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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What's a good intro book to stochastic processes?

I'm taking stochastic processes now (I'm an undergrad math major), and feel the book we're using is inadequate and lacks detail. Here is a link to the book: ...
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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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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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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 ...
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1answer
13 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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1answer
25 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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1answer
31 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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1answer
99 views

Is the graph of a Brownian motion over an interval measurable?

Let $n \in \mathbb{N}_1 := \{1, 2, \dots\}$ and let $B:\Omega \times [0, \infty) \rightarrow \mathbb{R}^n$ be a standard, $n$-dimensional Brownian motion over the probability space $(\Omega, ...
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22 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 $k>0$, ...
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Non-Markovian processes bibliography [on hold]

I'm a grad physics student and I need bibliography to start in the field of non-Markovian processes. I'd like a book with an introduction to the subject. Thanks for your suggestions.
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279 views

Question about an exercise in Revuz/Yor

I'm solving exercise 2.28 in Revuz/Yor. I was able to prove 1). Unfortunately at 2) I got stuck. I have to show: Let $B$ be a d-dimensional Brownian motion and $A\in \mathcal{A}:=\cap_t ...
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1answer
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Solve the SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ [on hold]

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 > ...
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10 views

Limit distribution on return time $\tau = \inf\{k: X_k = X_m \text{ where }m<k\}$ [on hold]

Suppose there is a stochastic process ${X_i}_{i=1}^n$ where $X_i$ is distributed normally over $\{1,\dots,n\}$. As $n\rightarrow \infty$, the probability that any one value is repeated should go to ...
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1answer
295 views

Some basic questions about Stochastic Calculus

I have a transition function for a Markov process $X_t$. I want to find a density function for the stochastic process $Y_t := \int_0^t X_s \,ds$. Some questions about this: Is this the same as the ...
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8 views

Density of this quantity for a Geometric Brownian Motion?

If we define $X_T = X_t e^{(\mu-\frac{1}{2}\sigma^2 ) (T-t) + \sigma W_{T-t}}$ where $W_{T-t}$ is a classical weiner process. How would we go about deriving the density and expectation for $X_{max} - ...
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25 views

Empirical 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 $P(X_0=0)=1$, $E[(X_{t+s}-X_t)^2|X_t]=s$, ...
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1answer
27 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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17 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
442 views

Doob-like inequality

I am not sure that this is a proper question. I am looking for the verification of the proof rather than for an answer. There is one inequality I use pretty often and I would like to be sure that it ...
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1answer
32 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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2answers
23 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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38 views

Finding Conditional Expectation and variance E(Y|X=x)

I want to find the conditional Expectation and variance of random function Y for a given value of random function X, i.e. E(Y|X=x). Here X is x(t) and Y is x(t+τ). Also, x(t) is a stationary Gaussian ...
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1answer
26 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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37 views

What is the distribution of the area between a Brownian Bridge and the x-axis?

Lets say that we have a Standard Brownian Bridge ($\sigma=1$) with endpoints $(0,0),(1,0)$ Is there a way to derive the distribution of the area between a sample path of this bridge and the x-axis?? ...
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12 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', ${f_X}\left( x \right)$, ...
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26 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 ...
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1answer
732 views

Independent increments of Brownian Motion

Suppose we have the $(W_t)$ Brownian Motion and the filtration $F=(\mathcal{F}_t)$, where $\mathcal{F}_t:=\sigma(W_s;s\le t)$. I know that for any $n\in \mathbb{N}$ and $0\le ...
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86 views

Are irreducible, positiv-definite Markov chains aperiodic?

If $M$ is the transition matrix of a discrete Markov chain, and $M$ is both irreducible, symmetric and positiv-definite, is the resulting Markov chain necessarily aperiodic? In my intuition, ...
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401 views

Application of Monotone class Theorem in the proof of Kunita-Watanabe Inequality

The Kunita-Watanabe Inequality says: Let $X,Y$ be two continuous locale martingales and $H,G$ two product-measurable functions on $(0,\infty)\times \Omega$, then $$ \int_0^t|G_s||H_s|d|\langle ...
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33 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
43 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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10 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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22 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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Looking for solutions manual: Probability and Stochastic Processes for Engineers [on hold]

My first posting to this community. I am an engineer. I am trying to teach myself the elements of Stochastic Processes. I found the book "Probability and Stochastic Processes for Engineers" By C. ...
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11 views

detailed balance condition for coupled Langevin equation

Suppose $a$ and $m$ are real variables and they satisfy the following two coupled Langevin equations: $$ \dot{a}=F_a(a,m)+\eta_a(t);\quad\dot{m}=F_m(a,m)+\eta_m(t) $$ where $\eta_a$ and $\eta_m$ are ...
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1answer
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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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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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33 views

Measurability of the points of (strict) increase for Stochastic Process

Given a background space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ , I'm considering a stochastic process $X:=(X_{t})_{t\geq0}$ with distribution $X(\mathbb{P})$ on ...
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1answer
22 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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Is $X(u,t)=\int^t_0V(u,\alpha)d\alpha$ wide-sense stationary?

I got a problem. The velocity of a particle in the X-direction is $V(u,t)$, where $V(u,t)=\pm V.$ $V(u,t)$ changes direction on average of $\lambda$ times per seconds, according to a Poisson law, ...
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1answer
357 views

Is a Bernoulli process a Markov chain?

For a Bernoulli process, the outcome of a future trial is independent of the outcome of past trials. I.e., the future behaviour of a Bernoulli process is independent of its past, i.e. a Bernoulli ...
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1answer
216 views

Local martingale is locally uniformly integrable martingale?

Is a local martingale locally uniformly integrable martingale ? Here I define a local martingale to be the process with a localizing sequence $\tau_n$ such that the stopped process is martingale. ...
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36 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 $n \rightarrow \infty$. ...
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31 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 ...
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412 views

Solution to nonlinear Stochastic Differential Equation

$dX_t=\left(\sqrt{1+X_t^2}+\frac{1}{2}X_t\right)dt+\sqrt{1+X_t^2}dW_t, X_0=0$, where $W_t$ is brownian. I tried using $X_t=\sinh(W_t)$ but then when I apply Ito's lemma to it, I can't get the first ...
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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 ...
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1answer
38 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 ...
2
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1answer
48 views

Compute almost sure limit of martingale?

Let $Y_1, Y_2, \dots$ be nonnegative i.i.d random variables with mean 1. Let $$X_n = \prod_{m \le n}Y_m$$ If $P(Y = 1) < 1$, prove that $\lim_{n->\infty}X_n = 0$ almost surely. I feel like ...