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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Perron Frobenius Theorem and Markov chains and more

I came across few ways of calculating convergence rates of Markov chains but I am a bit confused as to how these differ from each other and what may be the best way to calculate. The second ...
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61 views

Monotonicity and Convexity of Stochastic Matrices

The definition of stochastic monotonicity and convexity is given by "Stochastic Orders and Their Applications" by Moshe Shaked and George Shanthikumar (1994) as: Let $P = \{p_{i,j} \}$ be a ...
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2answers
49 views

Convergence time of a Markov chain

We know that a regular Markov chains converges to a unique matrix. The convergence time maybe finite or infinite. My interest is in the case where the convergence time is finite. How can we accurately ...
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41 views

Distribution of Sum of Brownian Motion and Integrated BM

Let $W(t)$ be a standard Brownian motion (BM), in particular $W(t) \sim \mathcal{N}(0,t)$. Then it is easily shown that $\int_0^T W(t) dt \sim \mathcal{N}(0, T^3/3)$. Question: What is the ...
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85 views

How fast does this Markov chain converge?

Observe the above a Markov chain and limiting matrix of it. Finding the limiting matrix if it exists is easy but I am curious as to how fast this given matrix converges to its limiting matrix. Is ...
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16 views

Homogenous Poisson Process, calculating expectation

I'm trying to solve exercises from chapter 2 of Rick Durrett's Essentials of Stochastic Processes. Ex 2.22: Suppose $N(t)$ is a Poisson process with rate 3. Let $T_n$ denote the time of the $n$th ...
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36 views

How big a Brownian bridge can get? Confidence band.

If we know the endpoints of the Brownian path, is there any theorem telling us if it can be contained within a ball a.s. (with probability one)? For example contained in two big enough balls (call it ...
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47 views

Calculation of distribution of a gaussian process

Currently finishing the last year of PhD in statistics, we wonder if you could help us with the following. Let $T = [0,1]$ and $X = \left( X_{t}, t \in T \right)$ be a gaussian process with mean ...
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37 views

Brownian Bridge conditional probability

The problem is to show that the density $P[W_{t_1} \in dx_1,...,W_{t_n}\in dx_n | W_T = b]$ is the density of a Brownian bridge from $a$ to $b$. $W$ is Brownian motion. The density of a Brownian ...
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50 views

Variance of Integrated Geometric Brownian Motion

I'm just asking for verification that my derivation is correct, as I can't seem to find this result elsewhere. I'd like to calculate $Var(\int_0^T X(t) dt)$ where $X(t) = X_0e^{(\mu - ...
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22 views

questions on a property of ARCH model

When reading the book of Analysis of Financial Time Series, I have a question on the ARCH model, defined as follows Regarding this model, the author also states ...
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29 views

Expectation Involving Two Values of Geometric Brownian Motion

Not sure this is the best place to ask for verification, but I can't seem to find a derivation anywhere else. I want to calculate $\mathbb{E}[e^{\sigma(W_t + W_s)}]$, where $W_t$ and $W_s$ are two ...
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14 views

Long memory of stochastic differential equation

It is well known that the solution to an ordinary stochastic differential equation has the Markov property so that if one tries to model some kind of long memory process one has to instead consider ...
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3answers
90 views

Similarity between two probability distribution

I am not sure how to put the question. I am not even sure if this question makes sense at all. I know that the similarity of two discrete (or continuous) distributions can be quantified by ...
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1answer
27 views

Finite expectation - Difference between these two statements?

Main Question: Let $\{X_n, n=0,1,2...\}$ be a stochastic process. How are following two statements different? \begin{align} E[|X_n|]<\infty \text{ for all } n \end{align} and \begin{align} ...
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36 views

BMO martingale and exponential martingale

Consider the BSDE, $$ Y_{T}-Y_{t}=\sum_{i=1}^{n} \int_{t}^{T} Z_{s}^{i}dB_{s}^{i} - \frac{1}{2}\int_{t}^{T} \left| Z_{s}\right|^{2}ds $$ where $B$ is a standard Brownian motion on a complete ...
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46 views

Stationary distribution for random walks on directed graph

There is an equation (Eq. (2)) in reference by Lovasz and Winkler about the stationary distribution of a random walk on directed graphs that I would like to find references for where the equation is ...
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1answer
39 views

continuous RV from discrete RV

So I am reading some notes in stochastic processes and I don't really understand the solution of this problem: Problem: Let $(\Omega,F,\mathbb{P})$ be a probability space where $\Omega$ is the set ...
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27 views

Power spectral density of convolution of stochastic processes

I was wondering what it is the result of convolving two WSS processes in terms of power spectral densities. I know that, the output $Y(t)$ of a generic linear time invariant system with impulse ...
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Finding a pre-visible process

Question: Let $W_t$ be a standard brownian motion under P with filtration $\mathscr F_t$. Let: $$ M_t=\mathbb E[W_T^2|\mathscr F_t] $$ Show that $M$ is a P martingale. This is simple enough using ...
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16 views

Distribution of integral with two ito integrals multiplied in the integrand?

I'm trying to calculate the distribution of $\int_{t_{n}}^{t_{n+1}}\int_{t_{n}}^{\tau}\sigma_{1}(t)dW_{t}^{(1)}\int_{t_{n}}^{\tau}\sigma_{2}(s)dW_{s}^{(2)}\sigma_{3}(\tau)d\tau$ where $W_{t}^{(1)}$ ...
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1answer
160 views

Lookback option with floating strike: boundary condition

I am trying to make sense of one of the boundary conditions of a look-back option with floating strike. Some notation first: let $v(t,x,y)$ denote the price at time $t$ of the option under the ...
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39 views

Fubini Question in context of Independence

I am trying to show that if $X_t$ is some process and there is a function $p$ such that $$P[(X_{t_1},...,X_{t_n}) \in A_1 \times...\times A_n] = \int_{A_1 \times...\times A_n} ...
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characterising attractors for master equations

I have a master equation for $(x,y,z)$ with the constraint $x+y+z=N$. $x$ can be regarded as the number of animal of a certain species in the whole system. In other words, I have a differential ...
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35 views

Random process $X(t) = 10 \cos(Wt + A)$.

I am doing some exercises based on random process, but I can't find a way out on this: Let $X(t) = 10 \cos(Wt + A)$, where W is a Gaussian aleatory variable with parameters $N(10,2)$ and $A$ is ...
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1answer
68 views

Probability of getting SUCCESS AND FAILURE at number n-1 and n trial

In a sequence of Bernoulli trials let $u_n$ be the probability that the combination SF occurs for the first time at the trials number n-1 and n. To find the generating function I wrote the following ...
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1answer
89 views

Problem on Solving Stochastic Differential Equation

Let $(Xt)$ be a solution to the equation $dX_t = aX_t dt + \sqrt{(1+X_t^2)} dW_t$ where $W_t$ is a Brownian motion process at time t Let $Y = F(X_t)$ for a certain function $F$. Find $F$ for which ...
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32 views

American Put question

If the interest rate is zero. Then show that the optimal exercise for an american put option is always the terminal time. That is, it is equivalent to a european put option.
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27 views

limite presque sure

I just want to know why for a continue process X such $X_{t} \rightarrow Z$ p .s when $t \rightarrow \infty$ then lim inf $X_{s}^{2}$=Z when p .s $t \rightarrow \infty$ inf is on $\frac{t}{2}\leq s ...
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49 views

Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail. Just to avoid misunderstanding, ...
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97 views

Probability of a trajectory in Markov processes

I need help with a simple formula! (My question is taken from here, pag 26 eq 1.112. ) Consider a Markov Process with associated Master Equation: \begin{equation*} ...
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Poisson process different type of events

Suppose that it arrives people to a store according to a poisson process with rate $\lambda = 6$/hour , females arrive with probability $0.6$ and male with $0.4$. What is the probability that there ...
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38 views

Probability Density of Convolution of Two Random Processes or Variables

Suppose that we have two stationary random processes $x(t)$ and $y(t)$ with probability density functions $f_{x}(x)$ and $f_{y}(y)$ respectively. Now suppose we form: $z(t) = x(t) \ast y(t)$ What is ...
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36 views

SDE with no weak solution

I'm facing the followingd d-dimensional SDE: $$dY_t=\sigma(h_t)\,dB_t$$ In addition it holds, that: $h_t\in H$ and $H$ is compact (for example the simplex of $R^n$) the proces $h_t$ is progressivley ...
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35 views

Brownian Motion with drift (stupid question)

How do you prove that $$ \lim_{t\to +\infty} (B_t+ct)=+\infty $$ almost surely? $(B_t)_t$ is the standard Brownian Motion starting from $0$.
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Differentiating Exponential matrix Expression

To give the scalar version first: For the well known Ornstein-Uhlenbeck process: $dr(t)=\alpha(b-r(t))dt+\sigma dW(t)$ It is well known that the variance is: $\sigma_r^2=\sigma^2 \int_u^t\exp^{-2 ...
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34 views

Kolmogorov theorem

I just want to know why for a gaussian process X this inequality lead to apply Kolmogorov Centov Theorem. Thanks. $\mathbb{E}(X_{t}-X_{s})^{2}\leq c \vert t-s\vert^{2}$
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Decomposition of noisy measurements

What can be a good intuition behind decomposing a sequence $\{Y_n\}$ of noisy measurements (i.e. random variables) into two parts: one unpredictable and the other depending on the past. $$Y_n = ...
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1answer
53 views

The Vacisek Model and the short rate process

I am trying to do some calculations related to the Vacisek model, but I think I am mixing up concepts and I'm not getting to any solution. Let me explain what the problem is. The Vacisek model ...
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2answers
30 views

Markov property for a Stochastic Process

My question: Every Stochastic Process $X(t), t\geq 0$ with space states $\mathcal{S}$ and independent increments has the Markov property, i.e, for each $\in \mathcal{S}$ and $0\leq t_0\leq< ...
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1answer
33 views

Property of Brownian Motion's paths

We are considering a Brownian Motion $(B_t)_t$ with values in $\mathbb{R} $ starting from $x$ defined on the stochastic basis: $$(\Omega,\mathcal{E},(\mathcal{F}_t)_t,\mathbb{P}^x)$$ Then, let's ...
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1answer
50 views

Derive the Black– Scholes formula for the European call option.

Consider the standard Black–Scholes model. Derive the Black– Scholes formula for the European call option. thanks for help.
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20 views

Integrated gaussian process

I just want to know what kind of phenomenon a integrated gaussian process ($Y_{t}=\int_{0}^{t}X_{s}ds$ where X is a gaussian process) can modelize. Thanks.
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30 views

marginal distribution of Ornstein Uhlenbeck process

I am learning the OU process. For now, what I can understand is that the OU process is the strong solution of a SDE $d\sigma²(t)=-\lambda \sigma²(t)dt+dz(\lambda t)$ where z is the compound possion ...
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Justification of Poisson postulates

This may be a dumb question. The Poisson postulates are: $P(n=1,h) = \lambda h + o(h)$ $\sum\limits_{i=2}^{\infty}P(n=i,h) = o(h)$ Events in nonoverlapping intervals are independent What ensures ...
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Random node observation

The problem is as follows: In a two dimensional plane, nodes are randomly distributed with intensity $\rho$. Each node in the network swings between two states: available, non-avaialable for ...
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29 views

A question on recurrent events

In a sequence of Bernoulli trials let E occur when the accumulated number of successes equal to $c$ times the number of failures where $c$ is a positive integer. I need to show that E is persistent if ...
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1answer
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I want to show $\operatorname{Cov}(X(t),X(s))=\min(s,t)- \frac{st}{T}.$

i have this Equation with Condition $X\left(0\right)=a $ and $ 0\le t \lt T$ $$dX\left(t\right)=\frac{b-X\left(t\right)}{t-T}dt+dB\left(t\right)$$ I solved and got $$X\left(t\right)= ...
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Changes in the transition matrix of a Markov chain

In most or all Markov chain theories that I know of assumes that the transition matrix does not change over time. But what if certain changes are expected to occur at certain times in the transition ...
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19 views

Stochastic Approximation/optimization algorithms

Is there any busy discussion forum where research level questions of stochastic optimization/approximation algorithms can be discussed.