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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How to write the Hamilton Jacobi Bellman equation

We consider the following optimal control problem \begin{equation} V(t,x)=\max_{u}\mathbb{E} ( \log [\int_{0}^{T}u^{2}(t)dt + U(X(T))]) \end{equation} subject to the state process \begin{equation} ...
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34 views

Question regarding Notes on Strong Markov Property

I wrote the following notes from a lecture a couple of weeks ago and I don't understand a particular line. Suppose $B_t$ is a Brownian Motion. Now look at $B^x_t = x + B_t$ which is a BM starting ...
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25 views

IID implies Ergodicity

The environment space is given by $\Omega:=P^{\mathbb{Z}^{d}}$, where P contains the 2d-vectors serving as admissible transition probabilities. An Element $\omega \in \Omega$ is defined as ...
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30 views

Reconstruction of state covariance from output covariance

Let us be given an LTI system $$ \frac{d}{dt} x (t) = A x(t), \;\; x(0)=x_0 \\ y(t) = Cx(t) $$ where $x_0$ is a random vector (e.g. uncertainty). Then it is known that the expectation $\mathbb ...
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43 views

conditional expectation of the Brownian motion [duplicate]

$(B_t)$ is a Brownian motion and i assume that $s<t<u$ we have $$E[B_t |\sigma(B_s,B_u)] = G(B_s,B_u)$$ Does anyone knows the explicit expression of $G$ ? (the calculus is easy but ...
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22 views

the relationship between fractional difference and ACF of a time sequence

When reading the GARCH modeling part of book Analysis of Financial Time Series, I read the following statement. In specific, I do not understand how does the author ...
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84 views

Compute a probability in Random Walk by Martingales

Let $X_n$ be the state at time $n$ of a Markov chain with these transition probabilities : $$p_{i,i+1}=p_i\qquad,\qquad p_{i,i-1}=q_i=1-p_i$$ $(a)$ Show that $Z_n=g(X_n)\,;\,n\geq0$, is a ...
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17 views

Existence of increasing modification

I have the following, seemingly simple question: Consider a stochastic process $(X_t)$ satisfying $X_t\le X_s$ a.s. for all $t\le s.$ My question is: Does there exist a modification $\tilde{X}$ of ...
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34 views

Rescaling function for probability of $k$ adjacent ones in a binary string

Call $\xi$ a random variable taking values in $\{0, 1\}^{\{0, 1, 2, \ldots, n\}}$, where each character of the string has vaalue $1$ with probability $p$ and $0$ with probability $1-p$ independently. ...
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46 views

quadratic variation of $X_{t} = tB_{t}$?

Let $X_t = tB_t$ be a process where $B=(B_t)_{t>0}$ is the standard Brownian motion . evaluate $\langle X\rangle_t$ the quadratic variation of our process . I tried to calculate it using : ...
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24 views

Expression for $B_1$

I think that it is indeed the case that $$ B_1 = \int_0^1 \frac{B_1 - B_t}{1-t} dt, $$ where $B$ is a standard one-dimensional Brownian motion. Am I right? If so, how you we prove it? Thanks a lot ...
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18 views

Markov chains mixing time

Informally, the mixing time of a Markov chain is the time it takes to reach “nearly uniform” distribution from any arbitrary starting distribution. What does it mean by nearly uniform? I hope some one ...
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29 views

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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1answer
62 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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3answers
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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1answer
48 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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54 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
96 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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47 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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32 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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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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161 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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1answer
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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17 views

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
90 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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1answer
107 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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28 views

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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35 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 = ...