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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Independent random variable in the limit (need extension?)

Given a sequence of rv $(X_n)$ on $(\Omega,\mathcal{F},P)$ with values on $(E,\mathcal{E})$ where $\mathcal{F}=\bigcup \mathcal{F}_n$ and $\mathcal{F}_n=\sigma(X_s:s\leq n)$. Lets say there is a ...
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9 views

Bounds for transition density and its derivative

Suppose the process $X_t$ has a transition density $p(t,x,y)$, which is continuously differentiable w.r.t $y$. In my proof, I use the following properties of $p$ and $p_y$: There exist functions $\...
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1answer
22 views

Can anyone explain one step of derivation in a branching process example?

I am reading the branching process example from chapter 0 of Probabilities with Martingales by Williams. What confused me is the part between equation 0.9(c) and 0.9(d) on page 10 and 11. What does ...
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1answer
30 views

Thinning a Renewal Process - Poisson Generalization

If we have a Poisson point process with rate $\lambda$ and we keep each of its point with probability $p$, we obtain another Poisson point process with rate $\lambda p$. Does this result holds for a ...
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11 views

Estimate the drift and diffusion function numerically

I have a 1D problem as following $$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x} \Big[ \frac{1}{2} \frac{\partial (g(x) f)}{\partial x} -\mu(x)f \Big]$$ I have a time-series of ...
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67 views

Convergence of Markov process as some rates tend to infinity

Take the simple two state Markov process characterized by transitions $$ \begin{aligned} 0\rightarrow1 & \quad \text{ at rate } \quad \alpha\lambda \\ 1\rightarrow0 & \quad \text{ at rate } ...
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1answer
29 views

Correlation between lagged Brownian motions.

Say I have two Brownian motions $X^1$ and $X^2$. Say they have constant correlation $\rho$. Then of course I know the correlation between $X^1_t$ and $X^2_t$. Furthermore I know that correlation ...
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148 views

Limit distributions for Markov chains $X\to\sqrt{U+X}$

This question spawned from a recent, very interesting problem. Let $\varphi=\frac{1+\sqrt{5}}{2}$ and $T$ denote the map on the space of continuous probability density functions supported over $\...
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1answer
26 views

Formulating deterministic and stochastic production models (not solving them) [Beginner's Operations Class]

Question provided in picture This question has been troubling me as I am not used to questions without numbers as it is hard for me to visualise. I also find stochastic problems hard in general. &...
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14 views

how to check if a process satisfies the markovian property with continuous time?

as an example we have A source transmitting messages is alternately on and off. The off-times are independent random variables having a common exponential distribution with rate α and the on-...
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46 views

Computing $\mathbb{E}[Z_n\mid Z_0=1]$ for a branching process [closed]

I came across a question whilst revising material to do with Branching processes. I am looking for help with part $(iii)$, here is my working: \begin{align*}&\mathbb{E}[Z_n\mid Z_0=1]\\&=\...
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1answer
35 views

Doob Decomposition Theorem - submartingale iff increasing

Probability with Martingales To prove $b$ I tried: $$A_n \ge A_{n-1}$$ $$\iff E[X_{n} - X_{n-1} | \mathscr F_{n-1}] \ge 0$$ $$\iff E[X_{n} | \mathscr F_{n-1}] \ge X_{n-1}$$ That ...
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1answer
23 views

Doob Decomposition Theorem in Williams is working backward? Unique modulo indistinguishability?

Probability with Martingales This is my understanding of what is going on in the proof above: We first assume $X$ has such Doob Decomposition in order to figure out what $A$ to use in ...
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0answers
34 views

Uniqueness of the trajectories of the solution of an SDE

Consider an SDE \begin{equation} dX_t=f(X_t,t)dt+b(X_t,t)dW_t \end{equation} Suppose firstly that the coefficient are Lipschitz continuous. So by the theorem of existence and unicity I have that exist ...
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18 views

System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
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1answer
45 views

How to simulate a delta-correlated random process

I'm trying to do the simulation described in the paper attached, but there is something I don't understand. The author says that the random variables which satisfy the relation (Eq. (4) in the paper) ...
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63 views

Measuring the degree of convergence of a stochastic process

Consider a set of random variables $(X_1,X_2,X_3,...X_k)$ that are i.i.d. $Bernoulli(p)$ While I do not know $p$, I can estimate it using $$ Y(k)=\frac{1}{k}\sum_{i=1}^k X_i $$ Notice that $Y(k)$ is ...
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41 views

Help with change of measure and martingales

Consider two three stochastic processes $X$, $Y$ and $Z$ in probability space $(\Omega, (\mathcal F_t)_{t \geq0},\mathbb P)$ such that $$ X_t = \exp\left(\int_0^t f_s ds\right), $$ $$ Y_t = \exp\...
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41 views

Showing that the following process is a martingale [closed]

Let $Nf(x) = \frac{1}{\sqrt{2\pi}}\int_\mathbb{R} e^{-\frac{|x-y|^2}{2}}f(y)dy, \;\; f \in b\mathcal{B}(\mathbb{R}), x \in \mathbb{R}.$ Let $X = (X_t, \mathcal{F}_t, \mathbb{P}^x)$ a pure jumps Markov ...
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2answers
54 views

brownian noise and stochastic differential equations

Consider the SDE $$dx=3x(t)dt+dW(t)$$ Where we're dealing with Brownian noise. Now, dW comes from $$dW(t)=\int_0^{dt}ds\ \eta (s)$$ As far as I understood, $\eta$ is the noise distribution (...
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32 views

System of Stochastic Differential Equations (SDEs) from Diffusion on Manifold

I am looking at a system of SDEs due to Brownian motion on a 3d Riemannian manifold (see e.g. Ito, 1962, The Brownian Motion and Tensor Fields on Riemannian manifolds). I have reduced the associated ...
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1answer
24 views

Finding PDF of a converted random process

Let $s(t)$ be a periodic triangle wave as illustrated in the accompanying figure. Suppose a random process is created according to $X(t) = s(t − T)$, where $T$ is a random variable uniformly ...
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1answer
18 views

Prove that Wiener process is Markov process [closed]

Prove that the Wiener process $\xi(t),T\ni t $ that starts from $0$ is the Markov process. I have no idea...
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1answer
69 views

Compute $\int_1^2 B_t \; dB_t$

I have to compute the following Ito integral: $$\int_1^2 B_t \; dB_t$$ where $(B_t)_{t \geq 0}$ is the 1-dimensional Brownian Motion. In the definition of Ito integral, the integral is taken from $0$ ...
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1answer
31 views

Calculating PDF from Autocorrelation

I have a statement like this; A zero mean Gaussian random process $X(t)$ is wide sense stationary with the auto-correlation function $R_x(\tau) = 4e^{-2|\tau|}$ And I want to find the ...
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1answer
33 views

Does Change of Numeraire same as Change of Measure?

Does Change of Numeraire same as Change of Measure? It is a bit confusing since both looks same. Do they have same meaning, or just mathematically alike.
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1answer
37 views

Proof of inequality with local martingale and stopping time..

Let $M$ be continuous local martingale starting from zero. For $a>0$, let $\tau_a=\inf \left\{t \ge 0: |M_t|>a \right\}$. Show that for every $t \ge 0$ we have: $a^2\mathbb{P}(\tau_a\le t)\le\...
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1answer
42 views

A Markov process with right continuous trajectories and left limits

Let $Nf(x) = \frac{1}{\sqrt{2\pi}}\int_\mathbb{R} e^{-\frac{|x-y|^2}{2}}f(y)dy, \;\; f \in b\mathcal{B}(\mathbb{R}), x \in \mathbb{R}.$ Let $X = (X_t, \mathcal{F}_t, \mathbb{P}^x)$ be a pure jump ...
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1answer
49 views

Population in a Galton Watson process

Consider a Galton-Watson process, $W_0$, $W_1$, $W_2$ $\ldots$, where $W_0=1$ and the next random variables are defined by the following recursion, $$ W_t = \sum\limits_{i=0}^{W_{t-1}} \xi_i, $$ where ...
3
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1answer
51 views

Expectation value of stochastic process

For which $k>0$ process $X=(e^{kW_s^2})_{s \ge 0}$ belong to $\mathcal{L}^2_{\infty }(W)$ and for which belong to $\Lambda ^2_{\infty }(W)$. Set one localization sequence $(\tau_n)_{n \ge 0}$ for ...
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1answer
18 views

Vasicek equation [closed]

Vasicek interest rate stochastic differential equation is $$dR(t)=(\alpha-\beta R(t))dt+\sigma dW(t)$$ where $\alpha , \beta$ are positive constants. I need to use Ito-Doeblin formula to compute $...
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30 views

Error term in the definition of the transition rates of a continuous time Markov chain

I'm studying G.F.Lawler's stochastic process book. There he defines the transition rates $\alpha(x,y)$ from the state $x$ to state $y$ (the state space is countable) of a continuous time Markov chain $...
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1answer
35 views

Stopping times of random walk with time dependent absorbing barriers

I have a Bern$(p)$ random walk ($Y_i = 1$ with probability $p$ and Y_i = 0 with $1-p$) with two absorbing boundaries, $A: Y^i \leq t_i$ and $B:Y^i \geq d_i-t_i$. Now, both $d_i$ and $t_i$ are evolving ...
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17 views

Measure extension

Given a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},P)$ with convention $\mathcal{F}=\bigcup_{t\geq 0}\mathcal{F}_t$. Given a positive $(P,\mathcal{F}_t)$-Martignale $M_{...
2
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1answer
40 views

Oblique bracket, stochastic integral

Let $X_t=\int_0^tsW_s^2dW_s$. How to set $<\int_0^tW_scos(s)dX_s>$? is it: $<\int_0^tW_scos(s)sW_s^2dW_s>=\int_0^ts^2W_s^6cos^2(s)ds$
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7 views

How to calculate sigma algebra generated by random variables in stochastic process?

According to http://www.math.uah.edu/stat/processes/Stop.html, a stochastic process $X=\{X_t:t \in T\}$ is a stochastic process with state space $(S, \mathscr{S})$ defined on underlying probability ...
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1answer
94 views

Itô-isometry in the extended case?

It is shown when constructing the Itô-integral that if: $E[\int_0^T X_t^2dt]< \infty$. Then we have that Itô-isomtry: $E[\int_0^T X_t^2dt]=E[(\int_o^TX_tdB_t)^2]$. In the extended Itô integral, ...
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25 views

Quadratic covariation

I am trying to solve small task from stocastic calculus. It can be shown that for stochastic processes X and Y , the quadratic covariation satisfies the polarisation formula - $[X, Y](T) = 1/2 ([X + ...
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1answer
35 views

Using of Ito formula with martingales

We have exam test - $\alpha,\beta \in \mathbb{R}$ and $N(t)=e^{\beta t}cos(\alpha W(t)).$ It is necessary to calculate $\mathbb{E}[cos(\alpha W(t))]$. I know that $\beta$ can be chosen so that $N$ ...
3
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63 views

What's the variance of the following stochastic integral?

The stochastic integral is defined as $$u_t = \int_{t-1}^t e^{-\kappa(t-s)}\int_0^s e^{-c(s-r)} \, dW(r) \, ds.$$ where $W(t)$ is a standard Brownian motion, $\kappa$ and $c$ are both positive. I ...
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1answer
25 views

Ito integral for simple stochastic process

I need for $l=1,2......$ prove that $E[W^{2l} (t)]=$ $\frac{(2l)!}{2^l l!}$ and $E[W^{2l+1} (t)]=0$ I know that Ito integral for simple stochastic process satisfies $E[I^2 (t)]=E\int_0^t\Delta^2(u)...
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40 views

martingale square integrable

Let $X_t=\int_0^te^{W_s}dW_s$ and $Y_t=\int_0^tW_sdX_s$. How to show that $X$ and $Y$ are martingale square integrable? ($W_t$ - Wiener) It it enough to show that $\mathbb{E}X_t^2<\infty$, $\...
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161 views

Can we model this set of experiments as an stochastic process and estimate the sample size?

I have an image with the size 5575x9440 and I'm implementing a modified version of the algorithm used in this paper on it, but because the code performance is low ...
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1answer
27 views

Compound poisson process invariant measure

Let $\rho$ be a probability measure in $\mathbb{R}$, $(N_t)$ a standar Poisson process and $(X_i) \stackrel{\text{i.i.d.}}{\sim} \rho$. Then $$Z_t = \sum_{n=1}^{N_t} X_n $$ is call a compound poisson ...
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28 views

conditions in which the repair shop process is recurrent (null\positive) or transient

here's the Story: Let $\epsilon_1.\epsilon_2,... $ be i.i.d numbers of machines for repair to the repair shop on mornings of days $1, 2,...$ . Assume that the shop is capable of repairing exactly K ...
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1answer
18 views

Conditions for limiting distribution to equal stationary distribution of SDE

I have SDE of the form $$dX_t=a\mathopen{}\left(X_t\right)dt+b\mathopen{}\left(X_t\right)dW_t,$$ where $W$ is Brownian motion. If the stationary distribution of $X$ exist is it equal to the limiting ...
3
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64 views

Show that $\hat{Y}$ is an optimal linear estimator of Y

Relevant Information. Let $X(t)$, $t \in T$ be a second order process. Let $M_0$ be the set of random variables of the form $a + b_1X(s_1)+ \cdots + b_nX(s_n)$ for a positive integer $n$ and constants ...
5
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32 views

$\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}$

Let $U$ be uniform distributed in $[0,1]$ . Show that with probability $1$ there's maximum a finite amount of $n \in \mathbb N$, so that the inequality $\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}...