# stochastic-processes's questions - Chinese 1answer

8.291 stochastic-processes questions.

### Binary branching process with exponential distribution

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 ...

### -1 Find the transition probability function P(y,t,x,s) for Brownian motion with drift B(t)+t.

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。 ...

### How to construct a Poisson process not based on Lebesgue measure?

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 ...

### Joint probability density of fractional Brownian motion process

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 ...

### Doubt about a Corollary of Law of Iterated Logarithm for Brownian Motion

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 ...

### How do you test a sequence of random variables for independence?

0 answers, 12 views probability stochastic-processes
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 ...

### Determine the mean of the product of the first two arrival times?

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 ...

### 1 Probability approach in the expected payoff of a dice game

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 ...

### 1 What's the expected stopping time for a sum of random variables to cross a threshold?

1 answers, 16 views stochastic-processes stopping-times
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 ...

### How do you take an average and variance of a stochastic series defined by an moving average(1) process?

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 ...

### 2 Does $Z(t)$ have stationary increments?

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 ...

### 2 Martingale ganerated by random walk

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 ...

### Basic Probability with Stochastic Sum (Moving Average Process)

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 ...

### Reference request - Fokker-Planck equation for jump-diffusions

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-...

### -2 Branching process question.

2 answers, 332 views probability stochastic-processes
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 ...