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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If $Y_n=\min\{M_n,7\}$ and $\{M_n\}$ is a martingale wrt ${X_n}$, show that ${Y_n}$ is a supermartingale wrt ${X_n}$

If $Y_n=\min\{M_n,7\}$ and $\{M_n\}$ is a martingale wrt ${X_n}$, show that ${Y_n}$ is a supermartingale wrt ${X_n}$ I tried doing cases for $M_n<7$ and for $M_n>7$, but I couldn't get that ...
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29 views

Stochastic control HJB equation

I am trying to solve this optimal control problem : $ V(x,t) = inf( E[\int_{0}^{1}(x(t)^2 - \frac{1}{2}u^2(t))dt + x(1)^2])$ subject to $dx(t) = u(t)dW_t$ $x(0) = x_0 \in R $ $u(t) \in [-1,1] $ ...
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38 views

Simple application of Donsker's theorem

I am trying to do exercise 5.15 in Moerter's book "Brownian Motion". It seems quite easy, but I can't solve it somehow: Suppose $S(j)_j$ is a SRW on the integers, started at zero. Show that: $$ ...
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16 views

Coupling Brownian Motions

I want to simulate three freight rate indices which are naturally correlated. The freight rate dynamics ($X$) can be modeled as a geometric Brownian motion: $dX_{t} = \mu X_{t}dt + \sigma ...
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17 views

Finding a solution to the SDE of $dX_t = -2 (1-t)^{-1}X_tdt + \sqrt{2t(1-t)} dW_t$.

I am trying to find the solutions to the SDE: The solution of the following SDE $$dX_t = -2 \frac{X_t}{1-t} dt + \sqrt{2t(1-t)} dW_t, \quad X_0 = 0 $$ where $W_t$ is a Wiener process. I know that ...
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27 views

Getting different results with two methods for a Markov Chain

Given the below Markov transition matrix, calculate $p_{0,1}^9$ \begin{matrix} 0.5 & 0.5 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{matrix} Method ...
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Harris Chain and Stationary Distribution

Consider a Harris chain given by $\{X_n\}$ with the following transition function, $X_{n+1}=\max \{0,X_n-b\} $ with probability $p$ and $X_{n+1}=\max \{0,a-\tau\} $ with probability $1-p$, where $\tau ...
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24 views

Right-continuity of the expectation of a supermartingale

I start with a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t),P)$. I assume that the filtration is right-continuous. On this probability space I define a supermartingale $M$. Now ...
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15 views

Distribution of the first exit time of a one-dimensional diffusiom/ Brownian motion

I have a one-dimensional diffusion on $[0,1]$ and I need to calculate the distribution of the first exit time of the interval $(-\epsilon,\epsilon)$ for an $\epsilon > 0$. A good first step would ...
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24 views

M/M/1 queue derivation: how to “recursively solve in dependence on $p_0$”

I want to sketch out the derivation of the equations for an M/M/1 queue for a presentation I'm giving. I can understand most of the derivation from Willig but I don't understand this section from p10 ...
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41 views

How to improve Euler-Maruyama discretisation with analytical moments?

I'm trying to improve Euler-Maruyama discretisation by adding to it the analytical moments. To try it I made a very simple example on the stochastic process $X(t) = W(t)^2$, where $W(t)$ is a standard ...
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2answers
31 views

Construct a martingale with two given distributions.

This is a follow up of another post: Construct a martingale with a given distribution? Given two distributions $f_1(\cdot)$ and $f_2(\cdot)$ on $\mathbb R$, under what condition can we construct a ...
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13 views

ratio distribution of two dependent Poisson variables

Consider a homogeneous Poisson point process $\Phi$ in a disc with radius $R$, and the density of $\Phi$ is $\lambda$ per unit area. Denote the total number of points in the whole disc as N, and ...
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49 views

Standard Wiener process continuity

This will be one big question. Basically, I had a lecturer who, to put it mildly, was not able to explain anything well. I am preparing for the exam now. One sample exam question is the following: ...
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32 views

Show this one is a martingale?

On a fixed interval $[0,T]$, let $(W_t)_{0\le t \le T}$ be a Brownian motion, and $ (\gamma_t)_{0\le t \le T} $ a cadlag process. Let $$ M_t = exp ({\int_0^t\gamma_sdW_s - ...
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12 views

Dirichlet process mixture model

I'm reading Nonparametric Bayesian Inference by Peter Müller and Abel Rodriguez. In Chapter 3, there is no proof provided for some formulas but I think I need to know exactly how it was derived if I ...
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13 views

Some doubts in the definition of some random variables in stochastic process

There is this passage about Poisson process in the textbook Stochastics by Hans-Otto, Georgii. Here is a copy of the text: Let $\alpha>0$ and $(L_i)_i\geq0$ be a sequence of i.i.d random ...
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24 views

Finding the mean of $X_t = \int_0^t sW_sdW_s$

For the stochastic integral, where $W_t$ is a Wiener process, I am trying to find the mean of $X_t = \int_0^t sW_sdW_s$. I have read before that any stochastic integral with $dWt$ has mean zero, but I ...
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42 views

Probability that a male line of descent will die out…

One quarter of couples in a society have no children. The other three quarters have exactly three children, with each child being equally likely to be a boy or girl. What is the probability that the ...
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Express PGF of Z in terms of PGF of $\xi$ with Z having the same distribution as $\xi$ but is conditional on $\xi$ being nonnegative [closed]

Let $\phi(s)$ be the generating function of an offspring random variable $\xi$. Let Z be a random variable whose distribution is that of $\xi$, but conditional on $\xi>0$. That is, ...
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27 views

solving two dimensional diffusion advection equation.

I know that the solution to one dimensional diffusion advection equation is easy to obtain. However, was wondering if the same is true for two dimensional linear diffusion advection equation, i.e., ...
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29 views

Soft Question - book recommendation - Stochastic Processes

My mother language book on stochastic processes is pretty much complete(~500 pages) but would like another one in English, to have in my library. I'm looking for a similar book containing the ...
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26 views

Will Ito's Isometry result in $E\left(\int_0^t \cos(u)\,dB_u \int_0^t \sin(u)\, dB_u \right) = E\left(\int_0^t \cos(u) \sin(u)\, du \right)$?

If I have two integrals, $X_t = \int_0^t \cos(u)\,dB_u$and $Y_t = \int_0^t \sin(u)\, dB_u$ , where $B_u$ is a Wiener Process and I am trying to find: $$ E\left(\int_0^t \cos(u)\,dB_u \int_0^t ...
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35 views

Construct a martingale with a given distribution?

Given a random variable Y, is it possible to construct a martingale M such that $$M_1 \stackrel{D}{=} Y$$ I'm not sure how to go about proving that such an M exists under such general conditions, but ...
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28 views

How many types of martingale related stochastic processes are there?

Previously I had thought that the only concepts in this direction were martingales, submartingales, and supermartingales. However, at least when discussing quadratic variation and stochastic ...
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32 views

Proof of normal distribution property used in Levy's construction of the brownian motion?

I have been trying to follow the construction of Brownian motion by Levy. I need a property about the conditional distribution of the Brownian process in an interior point of an interval given its ...
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24 views

Can we apply an Itō formula to the solution of a SPDE?

Let $V\subset H\subset V^\ast$ be a Gelfand triple $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(W_t)_{t\ge 0}$ be a ...
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58 views

Rephrase a multiparameter SDE indexed by time and space as an infinite dimensional SDE indexed by time

Let $\mathcal V_t\subseteq\mathbb R^3$ be the bounded space occupied by a closed particle system at $t\ge 0$ and $$[0,\infty)\ni t\mapsto X_t\in\mathcal V_t\tag 1$$ be the movement of a fixed particle ...
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43 views

Coefficient matching proof that $e^{\alpha x-\frac{1}{2} \alpha^2}=\sum_{n=0}^{\infty} \frac{1}{n!}H_n(x)\alpha^n$, where $H_n(x)$ are Hermite poly.?

Hermite polynomials can be defined as (from wikipedia): $$ H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n} e^{-x^2/2}. $$ I am trying to show that: $e^{\alpha x-\frac{1}{2} \alpha^2}=\sum_{n=0}^{\infty} ...
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33 views

Stochastic modeling. A bidding Model

The Question: Let $U_1,U_2,...$ be independent RVs, each uniformly distributed over $(0,1]$. These random variables represent successive bids on an asset that you're trying to sell, and that you must ...
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34 views

Probability of exponential growth event

Under the assumption of exponential growth of a population of cells, the population size at time $t$, $N(t)$, is: $$N(t) = N_0\exp(rt)$$ where $r$ is the rate of division and $t$ is time. What is ...
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15 views

Squared sum of Rayleigh Distributed Random variables

I would like to evaluate the distribution of $|x.h_1 +y.h_2|^2$ where $h_1$ and $h_2$ have Rayleigh Distribution. I shall be grateful if someone can help me is this regard. One thing I would like to ...
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35 views

Prove Wald's identities for Brownian motion using stochastic integrals

The question is as follows: Let $W$ be Brownian motion and $T$ a stopping time with $\mathbb{E} T < ∞$. Show (use stochastic integrals) that $\mathbb{E}W_T = 0$ and $\mathbb{E} W^2_T = \mathbb{E} ...
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32 views

Do we have integrability in these cases?, a version of Doob-Meyer decomposition.

There are as I can see several version of the Doob-Meyer decompositions, I am looking at this version where we do not talk about class D or uniform integrability. Let Z be a cádlág submartingale. ...
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18 views

inverse Stochastic differential equation

SDE are really new for me, so I'm sorry if this is a silly question. Let $W_t$ be a Wiener process and let $x_0$ denote the initial value of the process. If I'm correct, for $\text{d}X_t = -(\beta X_t ...
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35 views

Generating a list of numbers

A set of numbers is generated starting from $0$ in the following way: Add the current number to the resultset In a chance of 50:50, do Either add $2$ to the current number Or subtract $1$ from the ...
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iid measure of a stationary stochastic process

If I have some continuous path $x(t)$ which runs from time $t=t_0$ to time $t=t_n$ which has some, let's say Markov, generating measure $p(x)$ such that in some sense it is the limit ...
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42 views

Construction of continuous-time markov chain and finding stationary distribution

There are 15 lily pads and 6 frogs. Each frog, with rate 1, jumps to one of the other 9 unoccupied pads chosen uniformly at random. What is the stationary distribution for the set of occupied lily ...
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35 views

Average waiting times

I have the following exercise, which I would like to solve: Company A run buses between New York and Newark, their bus leaves New York every half an hour starting from 0:00, 24h a day. Company B also ...
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59 views

Prove that $Y_n=X_{n-1}X_n$ is a markov chain

Let $\{X_n\}_{n=0}^\infty$ a sequence of discrete random variables independent identically distributed. Let $Y_n$ such that $Y_n=X_{n-1}X_n$ for all $n\ge 1$ Is $\{Y_n\}_{n=0}^\infty$ a markov ...
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Unsure of Probability Terminology

I am not sure how to call the following property (I have forgotten my probability theory!). I'd be grateful if someone can tell me the keywords: Suppose I have the following cartoon: I have a ...
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14 views

What is an example of a stochastic nonlinear dynamic system with 2 separated stable orbits

I have some social science data to which I would like to fit a stochastic difference or differential equation in two variables. (I observe the system only at discrete intervals). This system that has ...
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1answer
48 views

Poisson process of satellite launches

Satellites are launched into space at times distributed according to a Poisson process with rate $\lambda$. Each satellite independently spends a random time (having distribution $G$) in space ...
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62 views

Ergodicity of stochastic recursive process

Does anyone know how one can show ergodicity for a recursive stochastic process determined by the following equation: \begin{equation} X_n = f(\varepsilon_{n-1},Y_{n-1})X_{n-1}, ...
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20 views

Ito Formula for increments of Ito Processes

Let $X_{t}=X_{0}+\int_{0}^{t}a_{s}ds+\int_{0}^{t}\sigma_{s}dW_{s}$, $W_{t}$ is a standard BM. How can I apply Ito formula to $(X_{t}-X_{s})^{2}$? Should I use a multidimensional version?
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44 views

Coarsest filtration

Let $(\Omega,\mathscr F, \mathsf P)$ be some probaility space, $T = [0,+\infty)$ and $\eta = (\eta_t)_{t\in T}$ be some real-valued stochastic process. I say that a stochastic process $\xi$ is good if ...
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29 views

Find the conditional density of the ratio of events described by a Poisson Process?

Arrivals are described by a Poisson process with a constant intensity $\lambda.$ We are asked to describe the conditional density of the ratio $\frac{W_9}{W_{10}}$, given that at the time $T$ the ...
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31 views

Question regarding proof of property related to the stopped sigma-algebra.

I have a proof of a property regarding the stopped $\sigma$-algebra, where one part I do not understand, I'll highlight what I do not get, can you please help me? We have a probability space ...
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1answer
42 views

Definition/Construction of Wiener Measure

I want to make sure I understand this rigorously: Assume we already know that Brownian motion $B_t$ on $[0,\infty)$ exists/how to construct it. Every $\sigma$-field considered is implicitly assumed ...
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33 views

Expected value of a series of random variables

There are $K$ checkout counters in the mall, and there are $N$ shoppers in the queue waiting for a checkout counter. Initially all counters are empty. Whenever a counter is empty, the next shopper in ...