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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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filtration of a stochastic integral

1) Consider the Ito-integral: $S_t = \int_{0}^{t}f(s)dW_S$, where f is a borel bounded function and $W$ is a brownian motion. Is the $\sigma$-algebra, $\sigma(S_t) = \sigma(W_s, s\leq t)??$ For ...
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A Conditional Probability Problem

I am interested in finding the following problem: Let $\tau_1$ and $\tau_2$ are ordered statistics from a set of 2 independent uniform $(0,t)$ R.V. and let $Y_1,Y_2,Y_3$ are nonnegative iid R.V. that ...
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Solve the SDE $dX_t=\sqrt t(X_t+\sin t)dW_t$

I am new to stochastic differential equation and ran into a question of solving $$dX_t=\sqrt t(X_t+\sin t)dW_t$$ where $W_t$ is the standard Wiener Process and $X_0 \equiv K\in \mathbb R$. I know Ito'...
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A simpler proof for a probability problem

I am trying to prove the following lemma 2.3.5 from Stochastic Processes, Sheldon Ross, 2nd ed, page 77 which I have provided it here for convenience. The proof is provided in the book based on a ...
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29 views

Reference request in optimal stopping

I am given the following task. Distributed over a trading day, I am supposed to buy a certain quantity of a good. The price of this good changes during the day. The goal is to buy the required ...
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11 views

Joint distribution of Poisson process

$\Pi(t)$ is a Poisson process I want to calculate joint distribution of $(\Pi_{t_1}\Pi_{t_2}...\Pi_{t_n})$ Please check my solution: Lets define random variables $X_1 = \Pi_{t_1}, X_2 = \Pi_{t_2}- ...
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Non-unique stationary distribution but all states inter-communicate?

I have a Continuous Time Markov Chain with the following probability transition matrix: $$P_t= \begin{bmatrix} 1-\lambda t e^{-\lambda t} & \lambda t e^{-\lambda t} \\ \mu t e^{-\mu t} &...
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Why is $P^b(B_{T-T_b} \in (-\infty,b))=1/2$

Why is $P^b(B_{T-T_b} \in (-\infty,b))=1/2$ on the set $\{T_b<t\} $ where $T_b=\inf\{t \ge 0 :B_t=b\}$ and $T=t 1_{\{T_b<t\}}+\infty 1_{\{T_b \ge 0\}}$. I am trying to understand Proposition 2....
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Question about stochastic, (change of measure) Many thanks! [on hold]

Can any one give some hint for this question? Let $\{S_t\}_{t=0}^\infty$ be an asset price process defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$. Assume that the log-return of $...
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Derive the Conditional Distribution of a Brownian Motion Process

Let $\ X(t),t \ge 0$ be a Brownian motion process. That is, $\ X(t)$ is a process with independent increments such that: $$\ X(t) - X(s) \sim N(0,t-s), 0\le s \lt t $$ and $\ X(0)= 0$. ...
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Conditional expectation of correlated processes

Consider the known $C^1$ functions $f^1, f^2$ and the continuous semimartingales $X^1,X^2,S^1,S^2$ (solutions of a non-linear SDE). Suppose that $X^i$ is correlated to $S^1$ and $S^2$ with correlation ...
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When is it necessary to solve Kolmogorov forward equations (KFE) for a Markov Chain?

Say I have a continuous time markov chain, time homogeneous $X$ with a few states (say, 2). I want to know the distribution of where $X$ is at time $t$, call it $\mu_t$, which will be a vector of 2 ...
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If $f(t,x)$ is continuous and $B_{t}$ has continuous paths, then $f(t,B_{t})$ converges almost surely

Let $f \colon [0,\infty) \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a function which is continuous in both variables $t$ and $x$. Let $(B_{t})_{t \in [0,\infty)}$ be a stochastic process ...
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composition of convergence in probability and in distribution [on hold]

I ran into a problem in my stochastic process work. I have simplified it into the following statement. Consider a continuous random process $X_n(t)$, for $t\in [0, 1]$. Suppose that $X_n(t_0)\to t_0$ ...
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Absolute value of a martingale

Given a martingale $M(t)$, can I use Doob's inequality on $|M(t)|$ to achieve the following upper bond? $$P(\sup_{0\leq t\leq T}|M(t)|>\epsilon)=P(\sup_{0\leq t\leq T}|M(t)|^2>\epsilon^2)\leq\...
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Brownian motion in random landscape

I have some trouble with an exercice: Take $(W_t)_{t\ge0}$ and $(B_t)_{t\ge0}$ two independent standard Brownian motions started from 0 and define the process $$X_t=\int_{\mathbb{R}}\sqrt{L_t^x(B)}...
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If $Y$ is a Markov chain and $h>0$, why is $(Y_{\lfloor t/h\rfloor})_{t\ge0}$ not a Markov process?

Let $\left(Y^{(n)}_k\right)_{k\in\mathbb N_0}$ be a Markov chain for $n\in\mathbb N$, $(h_n)_{n\in\mathbb N}\subseteq(0,\infty)$ with $h_n\xrightarrow{n\to\infty}\infty$ and $$X^{(n)}_t:=Y^{(n)}_{\...
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How to find probability transition matrix for continuous time markov chain?

In Grimmet and Stirzaker, on page 258 it explains how to find transition probabilities, given a generator matrix: (a) nothing happens during $(t,t+h)$ with probability $1+g_{ii}*h+o(h)$ (b) ...
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22 views

Reference Markov martingale Harmonic function

I've just finished a course of stochastic process (discret martingale and markov chain). I would like to go further, I heard it exists a link between martingale markov process and harmonic functions. ...
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time series data : Predict $Y$ with $X$, where $X$ is dependent on $Y$

Let $Y$ be a target variable which you want to predict on using $X$ (e.g with a statistical model), where $X,Y\in \mathbb{R}$. You are given data which looks like this : $$ data_t = (X_t, Y_t), \ t\...
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Calculate limit: Make a stopping time $T$ bounded $T\land n$ and take the limit $n \to \infty$

Say we have a martingale $X$ and a stopping time $T$. Instead of directly studying the stopped process $X_T$, many proofs employ a trick, namely, one considers the bounded stopping time $T\land n$ ...
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Proving that a process is a brownian motion: How do I show independent increments?

$W_t$ is a brownian motion and I want to show that $W^*_t=(-W_t)$ is also a brownian motion. I can easily show the distribution the new variable: $W^*_t \sim N(0,t)$. But one of the properties of ...
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Interarrival Times of Poisson Process

Let $(N_s)$ be a homogene Poisson Process with rate $\lambda>0$ and let $t>0$ be fixed. $T_{N_t}$ is then the last arrival time before time $t$ and $T_{N_t+1}$ is the first arrival time after ...
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How is a spectra function related to the mean value and standard deviation?

everyone. I'm starting to research stochastic optimization algorithms, and my simulation model is an aircraft which endures turbulence. I've read a document on the atmosphere environment provided by ...
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1answer
23 views

Period of each state

I am trying to determine the period of each state $ j = 0, 1, 2$ for this irreducible Markov Chain with transition probability matrix $$P=\begin{bmatrix}0&0&1\\1&0&0\\\frac{1}{2}&\...
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42 views

Failure Time of n independent continuous random variables

Let $X_j$ , for $j$ between $1$ and $n$ be independent continuous random variables with failure rate function $r_j(t)$ . Now consider $T$ to be independent of this sequence. Find the failure rate ...
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Random time change for a Poisson process and convergence with respect to the Skorohod topology

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\left(Y^{(n)}_k\right)_{k\in\mathbb N_0}$ be a time-homogeneous Markov chain on $(\Omega,\mathcal A,\operatorname P)$ and $$X^{(n)}_t=...
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Interpolation of martingale

Consider a non-constant martingale $(M_n)_{n\ge 0}$ with the filtration $(\mathcal{F}_n)_{n\ge 0}$, where $\mathcal{F}_n=\sigma(M_m:m\le n)$. Define for $t\in\mathbb{R}_{\ge 0}$: $$X_t:=M_{\lfloor ...
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1answer
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Predictable process, measurable process, filtration (continuous and discrete)

1) Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and $(\mathcal F_n)$ for $n\in \mathbb N$ a filtration. A) First of all, I'm not clear with filtration. Why are they important ? Indeed, ...
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how to prove a supermartingale stopped at a stopping time is a supermartingale? [on hold]

I'm reading Shreve's Stochastic Calculus for Finance, vol 2. In Ch 8, after define stopping time as Def 8.2.1 A stopping time $\tau$ is a random variable taking values in $[0,\infty]$ and ...
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Brownian motion reflection principle result

I'm studying about the reflection principle of the brownian motion, and I found that this result is a direct consequence of this principle: Let $B_t$ a brownian motion, then for every $a \in \mathbb{...
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Girsanov's theorem and likelihood for random initial conditions

Consider as an example the Stochastic Differential Equation $$ \text{d}Y(t) = -\kappa Y(t) \text{d}t + \sigma \text{d} B(t), \qquad t \geq 0 $$ where $B(t)$ is a standard Brownian motion, $\kappa$...
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Stochastic Processes problem

I have already solved the problem but only up to the 2nd part, I can't equate directly the 3rd part but i don't know how to explain the reason behind it where k=0 then the greater than or equal to ...
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Generalized Polya's Urn

In Polya's urn, we have $b$ black balls and $w$ white balls at time $t$. At time $t+1$, we have $b+1$ black balls with probability $\frac{b}{w+b}$ and $w+1$ white balls with probability $\frac{w}{w+b}$...
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Covariance of two Ito / Diffusion processes

Let $B_t$ denote the standard Brownian motion process. $X_t$ and $Y_t$ are Ito diffusions with the following SDEs: \begin{align} dX_t &= \mu(t,X_t) \; dt + \sigma(t,X_t) \; dB_t \\ dY_t &= \mu(...
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How to show that $X_n$ and $Y_n$ are Markov chain and how to find their transition probabilities.

The Sujata store across our campus stocks Lyril soap. He follows $(1/5)$ inventory schedule. This means, if he has $\le 1$ soaps by the closing time today, he will get some more from his godown to ...
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133 views

What is the stationary distribution of the following Markov chain?

Consider a chain with state space $\{1,2, \cdots \}.$ If you are at $1$ go to state $j$ with probability $p_j$ $($$ j=1,2,\cdots$ $) ,$ where these are non-negative numbers adding to $1$. If you are ...
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1answer
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Measurable Version of a Banach Space Valued Process

I'm trying to convince myself on the existence of a measurable version of a process, $X:[0,T]\times \Omega\to E$ which is assumed to be stochastically continuous with $E$ a separable Banach space. ...
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Delta Dirac formula - Ito-Tanaka

I am trying to understand the definition of a Local-Time as appearing in Wiki, https://en.wikipedia.org/wiki/Local_time_(mathematics). In particular I am trying to understand how the dirac function is ...
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probability distribution of discrete time continuous state markov process

Consider the sum $$ y = \sum_{i=0}^n x_i, $$ where $n$ is a discrete random variable with pmf $q_n$, normalized via $\sum_{n=0}^\infty q_n = 1$, and $x_i$ is a continuous random variable with pdf $p(...
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What is the difference between ergodicity and the law of large numbers?

I want to begin by saying that I know absolutely no measure theory. To my knowledge, roughly speaking a stochastic process is ergodic if its time average converges to the expectation (space average) ...
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Sum of $n$ coin flips, where the probability of heads at experiment $t$ is dependent on the outcomes of the experiment at $t-1$

Suppose I play a game of $n$ coin flips, where heads is $1$ and tails is $0$. If each coin flip was independent, the expected sum of all $n$ coin flips is trivial. What if there is dependence? How can ...
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Confusion about given proof of the compensated Poisson process being a Martingale?

Given the following proof of the compensated Poisson process being a Martingale Why does the proof start with $E[N(t)-\lambda t|N(s)]$ when the question asks to prove that $X(t)$ is a Martingale? ...
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The solution to the mean reverting-process is Gaussian [closed]

What does it mean when my teacher says that the solution to the mean-reverting process is Gaussian? Thanks
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Steps of a Markov chain subordinated to a Poisson process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\left(Y^{(n)}_k\right)_{k\in\mathbb N_0}$ be a time-homogeneous Markov chain on $(\Omega,\mathcal A,\operatorname P)$ $D([0,1]):=\...
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Martingales for a random walk

Firstly, I just want to check my understanding: If we have a symmetric random walk such that $P(S_{n+1} = S_n + 1|S_n)=1/2$ $P(S_{n+1} = S_n - 1|S_n)= 1/2$ then $S_n$ is a martingale, $(S_n)^2$ ...
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Bounded moments for continuous processes

I am trying to prove that an $a.s$ continuous $\mathbb{R}$-valued process have bounded moments on a compact set. Does anyone know how to go about this?
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Stochastic processes with same distribution, which are no modification

Consider stochastic processes $\mathcal{X}$ and $\mathcal{Y}$. How can I find an example in which $\mathcal{X}$ and $\mathcal{Y}$ having the same distribution but are no modification of each other? I ...
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27 views

Probability of overlap of random intervals dropped on unit circle

Suppose I have a circle with circumference $A$. Along the circumference of this circle, I randomly drop $N$ arcs with fixed length $a < A$. Now suppose I drop a single additional arc ($N+1$). What ...
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28 views

Backward Kolmogorov equation for simple markov process

The following exercise is from a course on SDE's and I am a bit stumped. Consider the process. $dX_t=\lambda\left(\xi-X_t \right)dt+\gamma\sqrt{|X_t|}dB_t$ $\lambda,\xi,\gamma>0$ Find $\mathbb{P}...