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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38 views

Is the following process bounded (iterative normal sampling)

We define the following stochastic process: $X_0=1$ $\forall i\geq i:X_i\sim\mathcal N(0,X_{i-1}^2)$ That is, we first sample $X_1$ from the normal distribution with variance $1$, then in the $i$-...
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1answer
86 views

Expected value of time integral of a gaussian process

While working on a problem I've stumbled upon some expected values of time integrals of Gaussian stochastic processes. Some of them were addressed this question, but I have found also this one $$\...
2
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1answer
103 views

Integral of a Brownian bridge with respect to time

Let $(W_s)_{s\geq 0}$ be a Brownian motion and $t$ a fixed point in time. What is the distribution of $$\Big.\int_0^tW_sds\Big|W_t$$ i.e. the integral of a Brownian bridge with respect to time? Is it ...
3
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1answer
377 views

A question about independence of sigma algebras (generated by random variables)

Let $X_1, X_2, \ldots$ i.i.d random variables. Is it possible that $$\{X_{n+1} \in B\} \in \sigma({X_1, \ldots, X_n})$$ for some $B$? Why yes/not? I want to show that $\sigma(X_{n+1})$ and $\sigma(...
2
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2answers
176 views

Equilibrium distribution of Ehrenfest's urn

(I'll post my own answer to this, but others may be of interest, so post your own if you have one.) (PS: In reply to comments posted below: Stackexchange encourages posting an answer to one's own ...
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1answer
66 views

Showing that the Brownian Bridge is Gaussian

Take $X_t = (1-t)B_{t/(1-t)}$ for $t\in[0, 1)$ where $B_t$ is a $1$-dimensional Brownian motion. I want to show that $X_t$ is Gaussian. I have actually never been able to find a precise definition ...
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1answer
42 views

Data distribution for sine time series

Suppose we have a time series $x_t=\sin(0.02\pi t)$. Although this time series is totally deterministic, we can treat it as one realization of a proto/quasi/pseudo-stochastic process and estimate the ...
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1answer
136 views

Position of Brownian motion at exit time from the upper half plane

I am currently reading some books on SLE and struggling on some problems regarding Brownian motion. For a Brownian motion in $\mathbb{R}^2$ starting from $(x,y)$, I don't know how to find the ...
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1answer
76 views

Verifying that a certain process is not a Brownian motion

Let $B$ be a standard Brownian motion in $1$ dimension. Define \begin{equation} \tau = \inf \bigg\{ t \geq 0 : B_t = \max_{0 \leq s \leq 1} B_s \bigg\}. \end{equation} We want to show that $(B_{t+ \...
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38 views

What is the state space of this markov chain?

Consider a system where two persons sit at a table and share three books. At any point in time both are reading a book, and one book is left on the table. When a person finishes reading his/her ...
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2answers
115 views

probability of a brownian motion being equal to the running maximum

Let $B$ be a standard Brownian motion on $\mathbb{R}$. I would like to show that $$ \mathbb{P} \bigg\{ B_1 = \max_{t \in [0,1]} B_t \bigg\} =0 .$$ I argue that since $\max_{t \in [0,1]} B_t $ has the ...
2
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1answer
89 views

Solution to truncated renewal function

Let's begin with some theory on the renewal process. In a renewal process $N(t)$, let $t$ denote the interarrival time, and $f(t)$ and $F(t)$ denote the PDF and CDF respectively. Let $M(t)=E[N(t)]$, ...
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2answers
46 views

Prove that lim sup of a function belongs to a certain sigma algebra

I am so baffled with this problem: Let $B$ be a standard Brownian motion, $\{ \mathcal{F}_t \}$ be the filtration generated by the Brownian motion. I would like to show that for any $k>0$, \begin{...
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1answer
214 views

Solve the SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ [closed]

Solve the following stochastic differential equations $ dX_t = \frac{1}{2 X_t} dt + dB_t$ or equivalently with a transformation $Y_t = X_t^2$ $ dY_t = dt + 2 \sqrt{Y_t} dB_t$ with $Y_0 = y_0 > ...
2
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1answer
438 views

Joint distribution of arrival times in Poisson process

I need to compute the following joint distribution in a Poisson process: $f_{S_A S_{A+B}}(t_1, t_2), t_2\ge t_1$ $S_A$ and $S_{A+B}$ are the arrival epochs of the $A^{th}$ and ${A+B}^{th}$ arrivals ...
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220 views

A simple characterization of the Brownian Motion

A well-known characterization of the Brownian Motion says that it is the only continuous process $X_t$ (defined on $[0,\infty)$) such that $P(X_0=0)=1$, $E[X_t^2]=t$, $E[X_t]=0$ for any $t\ge 0$ the ...
3
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1answer
113 views

Expected value of integrals of a gaussian process

I have limited knowledge of the theory of stochastic processes. While working on a problem I've stumbled upon some expected values of time integrals of Gaussian stochastic processes. Before starting ...
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0answers
107 views

continuous random walks, wiener process, ito process: “snowballing” for high enough volatility?

I'm finishing a project for my ODE class and ran into some strange behavior involving a SDE (not exactly sure how to say this, but...) generated by an Ito process, using the Wiener process. I guess ...
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1answer
390 views

Variance of integrated squared wiener process

So I'm trying to figure out the mean and variance of $X = \int_{0}^{1} W^2(t) dt $ where $W$ is the Wiener process. The mean I've worked out easily to be $\frac{\sigma^2}{2}$ but I'm having ...
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94 views

Are these transient or recurrent states in a Markov chain?

I have the following transition matrix for a Markov chain with states $A, B, C, D, E$ $ \left| \begin{array}{ccc} 0 & 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & \...
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1answer
100 views

The smallest filtration for which a sequence of random variables is adapted

Let $X_1, ..., X_n$ be a sequence of random variables. Show that $\hspace{60pt}$ $\mathcal{F}_n$ = $\sigma(X_1, ..., X_n)$ is the smallest filtration such that the sequence $X_1, ..., X_n$ is ...
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84 views

How to determine the probability density function, ${f_{\dot X}}\left( {\dot x} \right)$, for the derivative process of a stochastic process?

I would like to calculate the up-crossing rate ($\nu _a^ + $) for a stationary stochastic process, $X(t)$, given by the probability distribution function of its 'intensity', ${f_X}\left( x \right)$, e....
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1answer
110 views

Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?

Suppose we have an sde of the form: \begin{eqnarray} dX_t=b(X_t)dX_t + \sigma (X_t)dB_t \end{eqnarray} where $b$ and $\sigma$ are Lipschitz. Then we have existence and uniqueness of the solution $X_t$...
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2answers
41 views

I can't understand this difference equation step

I am working on birth-death processes and I can't understand a step that is taken in a proof. The mean of a process is defined as $$\mu(t) = \sum_{n=1}^{\infty}np_n(t)$$ At certain stage in the ...
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2answers
59 views

Probability of ultimate extinction? Need to show that an infinite series is less than $1$

I have the following probability generating function for a branching process - $$G_n(s) = \frac{n}{n+1} + \sum_{r=1}^{\infty}\frac{n^{r-1}}{(n+1)^{r+1}}s^r$$ It says in a book that extinction is ...
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1answer
178 views

Do we need Feller condition if volatility jumps?

Consider the SDE: \begin{equation} dv_t = k(\theta - v_t) dt + \xi \sqrt{v_t} dW^{v}_{t} \end{equation} It describes a process $v_t$ which is a strictly positive if the drift is stronger enough, i.e. ...
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1answer
55 views

Brownian brigde, brownian motion and independence.

Let $\{W(t)\}_{0 \le t \le 1}$ a Brownian motion. Then $\{B(t)\}_{0 \le t \le 1}$ with $B(t)=W(t)-tW(1)$ is a Brownian brigde. My goal is to prove that $B(t)$ and $W(1)$ are independent. Since their ...
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43 views

Proving an equality for a Markov Chain

Let $X_n$ be an irreducible Markov chain taking values from the natural numbers (including $0$). Let $g,f$ be functions with $\mathbb{N}$ as domain (including $0$) such that $f = g + Pf$, where $P$ is ...
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43 views

two definition of the Poisson process

I read the definition of Poisson process in Shereve's "Stochastic Analysis" which is constructed explicitly: (A) step 1:construct iid exponential r.v.$\tau_i$ with parameter $\lambda$ step 2:define ...
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1answer
91 views

Bounding an expected hitting time

Consider a stochastic differential equation: $$dX_t = dW_t + \sin(X_t) dt, \, X_0 = x$$ where $W_t$ is a Wiener process. Define $$\tau_1 = \inf \{ t : X_t \in 2 \pi \mathbb{Z} \} \\ \tau_2 = \inf \...
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1answer
58 views

Can a chain with repeated nodes still be considered a Markov chain?

The well-known Markov Property is that $$P(X_n = i | X_{n-1} = k_1, \dots, X_{n-j} = k_n ) = P(X_n = i | X_{n-1} = k_1) $$ Suppose we lay out some stochastic model in the following transition ...
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98 views

Master equation of chemical reaction

I have about the construction of master equation for chemical reaction i.e. I have to construct differential equations for the probability mass function for the number of particles A, B and C. When ...
4
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1answer
153 views

Uniformly integrable martingale in a finite time horizon

Let $\{ M (t) \mid t \in [0,T] \}$ be a martingale and $\{ \tau_n \mid n = 1, 2, \ldots\}$ be an increasing sequence of stopping times such that $\tau_n \rightarrow \infty$ as $n \rightarrow \infty$. ...
2
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1answer
42 views

Density of $\int_{0}^{t}W'(B_{s})ds$ where $W'$ is smooth and compactly supported.

Only hints please Density of $\int_{0}^{t}W'(B_{s})ds$, where $B_{s}$ is 1-d Brownian motion. The density of $Y_{s}:=W'(B_{s})$ is $g_{Y}(y)=p_{B_{s}}((W')^{-1}(y))|\frac{d(W')^{-1}(y)}{dy}|$. How ...
2
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1answer
84 views

Conditional Gambler's Ruin

I've learned about the most canonical gambler's ruin problems, but what if winning or losing on a previous turn changes the probability of winning or losing on the following turn? Say each turn I ...
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1answer
31 views

Robustness of Markov Chains

A Markov Chain on a measurable space $X$ is uniquely determined by a stochastic kernel $P$ on $X$. Let $\mathsf P_x$ denote the probability on paths generated by $P$ and the initial condition $x\in X$....
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69 views

Question about calculate expected value

Assume $X(t)$ is a Brownian motion. Find $E[X(u)X(u+v)X(u+v+w)]$, where $0<u<u+v<u+v+w$ I have an idea to solve this problem, as follows: $E[X(u)X(u+v)X(u+v+w)]=E\{X(u)X(u+v)[X(u+v+w)-X(u+...
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2answers
133 views

Continuou Time Markov Chains - Poisson Distribution

Suppose $X_t$ and $Y_t$ are independent Poisson processes with parameters $\lambda_1$ and $\lambda_2$, respectively, measuring the number of calls arriving at two different phones. Let $Z_t=X_t+Y_t$. ...
3
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1answer
122 views

Compute almost sure limit of martingale?

Let $Y$, $Y_1$, $Y_2$, $\dots$, be nonnegative i.i.d random variables with mean $1$. Let $$X_n = \prod_{1\le m \le n}Y_m$$ If $P(Y = 1) < 1$, prove that $\lim\limits_{n\to\infty}X_n = 0$ ...
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1answer
72 views

Ito integral's zero mean

My Sto Cal prof gave a long proof for the fact that $E[\int_{0}^{t} f_s dW_s] = 0$ where W is Brownian and f is Borel x $\mathscr{F}$-measurable, adapted and satisfies some integrability condition. ...
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113 views

Learning Stochastic Processing, Modeling, and Analysis: Any Available Workbooks?

Motivation behind the question: I took the upper-level probability course at my college, and did pretty well. Most of the time throughout the class, I found myself intuitively understanding the ...
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1answer
31 views

Asymptotic behaviour of absolute different of two independent Poisson processes

Suppose we have $X_1,X_2\sim Po(\lambda)$ ($X_1$ and $X_2$ are independent). Consider the interval $[0,1]$ with 100000 subintervals of length $\Delta=\frac{1}{100000}$. I can calculate: $$E|X_1-X_2|=...
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38 views

markov property in Durrett's textbook

Assume $B_t(\omega)=\omega(t),\omega\in (C,\mathcal{C},\mathbb{P}^x)$ is a B.M.(C is the continuous function space ,$\mathcal{C}$ is generated by the coordinate maps) In Durrett's textbook,he proved ...
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1answer
229 views

How to solve a discrete SIR epidemic model?

Let $(S(t), I(t), R(t))$ be a continuous time Markov chain SIR model with discrete space, where $S(t)$ stands for the number of susceptible people at time $t$; $I(t)$ stands for the number of ...
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166 views

Brownian motion is almost surely continuous

Why is Brownian motion required to be almost surely continuous instead of merely continuous? For example, this is stated as condition 2 in this article in section 1, Characterizations of the Wiener ...
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1answer
47 views

Applications of Singular Functions

For our purposes here, a singular function is a continuous function such that the part which is absolutely continuous with respect to Lebesgue measure is zero. For example, the Cantor function or "...
2
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1answer
161 views

Brownian motion on the circle and Itô processes

Consider the differential system \begin{cases} dX_t &=& -\frac{1}{2}X_t dt - Y_tdB_t, \\ dY_t &=& -\frac{1}{2}Y_tdt + X_tdB_t, \end{cases} $X_0 = 1$, $Y_0 = 0$. Let $X_t$ and $Y_t$ ...
2
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2answers
172 views

What are the assumptions for applying Wald's equation with a stopping time

I am trying to understand the assumptions under which I am allowed to apply Wald's equation for a sum of a random number $N$ of random variables $X_n$, $1\leq n\leq N$. There seem to be several ...
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1answer
60 views

A queueing model issue.

I am very beginner in Queueing Theory and I am learning in my own. I am struggling in the following situation. Suppose in a service center if a job arrives it will immediately being processed if a ...
1
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1answer
28 views

Basic question on application of Itô's formula to a stochastic process

I am working on a problem where I now find myself wanting to apply Itô's formula to: \begin{equation} X_t = \exp(W_t -W_0-\frac{t}{2}+\int\limits_0^tX_sds) \end{equation} where $W_t$ is 1D Brownian ...