**8.291 stochastic-processes questions.**

Consider a particle $X_t^1$ which moves in space and assume that we have a countable family of exponential times ($exp(\nu)$ distributed) $T_1,....,T_N$ $N \in \mathbb{N}$, where $T_1$ is the time ...

Find the transition probability function P(y,t,x,s) for Brownian motion with drift B(t)+t.
I have already know the standard Brownian motion transition fuction is N（0，t），whose drift term is constant。
...

Let X(t) be a Gaussian white noise process with mean zero and variance $\sigma^2 \in \mathbb{R}$. Let $\tau \in \mathbb{R}$ be a constant. How would I identify the distribution of $$I = \int_{0}^{t} X(...

How can I show that the $p$-variation of a standard Brownian motion is infinite almost surely for any $p > 1/2$. By this I mean the total variation,
$lim_{\delta\to0} (sup_{\pi:\delta(\pi)=\delta}...

Here the definition of wikipedia of Branching process. Let $z_n$ the size of the generation $n$. So, $$Z_{n+1}=\sum_{k=1}^{Z_n}X_{n,i}$$ where $X_{n,i}$ is the number of offspring of the $i-$th ...

For example, we have $P(X_i=1)=0.5,P(X_i=-1)=0.5,(i\ge 0)$, $Y_n=X_nX_{n-1},(i\ge 1)$. I want to examine whether $Y_n,(i\ge 1)$ is a martingale.
My thinking are as follows:
$E[Y_{n+1}|X_1,...,X_n]=E[...

Let
$X=(X_t,t\ge 0$ be a real-valued stochastic process on a measurable space $(\Omega,\mathcal{A})$ with almost surely right-continuous paths
$\mathbb{F}:=(\mathcal{F}_t,t\ge 0)$ be a filtraiton on $...

If $\{X_n\}$ is a Markov chain, $N(t)$ is a poisson process. Can we say that $X_{N(t)}$ is a continuous Markov chain? How to prove that?

having a bit of trouble with the following question:
Consider a two-server queue with Exponential arrival rate $\lambda$. Suppose servers 1 and
2 have exponential rates $\mu_{1}$ and $\mu_{2}$, with $...

Suppose we have the following two stochastic differential equations for $x_0$ and $x$ respectively
\begin{align}
dx_0 &= -k_0(t)(x_0-1)dt+\eta_0(t) x_0\,dB \tag1\\
dx &= -(k_0(t)+\epsilon ...

Please accept apology if this question is vague. It is related to the link which describes white noise theory to deal with stochastic differential equation.
https://www.duo.uio.no/bitstream/handle/...

Let $E$ be a normed $\mathbb R$-vector space and $$\left\|f\right\|_{0+\alpha}:=\sup_{\stackrel{x,\:y,\:x',\:y'\:\in\:E}{x\:\ne\:x',\:y\:\ne\:y'}}\frac{\left|f(x,y)-f(x',y)-f(x,y')+f(x',y')\right|}{\...

It is clear to me that I can build a suitable underlying probability space for a homogeneous Poisson point process.
It is enough to have a pobability space $(\Omega,\mathcal A,P)$ with on it iid ...

Suppose $(x_t)=(\dots,x_0,x_1,\dots)$ is a time series realisation of a fractional Brownian motion process $B_H(t)$ with a certain Hurst exponent $0<H<1$.
I am interested in what can be said ...

I'm reading about the Law of Iterated Logarithm for standard Brownian motion.
As a Corollary of this we have the next:
Suppose that $\{B(t)\}_{t\geq 0}$ is a standard Brownian Motion. Then almost ...

So I have a sequence of random variables and I'd like to use a theorem that requires them to be iid. I've got the identically distributed part. But I'm unsure about independence. Is there any such ...

Let {X(t) : t ≥ 0} be a Poisson process with rate λ
Suppose it is known that X(1) = 2. Determine the mean of S1S2, the product
of the first two arrival times.
I am not sure what to do with the first ...

Consider two independent Brownian motions $B^1_t$ and $B^2_s$, and denote $X_{t,s}=B^1_t+B^2_s$. Fix $\{(t_1,s_1), \ldots, (t_n,s_n)\}$. How to find the conditional distribution of $X_{t,s}|X_{t_1,s_1}...

I have a big problem in understanding and deriving partial derivatives. I would like to understand the logic behind deriving partial derivatives for some exemplary processes. I have a few examples ...

Is there some sort of intuition or a good ilustrative example for random variables being $\sigma$-algebra measurable? I understand the definition, but when looking at martingales, the meaning of ...

Let $U_1,U_2, \dots $ iid interarrival times. If $m(t)$ is the expected number of interrarival times and $\mu = \mathbb E(U)$ what is
$$
\lim_{t \to \infty} m(t) - \frac{t}{\mu}= \hspace{2pt}?.
$$

For $s \leq t \leq T$, I want to evaluate $E\left[\exp\left(iz(W_t-W_s)+izb(s-t) + bW_T-b^2T/2 \right)|\mathcal{F_s}\right]$, where $W$ is Brownian motion on $\mathcal{F}$, and $b \in \mathbb{R}$. ...

I recently noticed something about the covariance function of a Brownian motion that I don't quite understand, and I was wondering if anyone could help me.
Suppose $W$ is a Brownian motion, and we ...

Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F\subseteq\mathcal A$ be a $\sigma$-algebra on $(\Omega,\mathcal A)$
$(E,\mathcal E)$ be a measurable space
$f:\Omega\times ...

I am trying to understand the problem of expected payoff of a dice game explained here. I can roll the dice up to three times, but after each one I decide if I want to try once again or not. The idea ...

Suppose we have a Ehrenfest Markov chain with $d + 1$ states $0, 1, \dots, d$. With transition probabilities $p(i, j) =1 - (i/d)$ if $j = i + 1$ and $p(i, j) = i/d$ when $j = i - 1$ (otherwise $p(i, j ...

Consider the following bivariate O-U process:
$\mathrm{d}X_t=k_{11}(\theta_1-X_t)\mathrm{d}t+\rho\mathrm{d}W_{1t}+\sqrt{1-\rho^2}\mathrm{d}W_{2t},$
$\mathrm{d}Y_t=(k_{21}(\theta_1-X_t)+k_{22}(\...

I have a sequence of non-negative identically (and I believe independently) distributed random variables $X_i$. I know their expectation $\mu = \mathbb{E}[X_i]$. And I'm waiting for the sum to cross ...

I am trying to compute, given the following recursive stocastic process (which I proved to be a submartingale wrt the filtration $\mathcal{F}_n = \sigma(Y_1,\ldots,Y_n)$) :
$$X_0=0, X_{n+1}=|X_n+Y_{n+...

Consider a random walk $S_n= \sum_{k=1}^n X_k$, where $\{X_k\}_{k=1}^\infty$ are independent and identically distributed random variables. Assume that $S_n \rightarrow \infty$ almost surely as $n \...

A flea hops on the vertices $A$, $B$, and $C$ of a triangle. Each hop takes it from one vertex to the next and the times between sucessive hops are independent random variables, each with an ...

Let W represent a standard Brownian Motion in one dimension. For some $b > 0$, let $S_b$ denote the first time that $|W_t| = a$. How do we show that the Laplace transform of $S_b$ is
$$E[\exp(\...

In coin tossing example (+1 win, -1 loss) your average winnings on last four tosses is:
$ .25\epsilon_t + .25\epsilon_{t-1} + .25\epsilon_{t-2} + .25\epsilon_{t-3}=w_t$
The sequence MA process such ...

Let be $$Z(t)= \int_t^{t+1}B(s)\,ds-B(t)$$ a process where $\{B(t), t \ge0\}$ be a standard Brownian Motion. I have to show that $Z(t)$ has stationary increments.
$\textbf{My very poor ideas: }$
I ...

I have the following Schwartz model:
$$dS_t=a(\mu-\ln S_t)S_tdt+\sigma S_tdW_t$$
$$X_t=\ln S_t$$
$$dX_t=a(\hat{\mu}-X_t)dt+\sigma dW_t$$
with $\hat{\mu}=\mu-\frac{\sigma^2}{2a}\sigma$
$$F_t(T)= \exp\...

Given two SDE's with random diffusion coefficients, if the diffusion coefficients are pathwise uniformly close, can we say the same about the solutions to corresponding SDE's?
More precisely, ...

This question is related to rough path theory. A link to wikipedia is: https://en.wikipedia.org/wiki/Rough_path
As I understand a signature determines a path. For this question consider a one ...

http://www.hairer.org/notes/RoughPaths.pdf here is a textbook, but I am completely lost at the definition. It is defined on page 13, chapter 2. A rough path is defined as an ordered pair, $(X,\mathbb{...

I have a question regarding Markov process with poisson attached. I have attempted the section A and C parts to this problem but have no idea how to even interpret the following question.
Q. A simple ...

Let $X(n)_{n \geq 0}$ be a homogeneous Markov chain with state space $S$. Define
$$p_{ij}(n):= \mathbb{P}( X(n+m)=j|X(m)=i),$$
where $i, j \in S$, $n \geq 1$, and $m \geq 0$.
Let $d(i)$ be the period ...

The task
Let $\xi_i, i \in \mathbb{Z}_+$ be i.i.d. random variables on $\mathbb{R}$ with a probability density function $\rho(x) > 0$. Denote by $\eta_n$ the minimum of r.v.s $\xi_i$ for $i \leq n$...

Given standard brownian motion, how do we evaluate the following in the easiest way for some range of $c's$
$\mathbb{E}e^{\frac{cB_{t}^2}{2}}=\frac{1}{\sqrt{2\pi}t}\int e^\frac{cx^2}{2}e^\frac{-x^2}{...

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal{A},\operatorname{P})$. By definition of $B$, for $\operatorname{P}$-almost every $\omega\in\Omega$ $$[0,\infty)\to\...

Let $(\Omega,\mathcal{F},P)$ be a probability space. Moreover let $\tau_x\colon \Omega \to \Omega$ for $x \in \mathbb{Z}$ be an ergodic group of tranformations that preserves $P$. By ergodic we mean ...

I'm covering stochastic time series and am rusty with probability theory. I was hoping this could be rather easily explained.
In illustration of a typical moving average, you win +1 USD if a fair ...

Is there a proper reference for the Fokker-Planck equation for jump-diffusion?
I have seen it in, for instance
https://inordinatum.wordpress.com/2013/11/20/fokker-planck-equation-for-a-jump-diffusion-...

Suppose we have a branching process, where at each time $n$, each individual produces offspring independently with the distribution $\{p_k\}$ and then dies with probability $0 < q < 1$.
For ...

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