $P(x_0), P(x_1)$ respectively. When $x_i$ is in its on state it sends probability $u_0$ to some other variable, let's call it $y$. We define the value of $y$…
redcalx
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Why $\lambda\equiv\nu(\mathbb{R})$ for compound poisson process?

I've seen this notation $\lambda\equiv\nu(\mathbb{R})$ in the book of Tankov and Cont for compound poisson process. I thought before that $\lambda$ (jump intensity) can be choosen independently of jump measure $\nu$. What does this notation imply?
learningmath
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Hazard rate of the Pareto distribution

I need to calculate the hazard rate of the Pareto distribution. I know that the residual life distribution looks at the remaining waiting time given that you have already waited for a certain amount of time x, and the hazard rate can be thought of…
endlessend2525
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Mean Square value of stochastic process given the spectral density

A stationary stochastic process have a spectral density of $$ S_{XX}(\omega) = 1 - \frac{|\omega|}{8 \pi}. $$ What is the mean square value of the process?
Adriano Gonçalves
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extinction process problem in branching process

I am trying understand solution to the branching process extinction problem as under:- "Suppose in a branching process the offspring distribution is as follows $p_k$ = $pq^k$, $q=1-p$, $0
SAK
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Probability Generating Function Formula derivation-branching process

I am trying to understand the derivation of following PGF Formula in branching process $$H_n(s)=H_{n-1}(H(s))$$ I presume that the initial steps for this derivation are standard...so am avoiding entering them. The derivation is clear up to the…
SAK
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Stopped optional process

Consider the following claim in a book. Let $X$ be an optional process and $T$ a stopping time. Then $X^T$ is also an optional process. The proof is based on the fact, that $X_T 1_{T<\infty}$ is $\mathcal{F}_T$-measurable. By a monotone class…
peer
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Finite distribution of a homogeneous Poisson point process

I need help to find the finite distribution (fidi) of a homogeneous Poisson point process for overlapping set. For example: Let $\Phi$ be a uniform Poisson point process of intensity $\lambda$ on $\mathbb{R}$. I want to determine…
ksank43
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Intuition of Doob-Meyer decomposition ( case of totally inaccessible jumps)

I try to understand Theorem 10 on page 107 of Protter's Stochastic integration and differential equations. The proof is really long, and for now, I just want to get an intuition. Here is the theorem: Let $Z$ be a cadlag supermartingale with $Z_0=0$…
Jay.H
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What is the proof of this equation? (Stochastic process)

$$\mathbb{E}[g(X)h(Y)]=\mathbb{E}[h(Y)\,\mathbb{E}[g(X)|Y]]$$ I am reading the book "An Introduction to Stochastic Modeling". This equation appears a lot but I can not see why. Can anyone please provide some proofs and examples? Thanks a lot.
Yilin Wang
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Continuous mapping theorem counterexample

let g be a continuous function. Then from the continuous mapping theorem We have the following result. If xt converges to x* wp1 as t goes to infinity then g(xt) converges to g(x*) wp1 Now suppose g is only piecewise continuous. Do we have an…
RMG
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Definition question : Two Stopping Times

I have encountered the following question for homework, with our lecturer only requiring us to have a basic idea about stopping times. The question is as follows: Let $X(t)$ be an Ito process and suppose that $a\neq b$ and: $$\tau_{a} = inf\{ t>0:…
Lindah
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Finding the probability function of a random sum of Bernoulli variables (stochastic processes)

Edit: Sorry for the Latex, I'm new to it and trying to fix it right now I am trying to find the probability function of a random sum of Bernoulli variables in this scenario: Starting from 9AM, clients arrive (independently), before a store opens,…
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