# Tagged Questions

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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### Showing Measurability of empirical process (with respect to ball measurability)

I'm currently working on a problem in a certain proof which i do not fully comprehend, so i'm asking here to hopefully get some help for understanding :-) The situation of the problem is the ...
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### Ergodic Process: Does it visit all state?

I read in this article: " Ludwig Boltzmann, coined "ergodic" as the name for a stronger but related property: starting from a random point in state space, orbits will typically pass through every ...
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### What are some good books about martingales?

I'm looking for suggestions for well written books dealing with martingale theory, not necessarily exclusively. I'm also looking for a nice compilation of problems, preferably with answers, on this ...
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### The random walk of two drunks

The problem is such: two drunks start at either end of an alleyway of length n. Apart from at the ends, they each move one step forwards or one step backwards randomly. At the ends of the alley they ...
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### Feynman Kac solution discontinuity at 0

In most exposition of the Feynman Kac formula $$\frac{\partial u}{\partial t}(x,t) + \tfrac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\partial x^2}(x,t) -V(x,t) u(x,t) = 0$$ the condition of the ...
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### Expectation of brownian motion at hitting time

Am i correct in my derivation? I want to calculate $\mathbb{E}B_{\tau_a}$. From the definition of the hitting time i get $B_{\tau_a}=a$, so $$\mathbb{E}B_{\tau_a}=\mathbb{E}a=a$$ I am new to the ...
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### Distribution over the time it takes for a random process to reach an upper threshold

I am trying to figure out a way of determining the distribution over the time it takes for an arbitrary random process to cross a threshold value. For example, a simple (solved) case would be the ...
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### Stochastic differential equation for $Y(t)=\sqrt{X(t)}$

Assume that $X(t)$ solves the stochastic differential equation $$dX(t)=\sigma(t)dW(t)+\mu(t)dt$$ with $\mu(x)=bx+c$ and $\sigma^2(x)=4x.$ Assume that $X(t)\ge 0$. Find the stochastic differential ...
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### How to show $Y(t)=\ln(\frac{X(t)}{1-X(t)})$ has a constant diffusion coefficient

A Process $X(t)$ on $(0,1)$ has a stochastic differential with coefficient $\sigma(x)=x(1-x)$. Assuming $0<X(t)<1$, show that the process defined by $Y(t)=\ln(\frac{X(t)}{1-X(t)})$ has a ...
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### Find the expected value of this random integral

How can we find the expected value of $u(t)$ in terms of the following information: $$u(t)=\int_{0}^{t}(f(s)+(T-s)Y)(t-s)X(s)ds$$ where: $X(s)$ is a wide sense stationary process with known ...
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### Let $X(t)=(1-t)\int_{0}^{t}\frac{dB(s)}{1-s}$ I want find $dX(t)$ [closed]

Let $X(t)=(1-t)\int_{0}^{t}\frac{dB(s)}{1-s}$, where $0\le t < 1$.Find $dX(t)$. thanks for help.
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### is continuity preserved under Expectation?

Let's say I have a random function $X(t)$ that is continuous in $t$, almost surely. Is it true that $$\mathbb E(X(t_1)) = \mathbb E\left(\lim_{t\to t_1} X(t)\right)?$$ This seems incorrect to me ...
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### Find probability of a Poisson process.

Given that $N=\{N(t)\mid t\geq 0\}$ is a Poisson process with parameter $\lambda>0$ I need to find $P(N(3)=2\mid N(1)=0, N(5)=4)$ So this is a conditional probability (can anyone clarify if this ...
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### Proving The relationship between The Gamma Distribution and The Poisson Process.

How would one prove the relationship between The Gamma Distribution and the Poisson Process. Namely, if X~Gamma($\alpha,\theta)$ and $\alpha\geq1$ is an integer, then $Pr[X>x]$ for the Gamma ...
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### questions on mean-square convergence for a AR(P) example

In the following example related to AR(P) process, I have two questions I marked these two questions with two different colors. Question 1) (I marked with yellow), why that sum is mean-square ...
### Convergence of Random Variables to $- \infty$
The next exercise is taken from Alan Karr: If $X_1, X_2,...$ are independent with: $$P(X_k=k^2)=\frac{1}{k^2}$$ $$P(X_k=-1)=1-\frac{1}{k^2}$$ Prove that \$\sum_{k=1}^{n} X_k \xrightarrow{a.s.} -\...