Mathematics Stack Exchange
Mathematics Stack Exchange

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 Process Markov Chains

The problem is as follows. Suppose that a baseball trading card for sale receives successive bids $$\Delta_1,\,\Delta_2,\cdots,$$ which are independent random variables with geometric distribution $$\mathbb P(\Delta=k)=0.01\cdot(0.99)^k$$ for…
Kyle
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homogeneous processes and homogeneous-increment processes

If I understand correctly, a homogeneous process must be homogeneous-increment, because for example, $L(X_{t_2} - X_{t_1}) = L(X_{t_2 + \tau} - X_{t_1 + \tau})$ can be proven by $$L(X_{t_2} - X_{t_1}, X_{t_1}) = L(X_{t_2}, X_{t_1}) = L(X_{t_2 +…
Tim
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Stochastic process for uniform probability on [0, 1]

Let $\Omega = [0,1] $, $ \mathcal{F}$ - $ \sigma$-algebra and $ P$ - uniform distribution on $(\Omega, \mathcal{F})$. For $t \in [0, \infty)$ we define $X_t(\omega) = \omega t$. What is the two-dimensional probability distribution? Is it that we…
M_T
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Probability that population goes extinct v.s. probability that population eventually goes extinct

Is there a difference between answering the questions: What is the probability that a population goes extinct? What is the probability that a population eventually goes extinct? For example, let $p_i$ be the probability that a parent has $i$…
sucksatmath
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What does it mean for a process to be adapted in terms of the algebras?

What does it mean that $X_{t}$ is "observable" at time $t$ in the context of an adapted process? My guess is that it is possible to determine which set the $\omega$ which is "under development" belongs to(and not belong to), which in turn imply a…
user485531
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Numeric solution for a stochastic DAE

I have the equations defined as follows: $$ \begin{align} \frac{dX}{dt}&=f(X,U), \\ U(t)&=g(X)+\xi_t, \end{align} $$ where $\xi_t$ is a Gaussian white noise, $f(X,U)$ and $g(X)$ are such functions as one could rewrite the first equation by…
Artem Zefirov
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Mean Square Estimate

Let $\left\{{y (n) , n = 1, 2, · · ·, N }\right\}$ be a real random sequence that represents $N$ observations of an unknown real random variable $x$. In the expression $$\sigma^2=E[(x-\hat{x})^2]$$ How will be able to start demonstrate that…
juaninf
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Discrete simulation of a continuous-time discrete state stochastic process - Convergence of simulation's cdf to the process's cdf?

Let $X_t$ be a continuous-time stochastic process on the state space {'off', 'on'}. Let $U$ be the cumulative duration spent 'on' during a time interval $[0,\tau]$ (that is, $U$ is the uptime during that interval). Suppose that we simulate $U$ by…
PtH
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Limiting probability distribution of total lifetime in renewal process

I am now very busy to preparing my stochastic modelling exam and I feel sad when dealing with some problems. 1.For a renewal process $ {N(t):t∈[0,∞)} $, how to get the limiting distribution of the total lifetime $\beta_t$? $\beta_t$ is defined as…
Garvin Cheng Hok Laam
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Brownian motion, Arcsin low of zeros

Let $V$ be the time of the last zero of $W(t)$ in $[0,t]$ How can I prove that $P(V>0)=1$ ? $V =\text{sup}\{s
Angela.M
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Binomial distribution with parameters p and N, p is distributed to a beta distribution with parameters r and s

For each given $p$, let $X$ have a binomial distribution with parameters $p$ and $N$. Suppose $p$ is distributed according to a beta distribution with parameters r and s. Find the resulting distribution of $X$. When is this distribution uniform on…
kira
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calibrate volatility parameter

Consider the simplest interest rate model $$r(t) = r(0) + h(t) +\sigma B(t),$$ where $r$ is an overnight interest rate, $r(0)$ is its initial level, $h(t)$ is a time-dependent drift and $\sigma$ is a constant volatility parameter, $B(t)$ is brownian…
hq286
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Probability sets

I am new in random processes and in one of my homework questions the Porfessor asks the following: Prove that if $\lim_{n\to\infty} P\{\sup_{k>n} |X_k-m|>= \varepsilon\} = 0$ the sequence of random variables $(X_n)_{n\in\mathbb{N}}$ converges almost…
user254769
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Application of Poisson Process

Am currently working on a Stochastic Poisson process on my project. I have thought and settled on the below scenario which I think is appropriate. However, solving it, I'm not getting what I expect. I want to cross a road at a spot where cars pass…
Ben
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Exponential independent Events

Am assuming a machine needs 2 types of parts to work which are type A and type B. The machine has one part of each type to begin, and there are also 2 spare A parts and 1 spare B part. I assume that, when a part fails, it is replaced by a spare part…
Ben
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