Questions tagged [stochastic-processes]

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 is the correct filtration?

Let $B\overset{\circ}{=}\left(B_{t}\right)_{t\geq0}$ denote a Brownian motion in a filtration $\mathcal{F}$. Are $X_{t}=\frac{1}{\sqrt{a}}B_{at}$ ($a>0$ constant) and/or $Y_{t}=tB_{\frac{1}{t}}$ ...
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How to compute the limit of $x_{t}$ given $x_{t+1}=K_{t+1}x_t+\xi_{t+1}$?

$x_t\in\mathbb{R}^n$ satisfies that $x_{t+1}=K_{t+1}x_t+\xi_{t+1}$, where $K_t\in\mathbb{R}^{n\times n},\xi_t\in\mathbb{R}^n. [K_t , \xi_t ]$ are i.i.d. with regard to $t$ and their statistical ...
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Is recycling samples better than drawing fresh ones?

At a high level, I am wondering if in a sequential process it is better to reutilize samples even if these samples have been used to make past decisions. Let me formalize my doubts in a toy example ...
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Question about how to construct Brownian Motion. [duplicate]

My first question was answered within minutes by Sangchul Lee, but after a week of asking people I am stuck again, and let me describe. I will try to write out the standard definitions carefully. Let $...
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1 vote
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proving Lenglart's dominated property for some function

$\textbf{Definition:}$ $X$ is $L-$dominated by $Y$ if $\mathbb{E}(|X_T|) \leq \mathbb{E}(|Y_T|)$ for every bounded stopping time $T$. Define $\mathbb{N}:=\{1,2,3,...\}$, i.e. the natural number ...
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Properties of sum of Poisson processes

Let $N_1$ and $N_2$ be a independent Poisson processes with intensities $\lambda_1=1$ and $\lambda_2=4$. Let $N=N_1+N_2$ and $S_n$ be a moment of $n$ event. I need to calculate the following: $P(N_1(...
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Two questions about inhomogeneous Poisson process

Let $N(t)$ be an inhomogeneous Poisson process with a intensity function: $$\lambda(t)=3t+1,\ for\ \ 0\leq t\leq 5,$$ $$\lambda(t)=3\ for\ \ t>5$$ I need to calculate: $E(N(6)-N(3)|N(2)=3)$ $P(N(6)...
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1 vote
1 answer
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Showing that a process is a supermartingale using Ito's formula

Consider a stock with price dynamics $$dS_t=S_t\sigma_tdW_t$$ where $(W_t)_{t\geq0}$ is a Brownian motion and $(\sigma_t)_{t\geq0}$ a bounded and continuous process adapted to the filtration $(\...
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Question on interchanging of random variables with the same distribution inside expectation

Let $T\in\mathbb{N}$ and let $(S_t)_{0\leq t\leq T}$ be such that the increments $S_1-S_0,\dots,S_T-S_{T-1}$ are independent and identically distributed, and let $(\mathcal{F}_t)_{0\leq t\leq T}$ be ...
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Are stochastic processes defined on product of probability spaces?

The following question has confused me lately. Suppose that you have a sequence of random variables $\{\xi_n(\omega)\}$ defined on some probability space $\left( \Omega,\mathcal{F},\mathbb{P} \right)$,...
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1 answer
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A statement about finite Markov chains

The following quantity $\tilde\pi$ is defined in the textbook Markov chains and mixing times by David A. Levin. Here $\tau_z^+ = \min \left\{t\geq 1| X_t = z\right\}$. Let $z \in \mathcal{X}$ be an ...
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1 vote
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Computing transition probabilities in continuous time Markov chain

How do we compute the transition probabilities in a continuous time Markov chain? Supposing $h$ is sufficiently small then how would I compute $p_{i,j}(h)$, I am aware of the relation to the generator ...
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(Stochastic) Time Series Reconstruction

Assume a linear stochastic dynamical system $$ \begin{align} \mathrm{d}x&=-J_{1,1}x+J_{1,2}y+\sigma_{1}\,\mathrm{d}W_{1}\\\mathrm{d}y&=-J_{2,1}x-J_{2,2}y-\sigma_{1}\,\mathrm{d}W_{1}+\sigma_{2}\...
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Uniform Integrability of a Martingale up to time T

I am trying to prove that a certain martingale $(R_t)_{t\geq 0}$ is uniformly integrable over a finite time interval $[0,T]$. Now I know that the definition of uniform integrability is that $\lim_{a\...
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Time series: ARMA characteristic polynomials have common roots

I have a question regarding the idea that if the roots of the characteristic polynomials of a time series (say some ARMA process) lie outside the unit circle, then the series will be invertible/causal ...
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M/M/1/10 queueing process with two different classes

I'm looking at a problem where we have calls queueing under two different classes, new calls and handovers. The number of calls arriving follow a Poisson process with $\lambda_{1} = 125$ per hour ...
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3 votes
1 answer
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Law of the square of a martingale divided by its bracket

Let $(M_t)_{t\geq 0}$ be a continuous martingale such that $M_0=0$ almost surely. There exists an increasing process $(\langle M\rangle_t)_{t\geq 0}$ which is called the bracket of $M$ such that $M^2-\...
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Non-linear backward Kolmogorov equation

The backward Kolmogorov equation (BKE): $$\frac{du}{dt} = A(x,t) \cdot \nabla_x u(x,t) -\frac12 \text{Tr}(BB^t(x,t) \text{Hess}_x u(x,t) - f(t,x,u, B,\nabla g), \;\;\; t<T$$ If $f\equiv 0$ then ...
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1 vote
2 answers
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Write probability of first return time in terms of first hitting time

For a time homogeneous Markov chain $(X_n)_{n\ge 0}$ with state space $I$ with no self loop . Given $X_0 = i \in I$ , define the first return time $T_i = \inf\{n\ge 1 : X_n = i\} $ and first hitting ...
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Are branching processes infinite state space markov processes?

Is it correct to view a branching process as an infinite state space markov process, with the states being the total population which can be 0,1,2,...?
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What is the cubic expectation (third-order moment) of a complex gaussian vector (say, E[$aa^{T}a$])?

For a real gaussian vector, an explicit formula for the cubic expectation can be found in Matrix Reference Manual (search 'Cubic Expectations' in this link for detail), i.e., say $a$ is a complex ...
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Stopping times for martingale

Here are some definitions. The nonnegative integer set is denoted by $\mathbb{Z}_+$. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space and $\{\mathcal{F}_{n}\}_{n\in{\mathbb{Z}_+}}$...
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why if $\{P\tau < \infty \} = 1$ then $\tau$ is a.s finite stopping time [closed]

why if $P\{\tau < \infty \} = 1$ then $\tau$ is a.s finite stopping time
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2 votes
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Showing that $\{S^x_{\tau \wedge n}\}_{n\in\mathbb{N}}$ is uniformly integrable

Let $\{X_n\}_{n\geqslant 1}$ a sequence of i.i.d. Rademacher r.v., that is $\Pr [X_1=1]=\Pr [X_1=-1]=\frac1{2}$ and for $n,m\in \mathbb{Z}$ with $n<m$ set $X_0:\equiv x$ for some $x\in(n,m)\cap \...
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Using the Girsanov theorem to construct a solution to an SDE

Suppose that $B$ is a standard Brownian motion and $b$ is a bounded, measurable function. Using the Girsanov theorem, construct a solution to the SDE $$dX_t=b(X_t)dt+dB_t.$$ I really don't know where ...
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3 votes
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61 views

Return of Brownian motion to zero

From chapter 4 of Bulinskiy & Shiryaev's Theory of Random Processes (ISBN 5-9221-0335-0): [Exercise 26] Let $W = \{W_t, t \geqslant 0 \}$ be a $m$-dimensional Brownian motion, where $m \geqslant ...
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Sampling distribution of GBM Maximum-Likelihood estimator

Given the geometric Brownian diffusion $$ X_t = \mu X_t dt + \sigma X_t d W_t$$ I learnt that its maximum likelihood estimators are the following as this web article suggests $$\hat \mu = \frac{\delta ...
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1 vote
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If $\tau ^{(n)}\downarrow \tau $ then $\{X_{\tau ^{(n)}}\}_{n\in \mathbb{N}}$ is uniformly integrable

Reading in the book of Bhattacharya and Waymire of probability theory I find the following assertion: Let $\{X_t\}_{t\in [0,T]}$ a right-continuous martingale adapted to a filtration $\{\mathcal F_t\}...
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How to determine the reflecting horizontal hidden barrier for Ito diffusion

The first article at the link below talks about that the Bi-Directional Grid Constrained (BGC) stochastic processes for a random variable X over time t is one in which the further it departs from the ...
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1 vote
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Fundamental theorem of calculus equivalent for stochastic integrals

Let $B$ be a standard Brownian motion in one dimension and let $H$ be a continuous, adapted, bounded process. Prove that $$\frac{\int_t^{t+h}H_sdB_s}{B_{t+h}-B_t}\to H_t$$ in probability as $h\...
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1 vote
1 answer
35 views

Distribution of solution to SDE

Let $X_0$ be a standard normal random variable and suppose that $$dX_t=-\frac{1}{2}X_tdt+dB_t.$$ $X_0$ is independent of the Brownian motion. Find the distribution of $X_t$ for $t\geq0$ and find $\...
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0 votes
1 answer
33 views

Two definitions of a Stochastic Process?

I have two supposedly equivalent definitions of a stochastic process. A stochastic process is an indexed set of random variables. Specifically $$ y = \{y(x) \; | \; x \in \mathcal{X}\}. $$ ...
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Finding Output of of a given linear system, with known nature of input of the random process and Autocorrelation function [closed]

A linear system with input Z(t) is described by $X'(t)+α*X(t)=Z(t), t>=0, X(0)=0.$ Find the output X(t) if the input is a zero mean Gaussian random process with auto-correlation function given by ...
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0 answers
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Calculating the transition density function by finding the partial derivative of a conditional probability. [closed]

I've been given a process $Y_t=(1+t)B_t^2,t\ge0$ where $B_t$ is a standard Brownian motion and asked to find the transition density function $f(y,t|x,s)$. I've been instructed that $f(y,t|x,s)$ can be ...
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Probability of a two-state continuous Markov chain

Consider a continuous-time Markov process ($\epsilon_t$) which takes two values ($\epsilon_t=0$ or $\epsilon_t=1$). Let $p_0$ denote the probability of switching from state 1 to 0 and let $p_1$ denote ...
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Probability and expected time a symmetric random walk hits the graph of $f(x) = x$

Suppose we have a simple symmetric random walk $X_n$ where $X_0 = i > 0$. Say we define the time $T=$ $\{n\geq0 |X_n=n\}$, i.e $T$ is the first time that the time index of $X_n$ equals $X_n$. I ...
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3 votes
1 answer
38 views

Question on use of Ito's formula with integral in the function

This question is being asked in the context of the Feynman-Kac formula. Suppose the real-valued process $Z$ satisfies the SDE $$dZ_t=b(Z_t)dt+\sigma(Z_t)dW_t.$$ Suppose we have functions $f:\mathbb{R}\...
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2 votes
1 answer
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Exponential submartingale inequality

In a paper I am reading I found the following: "Applying the exponential martingale inequality we derive that $$P\Big(\omega: \sup_{0 \leq t \leq k}[M(t)-1/2 \epsilon \langle M(t) \rangle] \leq 2 ...
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  • 1,771
2 votes
1 answer
31 views

$exp$ of Half-Normal Distribution

I know that the Half-Normal Distribution has moments of all orders - that is, if $X\sim\mathcal{N}(\mu,\sigma)$, then, $$ E[|X|^p]<\infty $$ However, do we also have $$ E[e^{|X|}]<\infty $$ ? ...
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1 vote
1 answer
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An identity about Probability of unions

Let $(\Omega,\mathfrak{A},P)$ be a probability space and events $A_1,...,A_n \in\mathfrak{A}$ with $P(A_{i_1} \cap...\cap A_{i_k})=P(A_1 \cap...\cap A_k)$ for all $k \in \{1,...,n \},i_k\in\{1,...,n \}...
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3 votes
0 answers
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Solution to a linear Backward SDE

In my last question, Jose and I discussed about the solution to a linear backward SDE. I followed his steps and made it clear. Besides, I read a paper from Professor Peng talk about the linear ...
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Distribution of a poisson process in an exponential random variable

I think I already solve this problem but I would like to know your opinion about the solution. Let $\{X(t)\}_{t \geq 0}$ be a poisson process with parameter $\lambda$, and T an exponential random ...
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  • 197
3 votes
1 answer
54 views

Black-Scholes model with a derivative with payoff $S_{T}^{3}$

Given a Black-Scholes Model and a derivative with payoff $S_{T}^{3}$ at time $T$. Check that the value of that derivative at time t is $V_{t} = g(t, T)S_{t}^{3}$, where $g(t, T)$ has to be determined. ...
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0 votes
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Nonhomogeneous (nonstationary) Poisson processes and splitting them

What I have learned from my recent readings is that if $\{N_t,t\geq 0\}$ is nonhomogeneous Poisson process (in $\mathbb{R}_+$) with intensity function $\lambda(t)$, and if an event occurring at time $...
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5 votes
1 answer
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Brownian motion with a stopping time

Let $x \geq 0,c<0,$ and a Brownian motion $(W_u)_{u}.$ Let $T:=\inf\{u \geq 0, B_u +cu\geq x\}.$ It follows that $Y:=\sup_{u \geq 0}(B_u+cu) \in ]0,\infty[.$ We want to verify that $\{Y \geq x\} \...
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1 vote
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Maximal inequality for Bessel process

Let $W = \{W_t, t \geq 0\}$ be an $m$-dimensional Brownian motion. The process $( \|W_t\|, \mathcal{F}_t )_{t ≥ 0}$ is then a Bessel process, where $\|\cdot\|$ is the Euclidean norm in $R^m$. Question....
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1 vote
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Brownian motion, submartingale

I'm trying to prove whether a given process is a submartingale. Let $W = \{W_t, t ≥ 0\}$ - m-dimensional Brownian motion. Prove that $( ||W_t||, F_t )_{t ≥ 0}$ is a submartingale (with a.s. continuous ...
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2 votes
0 answers
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Proof check for Problem 1.5.12 on Brownian motion and stochastic calculus

The problem is that the continuity condition seems to be a dummy condition. Here is my proof. Let $X$ be in $\mathscr{M}_2^c$ (Continuous square-integrable martingale and $X_0=0$), and $T$ be a ...
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4 votes
2 answers
108 views

Best strategy to reach $500 for a gambling situation in a casino

Suppose a gambler has \$100 to start with. Each time he/she has 0.4 chances of winning and 0.6 chances of losing a bet. If he/she wins he gets twice the money he put in and loses what he bet if he ...
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  • 477
-1 votes
0 answers
26 views

european call with binomial model in python

I have this exercise to resolve. Calculate the prices of a European and an American call option in an N = 3 period binomial model with $ S_0 = $ 1, $ u = \frac{1}{2} = $ 2 and $ r = - \frac{1}{4} $. ...
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