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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Birth-death process invariant distribution

Let $X_n$ be a birth-death process, with birth rates $\lambda_n$ and death rates $\mu_n$ (with $\mu_o=0$ and $\lambda_{-1}=0$). How do you show that the invariant distribution $\pi_i$ is: ...
7
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1answer
239 views

Reproducing Kernel Hilbert Space is dense?

Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. Let $E^*$ be a space of all continuous ...
3
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1answer
713 views

Dual of $C[0,1]$, Hilbert space and Riesz representation.

Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. I need help proving the following claim: ...
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808 views

Optional quadratic variation and predictable quadratic variation

What's the difference between optional quadratic variation (which sometimes is denoted by $ [M]$) and predictable quadratic variation (i.e $\ < M > $) of a stochastic process?
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53 views

Necessary condition(s) for transforms of Markov diffusion to stay Markov diffusions

I feel it always necessitates a certain amount of work before reaching the conclusion that the transform of a diffusion by a function is not a Markovian diffusion. I was wondering if there were ...
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151 views

Interpretation of Notation of Laplacian and Brownian Motion

Let $f: \mathbb{R}^d \to \mathbb{R}$ be a twice differentiable function. In particular, $\Delta f$ is well defined. Let $W := (W_t)_{t \geq 0}$ be a $d$-dimensional standard Brownian Motion. ...
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1answer
487 views

Difference between a Brownian Motion and the root of its square

Let $W_{t}$ be a Wiener Process (a Brownian Motion starting at $W_{0} = 0$). What is the difference between $W_{t}$ and $\sqrt{W_{t}^{2}}$? Using the Ito formula (in differential notation), ...
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2answers
142 views

Stopping time that depends on other stopping time

Consider a random variable $\tau$: $$\tau := \inf \left\{ {k = \sigma , \ldots ,T:S_k \ge u} \right\}\bigwedge T,$$ where $\sigma$ is a stopping time, $S_k$ - stochastic process, $T, u$ are ...
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289 views

Potential theory: discrete-time Markov processes

Recently I've found lecture notes on "Analysis on Graphs" where the potential theory methods were used to study discrete-time, time-reversible Markov chains (i.e. the state space is countable). ...
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2k views

Inter-arrival time distribution

It is known that for a Poisson process the inter-arrival time is exponentially distributed. My question, which may be nonsense, is this. Suppose you want to experimentally evaluate the distribution of ...
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1answer
242 views

Covariance of a Bernoulli process

I have a Bernoulli process $\Phi(t)$ with a symmetric distribution $p=1/2$. The random variable can take values $a,b$. My question is what is the covariance of this process ...
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197 views

Sufficient condition in terms of stopping times for a stochastic process to be a local supermartingale

(Question edited in response to Nate's comment) Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin ...
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651 views

A.s. finite hitting time with an infinite mean

Let $X$ be a discrete-time Markov process on some measurable space $(\mathscr X,\mathscr B)$. Let $B\in\mathscr B$ and $$ \tau_B:=\inf\{n\geq 0:X_n\in B\} $$ is the first hitting time of $B$. ...
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329 views

Brownian motion and Bessel process

Can you help me with the following: Prove that a geometric Brownian motion can be represented as a time-changed Bessel process $$ \exp(B_t+vt)=R_{A_t} $$ where $A_t= \int_{0}^t \exp(2(B_s+vs)) ...
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985 views

Drunkard's walk on the $n^{th}$ roots of unity.

Fix an integer $n\geq 2$. Suppose we start at the origin in the complex plane, and on each step we choose an $n^{th}$ root of unity at random, and go $1$ unit distance in that direction. Let ...
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2answers
207 views

Determining distribution of $X_t = \int_0^t W_s^2 \mathrm{d} s$

Premise Let $W_t$ be the standard Wiener process, and let $X_t = \int_0^t W_s^2 \mathrm{d} s$. I am interested in determining the distribution of $X_t$. What I did My line of attack has been to ...
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Dynamical programming principle - discrete case

Fix $\rho>0, C_2>C_1>0$ real numbers. Assume $dX_t=b(X_t,t)dt+\sigma(X_t,t)dW_t$ where $W_t$ is the standard Brownian motion, $X_0=x$, and $b_1\leq s_1\leq b_2\leq s_2\ldots\to\infty$ be a ...
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93 views

preliminary evaluation for forecasting models

Suppose I would like to use a method for data prediction, and that I have some empirical data (i.e., sequence of samples of the form [time, value]). Would it be possible to know in advance, based on ...
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211 views

Correlations between neighboring Voronoi cells

For a sequence $X_1,X_2,X_3,\ldots$ of random variables, what it means to say $X_1$ is correlated with $X_2$ is unambiguous. It may be that the bigger $X_1$ is, the bigger $X_2$ is likely to be. If, ...
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3answers
158 views

Prediction with time-series

I would like to use time-series to predict the behavior of a system with stochastic behavior. Since I am not quite familiar with this topic, could anyone point me to some good tutorial for this ...
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394 views

Probability distribution of sign changes in Brownian motion

Let us consider a 1d Brownian motion. Displacements in space will be positive or negative and this is a random variable $U(t)$ that characterizes a random process and that can take just the values ...
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1answer
355 views

Expectation of an integral of the minimum of a Brownian motion and a constant

I would like to compute the expectation of the following expectation $\mathbb{E}[\int_a^\infty e^{-rt}\min(x_t,c)\,dt]\,$ where a, r, c are constants, $dx_t = \mu x_t dt + \sigma x_t dW_t$ is a ...
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1answer
124 views

Good resources on branching processes

I'm trying to understand branching processes. Do you know any good and written in a simple way resources / web pages / books. Free resources are welcome :).
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125 views

Inequality for certain stochastic process.

Let $b: \mathbb{R} \rightarrow \mathbb{R}$ be a Lipschitz-continuous function and let $X_t$ be a real valued stochastic process satisfying the stochastic differential equation $dX_t= b(X_t) dt+ dB_t$, ...
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113 views

Show the limsup of $B_t/\sqrt{t}$ when $t\to\infty$ is positive

I am trying to prove the following statement about the standard Brownian Motion: $\varlimsup_{t\rightarrow\infty} \frac{B_t}{\sqrt{t}}>0$. I know that it is trivial to prove the above statement by ...
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1answer
198 views

Biased lower bounded random walk

I have a random walk with the following rules: It starts at 2 At each step it goes up by 1 with chance .4, down by one with chance .4 and up by 2 with chance .2 The walk ends if it reaches 0 I ...
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213 views

Conditional probability of a general Markov process given by its running process

I have a question as follow: "Let $X$ be a general Markov process, $M$ is a running maximum process of $X$ and $T$ be an exponential distribution, independent of $X$. I learned that there is the ...
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1answer
3k views

What is the Kolmogorov Extension Theorem good for?

The Kolmogorov Extension Theorem says, essentially, that one can get a process on $\mathbb{R}^T$ for $T$ being an arbitrary, non-empty index set, by specifying all finite dimensional distributions in ...
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194 views

Distribution of shapes of Delaunay triangles

Does anyone know the probability distribution of the shapes of Delaunay triangles in a constant-intensity Poisson process in the plane? Slightly later edit: One can imagine performing the experiment ...
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281 views

Is there a connection between the 3D random walk constant and the partition function?

In thinking about this question, I took a look at Pólya's random walk constants and was struck by the fact that an expression for the constant for a three-dimensional random walk, ...
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406 views

Does this modified random walk (2D) return with probability 1?

Pólya showed that a random walk (with the directions at each step uniformly distributed) on the integer lattice returns with probability 1. What if instead we consider the random walk where we are ...
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1answer
189 views

Finding a stochastic differential equation as limit of a discrete stochastic process

I stumbled upon a problem that seems simple but I cannot tackle it. Let $X_n$ be a discrete process defined by the following algorithm. Choose $X_0\in[0,1]$, set $\kappa>0$ small enough and ...
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1answer
152 views

If $X_t, t \in T$ is separable random process, is event that it's sample path be continuous on $T$ measurable?

If $X_t, t \in T$ is separable random process (with values in metric space), an event that it's sample path be uniformly continuous on $T$ $$ \bigcap_{m \in \mathbb{N}} \bigcup_{n \in \mathbb{N}} ...
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Quadratic Variation of Brownian Motion

Quadratic Variation of a Brownian motion $B$ over the interval $[0,t]$ is defined as the limit in probability of any sequence of partitions $\Pi_n([0,t])=\{0=t^n_0<\cdots<t^n_{k(n)}=t\}$ of the ...
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1answer
361 views

Basic Gaussian Conditioning

Suppose $X_t$ satisfies the SDE $dX_t = a \; dt + dW_t$, where $a$ is a constant and $X_0 = 0$. What is the distribution of $X_3$ given that $X_1 = -1, X_4 = 2$ and $X_5 = 1$? Give the type of ...
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1answer
930 views

confusion about the multi-dimensional Brownian motion

I am confused on the multi-dimensional Brownian motion. $B_t$ is a standard Brownian motion based on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})$ if ...
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1answer
316 views

Deriving SDE(s) and Expectation from Given PDE

We want to solve the PDE $u_t + \left( \frac{x^2 + y^2}{2}\right)u_{xx} + (x-y^2)u_y + ryu = 0 $ where $r$ is some constant and $u(x,y,T) = V(x,y)$ is given. Write an SDE and express $u(x,y,0)$ as the ...
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1answer
306 views

Difficulty in understanding this definition of Poisson process

I am having trouble in understanding this definition of Poisson process. Let $S$ be a random discrete subset of points of $\mathbb{R}^d$ and let $\lambda >0$. A partition $\mathcal{A}$ of ...
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2answers
471 views

Markov Chain: Pensioner Problem

A pensioner receives 2000 dollars at the beginning of each month. The amount of money he needs to spend during a month is independent of the amount he has and is equal to i (i.e. i thousand dollars) ...
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1answer
371 views

Martingales huh?

I am a bit confused in my reading of martingales. I am using Breiman's book and there is an example that doesn't quite make sense to me. Let us define a sequence of random variables in this way: Let ...
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170 views

Markov Chains - Using Gibbs & Metropolis algorithm.

Suppose $f_x,_y$ is bivariate normal distribution. I was given the parameters $(μ_1, μ_2, σ_1^2, σ_2^2)$ and $ρ=0.95$ the correlation coefficient. I want to generate $(x_1,y_1), ...
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307 views

Expectation of Stochastic Process Given First Hitting Time Information

Let $V_t$ satisfy the SDE $dV_t = -\gamma V_t dt + \alpha dW_t$. Let $\tau$ be the first hitting time for 0, i.e., $\tau $ = min$(t | V_t = 0)$. Let $s =$ min$(\tau, 5)$. Let $\mathcal{F}_s$ be the ...
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522 views

Quadratic variation and stopping time.

Let $X_t$ be a strict local martingale and let $S$ and $T$ be stopping times with $S \leq T$. Prove that $[X]_S=[X]_T$ implies that $X_t$ is almost surely constant on $[S, T]$, where $[X]_S$ and ...
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1answer
207 views

Independence of Thinning of a Poisson process

Consider a process $X_t=\sum_{k=1}^{\infty} \mathbf{1}_{\{T_k\le t\}}$, and a new process $\hat{X_t}=\sum_{k=1}^{\infty} z_k \mathbf{1}_{\{T_k\le t\}}$, where $P\{z_k=0\}=p, P\{z_k=1\}=1-p$, and ...
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1answer
103 views

Random Walks and Jumps

I am trying to understand why Random Walks' and Random Jumps', on a graph, transition matrix are also stochastic matrix. A stochastic matrix is a matrix the values of each row add up to 1 and no ...
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1answer
249 views

Kolmogorov Backward Equation Boundary Value Problem

I need to solve the backward equation $u_t - \gamma x u_x + \frac{1}{2} b^2 u_{xx} $ subject to the final condition $ u(x,T) = (x-a)^2 $. Here a and b and $\gamma$ are constants. I am given a strong ...
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How is the law of a stochastic process defined?

Let $(Ω, F, P)$ be a probability space, $T$ some index set, and $(S, Σ)$ a measurable space. $X : T × Ω → S$ is a stochastic process. Let $S^T$ be the collection of all functions from $T$ into ...
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140 views

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 ...
2
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1answer
286 views

Markov Decision Process - Utility Function

Reward R(S) in a Markov Decision Process is a mapping from a State S -> Bounded number. I want to know how a Utility Function is defined for an MDP. I think it has to be a mapping from a sequence of ...
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1answer
591 views

Exercise 7.7.1 in Grimmett & Stirzaker's 'Probability and Random Processes'

I'm having trouble solving exercise 7.7.1 in Grimmett & Stirzaker's Probability and Random Processes, which reads: Let $X_1,X_2,\ldots$ be random variables such that the partial sums ...