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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1answer
110 views

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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1answer
846 views

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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1answer
352 views

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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79 views

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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530 views

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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575 views

Is a predictable process adapted?

Let us consider a measurable space $(\Omega, \mathcal{F})$, with a filtration $(\mathcal{F}_t)_t$ of sub $\sigma$-algebras of $\mathcal{F}$. The predictable $\sigma$-algebra $\mathcal{P}$ is the $\...
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2answers
55 views

Conditional expectation equality

Does this statement hold and how to prove it correctly? $$ \mathbb{E}(\mathbb{E}(X\mid \mathbb{F})^2) = \mathbb{E}(X\mathbb{E}(X\mid \mathbb{F})) $$ Any help? Thanks. $\mathbb{F}$ is a sigma algebra....
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72 views

Proving a statement about stopping times

I'm getting stuck on the following statement Suppose that $\tau$ is a stopping time on some filtered probability space $(\Omega,\cal{F},\cal{F}_t,\mu)$ and that $f:[0,+\infty]\to[0,+\infty]$ is a ...
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892 views

Running maximum for Geometric Brownian Motion

Can anyone provide the expression and source for the running maximum $M_t$ for geometric Brownian motion $X_t$ as a function of the initial value $X_0$, drift $\mu$ and diffusion $\sigma$? $X_t$ ...
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1answer
335 views

Text on Probability Theory applied to Actuarial Science

I am a senior undergraduate who has passed the first three actuarial exams on probability (P), financial mathematics (FM), and models for financial economics (MFE). I am working on passing the life ...
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1answer
275 views

Is a compensated Poisson process uniformly integrable

Let $(N_t)_t$ be a Poisson process with intensity $\lambda$. Define $$ \bar{N}_t = N_t - \lambda t $$ which is clearly a martingale. My question is: is $\bar{N}$ uniformly integrable? I strongly ...
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832 views

Continuous local martingale of finite variation is constant

Is a continuous local martingale $M$ of finite variation constant? We know that there exists a sequence of stopping times $T_n\nearrow \infty$ a.s. as $n\to\infty$ such that the stopped process $M^{...
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0answers
99 views

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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1answer
44 views

Coupling between two CTMCs

Suppose I have two random processes $X(t)$ and $Y(t)$ starting at time $t=0$ and $X(0)=Y(0)=0$. The processes obey the following transition rates: $$ X(t):\begin{cases} 0\to 1,\text{at rate } A\\ ...
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58 views

Is a local martingale which is nonnegative at a deterministic time, nonnegative.

Assume $M$ is a continuous, local martingale s.t. for a single given $T$ we have $M(T)\geq 0$ and $P(M(T)>0)>0$. Can we then deduce $M(t)\geq 0$ for $t\leq T$? I'm trying to use the good old ...
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1answer
490 views

Proof that the square of a stochastic matrix is stochastic

We know that the square of a stochastic matrix is also stochastic, because the two-step transition matrix of a Markov chain is necessarily stochastic. However, in there another way to independently ...
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1answer
415 views

Quadratic covariation of Itô processes

I haven't found any similar question in the forum, so I trust some of you will find this thought-provoking (at the very least). Perhaps you can help me. Let's consider first the two following ...
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2answers
552 views

Local martingale is locally uniformly integrable martingale?

Is a local martingale locally uniformly integrable martingale ? Here I define a local martingale to be the process with a localizing sequence $\tau_n$ such that the stopped process is martingale. ...
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1answer
363 views

Equivalent defining Markov property

Consider the stochastic process $(X_t)_{t \in \mathbb{R}}$ and show the equivalence of the following two Markov properties: (a) $P(X_t \in A \mid X_u, u \leq s) = P(X_t \in A\mid X_s) \qquad \...
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2answers
43 views

a probability question related to computing the variance of a specific pattern

With respect to a given sequence of points $\{X_1, ... X_t, ...X_n\}$. I can understand why $E[S]= \frac{n-1}{2}$. But how to get that $Var[S]$.
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177 views

Differential equation with random variable

How can I derive analytically or compute numerically the solution to following differential equation $$ dy/dt = y\cdot X\cdot (y\cdot X - g(y,X))\cdot X $$ where X is a random variable (e.g. from a ...
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1answer
80 views

Mathematics of a Simple Counting Game

I wonder how can one think mathematically about the following game: People sit in a circle. One of them says "One!". Then somebody (no matter who - he/she can even be the former person) says "...
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1answer
70 views

Time to completion for a simple catalytic process involving two types of ideal gas molecules

Imagine we have two types of gas molecules diffusing in a chamber - those of type $A$ and those of type $B$. When two gas molecules collide, we have something akin to a simple catalytic reaction ...
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166 views

stochastic Birth model simulation vs deterministic exponential growth not equal

I am trying to simulate a simple birth model, where each birth event increase population by 1. Birth rate is $\lambda$. In equations it looks like this ${dN \over dt}=\lambda N$, then $N=N_0e^{\...
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1answer
26 views

regarding a proof of $\|\theta(e^{i\lambda})\|^2$

When studying the spectral representation of time series, I read the following formula, I am not clear how to prove the second equation. I expand the left side of the second equation with the $\...
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174 views

Comparing the stopping times of two stochastic processes

Let $f_1$, $f_0$, $g_1$, $g_0$ be $4$ distinct density functions on the real numbers $\mathbb{R}$ with the corresponding distribution functions $F_1$, $F_0$, $G_1$, and $G_0$, respectively. The ...
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1answer
792 views

Conditional Expectation of Poisson Process

I have a Poisson Process with stationary and independent increments. Therefore I know: $$P(N_T - N_t = r) = \dfrac{\exp(-\lambda(T-t))(\lambda(T-t))^r}{r!} \mbox{ where } T>t.$$ Now suppose I am ...
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1answer
132 views

A question regarding Markov Chains

Is it possible that we combine some states of a Markov chain, like in this figure? (All non-zero states combined) 1) If yes, what are the new transition probabilities, i.e. p1 and p2 and p3 in the ...
3
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1answer
73 views

A question about extensions of Markov semigroups

I've cross-posted this to MO, if a reply appears on that post I'll update this one. Suppose that $\{T(t)\}_{t\geq 0}$ is a Markov semigroup on the space of continuous bounded functions defined on $\...
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1answer
47 views

how to reformulate general markov property in discrete case

I read the wiki article on the markov property http://en.wikipedia.org/wiki/Markov_property#Definition and wondered how to work out this reformulation. It seems intuitively but I can not work it out. ...
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93 views

Two independent renewal processes

We have two urns (blue and red) that are connected, and two particles, $p_1$ and $p_2$, are traveling between these urns independently. The amount of time $Z_1$ that $p_1$ spends in blue urn is iid ...
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1answer
62 views

Error of apprximation

Have anyone read book "Paul Glasserman Monte Carlo MIFE", it's good, but i'm stuck in chapter 6 page 341 let $$ dX_t=a(X_t)dt+b(X_t)\,dW_t $$ they said that $$ \int_{t}^{t+\Delta t}a(X_{u}) \, ...
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167 views

simplified iid Galton-Watson-process — expectation and variance of population size

Let $Y_{ni}$ be iid and take on values in $\{0,1,2,\ldots\}$. Set $Z_0=1$ and define $Z_n:=\sum_{i=1}^{Z_{n-1}}Y_{ni}$ where by convention the sum is zero if $Z_{n-1}=0$. Let $E(Y_{11}) = \mu$ and $...
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1answer
68 views

Waiting time of two independent processes

Suppose that we have two independent alternating renewal processes such that both alternate between states "0" and "1" independently. The amount of time each of them is in state "1" and state "0" ...
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1answer
109 views

Optimization problem solved via dynamic programming

Consider a situation where decisions are made in stages. The outcome of each decision is not fully predictable but can be anticipated to some extent before the next decision is made. The objective is ...
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1answer
31 views

a representation question on AR(P) process

For AR(P) process, I once read the following property: P corresponds to the projection on the space $\bar{sp}\{x_2,...x_k\}$, which is defined as follows I do not know how does equation (3.4.1) ...
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87 views

Hypothesis testing: find the UMP test

Suppose $X_1,\dots,X_n$ are i.i.d. They are distributed as follows: $P(X_i > x)=(1+x)^{-\lambda}$ where $x\geq 0$ and $\lambda> 0$. I have to test the following hypothesis with level $\alpha_0$; ...
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1answer
108 views

Sum of exponential variables: Time from last arrival till a fixed time

Beginning at 8AM customers arrive at the bank at regular exponential intervals. Vivian arrives at the bank at 11AM and, being a VIP customer, is promoted to the head of the line. How long will she ...
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1answer
213 views

Integration of progressively measurable process

Let $X=\{X_{t},\cal{F}_{t}; 0\leq t<\infty\}$ be a progressively measurable process and $f(t,x):[0,\infty)\times \mathbb{R}^{d}\rightarrow \mathbb{R}$ be a bounded, $\cal{B}([0,\infty))\otimes \cal{...
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1answer
74 views

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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1answer
65 views

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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1answer
168 views

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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2answers
67 views

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.
3
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1answer
131 views

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 ...
3
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1answer
66 views

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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1answer
500 views

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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1answer
176 views

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 ...
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1answer
35 views

find the stochastic differential eqution with ito

I was trying to do some ito problems but I don't grasp how to apply the formula (which is the process). If somebody could give me a hand it would be great! Thanks so much in advance. I have the ...
2
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1answer
78 views

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.} -\...