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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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integration by parts formula

Below is from Liptser, Shiryaev "Theory of martingales", page 200: I have a question: How from eq. 3.5 and 3.8 they got the eq. 3.9? Since $G$ is of finite variation $\mathcal{E}^{-1}(G)$ is of ...
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Infinitesimal generator of diffusion process in one dimension

I consider a diffusion process $$ dX_t = b(X_t) dt + \sigma(X_t) dB_t. $$ From the general theory, we know that if $f\in C^2(\mathbb{R})$ and has a compact support, then the infinitesimal generator ...
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Poisson process with time exponentially distributed

While studying Poisson processes, I have found a problem I can't solve: Two people, A and B, arrive at a bank and wait to be attended. A and B are in the waiting queue and A is before B. The ...
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Multi-folds of Generating Functions

Stochastic Process Generating Function Practice I don't understand how $G_{N}(s) = G_{M}(G_{Y_i}(s))$. $N = Y_1 + Y_2 + Y_3 + ... +Y_M$, which to my understanding the main generating function should ...
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1answer
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Decomposition of a cadlag and of local bounded variation function $Z:\mathbb{R}_+\rightarrow\mathbb{R}$

Let $Z:\mathbb{R}_+\rightarrow\mathbb{R}$ be cadlag and of local bounded variation with $Z(0)=0$ and $V_Z(t)$ denotes the value of the total variation of $Z$ on $[0,t]$ for all $t\in\mathbb{R}_+$. I ...
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Is the sum of positive jumps from a jump process adapted again?

Let $(\Omega,\mathcal F,\{\mathcal F_t\}_{t\in [0;T]},\mathbb P)$ be a filtered probability space satisfying the usual conditions and let $\{X_t\}_{t\in[0;T]}$ be an adapted stochastic process, whose ...
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20 views

Proving this process is a martingale

Let $X_j, j \geq 1$ be $\mathcal{L}_{1}$ random variables and $\mathscr{F}_n = \sigma \left(X_j, 1 \leq j \leq n\right), n \geq 0$ be the natural filtration. Define the process $Z = \lbrace Z_n, n \...
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Upperbound on expectation of supremum

If $X$ is a supermartingale. I need an upper bound as: $\mathbb{E}[\sup_{-\tau\leq\theta\leq0}\|X(\theta)\|^k]\le K \sup_{-\tau\leq\theta\leq0}\mathbb{E}[\|X(\theta)\|^k], $ where $\tau>0$ and $...
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1answer
38 views

Why is $Y_n$ not a bernoulli process?

for $n \in \mathbb{N}$, let $X_n$ be a Bernoulli process with parameter $p = \frac12$, let $N = \min \{n \geq 2: X_1 \neq X_n \}$ for $n \in \mathbb{N}$, let $Y_n = X_{N +n -2}$. in a question it ...
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28 views

Quadratic Variation and Brownian Motion

Let $(X_n,F_n)$ be a martingale with $X_n \in L^2(\Omega,F,\mathbb{P})$. The quadratic Variation $(<X>_n)_n$ of the process $(X_n)_n$ is defined as $$ <X>_n := \sum\nolimits_{i=1}^{n}(\...
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How to I use Ito's formula to compute quadratic variation?

Let B be Brownian motion. Use Itos formula to compute the quadratic variation of $\left[X_t^i\right]$ for $\left[X_t^1\right]=e^B_t$, $\left[X_t^2\right]=ln(B_t^2+1)$ and $\left[X_t^1\right]=sin^2B_t+...
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Proving the Asymptotic Equipartition Property (AEP) for i.i.d. random variables

I'm currently working on some assignment and I'm not sure if my solution would be considered as correct in any way: Let $(X)_{n \gt 1}$ be a stochastic process of i.i.d. random variables with ...
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1answer
29 views

Show the map is measurable w.r.t. the product space

Let ${ ( X_t ) }_{ t \geq 0 }$ be an $\mathbb{R}^d$-valued stochastic process on $( \Omega, \mathcal{F}, P)$. I am trying to show that for any $A \in \mathscr{B} ( \mathbb{R}^d )$ the map $r : [ 0, \...
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14 views

Poisson process time of arrival proposition

This is taken from a stochastic processes text. Consider $N$ a Poisson process (number of arrivals at time $t$) with rate $\lambda$. Let $T_n$ denote an arrival time ($n$ a natural number). The ...
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Obtaining various bounds and statistics from a non-trivial random process

At a given time $t$, there is a vector of random variables $C(t) = \left(C_1(t), \cdots, C_n(t)\right)$ that has the following recursive relationship in time for all $i \in \lbrace 0, 1, \cdots, n \...
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1answer
29 views

Discrete Exponential Martingale - Properties

This question is about the discrete exponential martingale. Let $(Y_n)_n$ be a sequence of independent and identically distributed random variables with $m_{Y}(t) :=\mathbb{E}\left[e^{t Y_{1}}\right]&...
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Is there a deeper reason why the simple symmetric random walk on $\Bbb Z^D$ turns transient when increasing $D$ from 2 to 3?

Polya proved the following very well-known Theorem: A simple random walk on $\Bbb Z^D$ is recurrent if and only if it is symmetric and $D\le2$. Dropping simplicity (i.e. allowing jumps to non-...
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Joint PDF of two dependent exponential random variables [on hold]

I need help in following Let's assume we have two identically distributed exponential random variables $X$ and $Y$. What will be the joint pdf of those random variables, i need to consider two cases, ...
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Continuity in mean of a stochastic process

If $X$ is a stochastic process, a.s. continuos and such that $\forall t \geq 0, X_t \in L^1_\omega$, is its mean function $t \rightarrow E[X_t]$ continuos? I can show it if $X \in L^1_\omega L^{\...
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21 views

For which random variable is the spectral density a PDF?

The spectral density of a stationary random process is the inverse Fourier transform of the autocorrelation function of that random process. This spectral density is a probability density function for ...
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1answer
19 views

Distance of squared random variables and upper bound

I got two sequences of random variables, $(X_n)_n$ and $(Y_n)_n$, and I know that $| X_n - Y_n | \leq C a_n $, for some constant $C$ or equivalently $|X_n - Y_n | = \mathcal{O}(a_n)$. Now I want to ...
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Proof: the $\max{(B_t,0)}$ is submartingale without using convex function with Jensen's inequality

Proof: the $\max(B_t,0)$ is submartingale without using convex function with Jensen's inequality To prove it using convex function and jensen's inequality, we know the max function is convex and it ...
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1answer
48 views

Proof of Poisson process distribution

I'm confused about part of a proof of this theorem. Theorem: If {$N_t; t \geq 0$} is a Poisson process, then for any $t \geq 0$, $$P(N_t = k) = \frac{e^{\lambda t}(\lambda t)^k}{k!}$$ where $k = 0, ...
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Majority voting 9 Bits Error probability

i have 1 Bit that will send 9 times. Each bit has an error probabiltiy of 1/3. The received Bits will be interpreted with a majority voting. Do anyone have an idea how i do check the probability ...
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Is a vector of independent Brownian motions a multivariate Brownian motion?

Given a filtered probability space $(\Omega, \mathcal{F}, \mathcal{F}_{t\geq 0}, P)$: If $B_1, B_2, \dots, B_m $ are all real $\mathcal{F}_t$ Brownian motions, jointly independent. Is the resulting ...
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1answer
38 views

How to determine the transition matrix of the markov chain $X_n = f(X_{n-1}, \xi_n), n \geq 0$

I am having trouble to find the transition matrix of the following question: Let $X_0$ be a random variable taking values in a countable set $I \subset \mathbb{R}$. Let $(\xi)_{n \geq 0}$ be a ...
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Taylor on $h$ smooth

I'm struggling to proof that $\int_{\mathbb{R}}P(z,t|x)\sum_{n=1}^{\infty}D^{(n)}(z)h^{(n)}(z)dz$ (with $D^{(n)}(z):=\frac{1}{n!}\lim_{\Delta t\rightarrow 0}\frac{1}{\Delta t}\int_{\mathbb{R}}P(y,\...
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Strong Markov property and another stopping time

I'm trying to prove that given a regular continuous time Markov chain $X_t$ (pure jump process), its embedded chain given by $Y_n=X_{T_n}$ is a homogeneous Markov chain, where $T_n$ is the time of the ...
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Markov Chain holding time

Let $X$ be a continuous-time Markov chain. How does one justify $P(X(s)=x,0\leq s\leq t\mid X(0)=x)=\lim_{n\to\infty}P(X(kt/n)=x,k=0,1,\dots,n\mid X(0)=x)$ without prior knowledge of the ...
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Motivation of Milstein scheme

Milstein scheme is motivated by improving the convergence speed sigma terms of Euler scheme. where sigma and mu is globally Lipschitz continuous$$t_i=\frac{T}{n}i\quad dX_t=\mu(X_t)dt+\sigma(X_t)dW_t$$...
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Mean stopping time of a Brownian motion

I came across the following proof of the fact that the mean stopping time of a Brownian motion to hit $-1$ or $1$ is $1$: Let $B$ be a Brownian motion. We already know $B_t^2-t$ is a martingale. Let $...
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Find a distribution. Wiener proccess [on hold]

Find the distribution of $$\frac{1}{t-s} \left(W_t^2 + W_s \left[ \frac{t}{s} W_s - 2W_t \right] \right), \qquad 0 < s <t. $$ How do i do this? Where $$ W_t, W_s $$ - Wiener process
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Under which conditions the linear combination of two Wiener processes is still a Wiener process.

Let $\{W^{(1)}_t , t\ge0 \} $ and $\{W^{(2)}_t , t\ge0 \} $ be independent Wiener processes. Find all constants for which $\alpha W^{(1)}_t +\beta W^{(2)}_t, t\ge0$ is Wiener process. I already saw ...
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1answer
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Does time changed brownian motion have no-memory property?

Let $W=(W_t)_{t \geq 0}$ be a Browniwn motion. Do the processes $$X_t = W_{e^t} \quad \text{and} \quad Y_t = \exp \left(- \frac{t^2}{2} \right) W_{e^t}$$ have the no-memory property, i.e. are the sets ...
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21 views

Find covariance of a random process

Let $Y_t = e^{-\alpha t}W_{\beta \ \exp({2\alpha t})}$, where $W_{s} \ \text {is Wiener prosses,} \quad 0\le t, \quad \alpha , \beta\in \mathbb R^1$. Find $\text{Cov}(Y_t , Y_s).$ Here is my ...
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When is a linear recurrent process stationary?

Let’s call a sequence of random variables $\{X_n\}_{n = 1}^\infty$ stationary, if $\forall n, m, k \in \mathbb{N}$ $EX_n = EX_m$ and $Cov(X_n, X_m) = Cov(X_{n + k}, X_{m + k})$. Let’s call a sequence ...
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1answer
56 views

When can I cover all the unique items?

I'm totally a newbie in this community. I would like to ask for help in a modeling question. Thanks for your time and patience in advance. Assume I have N unique ...
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Monotonicity of the expectation of random variables

Define $r(t)=\frac{1+X_1+...+X_t}{1+Y_1+...+Y_t}$ $X_t\sim Binomial(Y_t,\alpha)$ $Y_t \sim Binomial(\lfloor 1+r(t-1)\rfloor,\delta)$ I would like to show that $E[r(t)]$ is a monotonic function of $t$...
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Martingales related to the square of a compensated Poisson process. [duplicate]

Let $N_t$ be a homogeneous Poisson process with rate $\alpha>0$ and $M_t=N_t-\alpha t$. Show that $M_t^2-\alpha t$ and $M_t^2-N_t$ are martingales. I computed \begin{align} \mathbb E[M_t^2\mid \...
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Meaning of $\int _0 ^T X_t dt$ when $(X_t)_t$ is a process

I am studying stochastic calculus (Ito integrals, to be precise) , and I am not sure if I got some things right. For instance, we have defined $\Lambda_B ^2 (a,b)$ as the space of progressively ...
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1answer
40 views

Black Jack Bust Probability [on hold]

Can you explain me how to calculate the probability that the dealer in black jack game will bust. (more than 21 points) There are different opinions in the relevant literature, so i would like to know ...
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29 views

What are some fields that require Stochastic?

I am a Pure Maths PhD student researching in functional analysis, particularly Banach space theory, Recently I came across stochastic control theory, which perform optimization using stochastic (...
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1answer
23 views

first-hitting-time and conditional probability

I have been struggling with the following problem. Let $\{X_{n}\}_{n\geq 1}$ be i.i.d. random variables with common distributional measure $\nu$. Let $B\subset\mathbb{R}$ be any Borel measurable set ...
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Determine parameters in moving-average model MA(1)

Let $x_t=(b_0\epsilon_t+b_1\epsilon_{t-1})$ be a MA(1) process. Assume that the autocovariance function $\gamma(k)$ is given. Use $\gamma(k)$ to determine the parameters $(b_1,b_1,\sigma^2)$. Without ...
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Is this application of a law of large numbers rigorous in this not identically distributed case?

Let $ \{ X_t \}_{t \in \mathbb{N} }$ be a sequence of indipendent random variables such that $X_t \sim N(u_t, 1)$ for all $t$ where the mean $u_t$ is given by the equation $$u_t = \theta u_{t-1} + \...
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1answer
21 views

Stochastic process vs regression for modelling data

why one would model something (anything?) with a stochastic process, such as Ornstein–Uhlenbeck (OU), rather than a regression, like $y(t) = \beta_0 + \beta_1e^{( - rt)}$? With regression, you can use ...
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31 views

Stochastic Optimization and Monte Carlo

Assume we are in a Brownian filtration where I denote $W$ the Brownian motion. My problem is to numerically compute $$ \min_X E (\int^1_0 X^2_tdt),\ \ \ \ (*) $$ where $X$ is adapted to the filtration ...
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1answer
29 views

Stopped uniformly integrable process in discrete time is uniformly integrable?

I'm studying for work the book: "Stochastic Calculus and Application" by Choen and Elliot 2 ed. In section $4.2$ (pg. 91) it states the discrete version of the Optional Stopping Theorem for bounded ...
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25 views

Is stochastic integral the same as Lebesgue integral of Banach-valued function?

I'm reading a book about stochastic processes and I've come to "stochastic integration." In this section stochastic processes are basically functions $$X:[0,\infty)\rightarrow L^2(\Omega, \mathbb{C})....
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Stochastic Proof Piece

As a part of the proof of recurrence criterion, (Page 22 here: http://web.math.ku.dk/noter/filer/stoknoter.pdf), it is shown that $(P^n)_{i,j} = \sum_{m=1}^n (P^{n-m})_{j,j}f_{ij}^{(m)}$. This makes ...