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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product of martingales bounded in $L^2$

Let $(M_t)_t$ and $(M_t)_t$ be two càdlàg martingales on the same filtered probability space. We know that $M_{\infty}$ and $N_{\infty}$ are orthogonal in $L^2$. Is it true that $(M_t N_t)_t$ is a ...
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84 views

bound on compound poisson process

I am trying to find a "good" bound for $P(C_t \geq a)$, where $C_t$ is the compound poisson process, i.e. $$ C_t = \sum^{N_t}_{i=1} Y_i. $$ $(N_t)_t$ is a homogeneous Poisson process with rate ...
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2answers
395 views

Wald's equation example controversy

I'm trying to get a grip of Wald's equation, applying it to the following example. Suppose, we have a simple sequence of fair coin flips, where heads wins us a dollar, while tails means loss of a ...
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25 views

How to estimate with random system matrix?

If the system is given by $y=Ax+z$ where $z$ is white Gaussian noise and $A$ is a random matrix with i.i.d. distribution with zero mean how can we estimate $x$ from received vector $y$? I tried ...
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136 views

Laws and Moments of two dimensional brownian motions

I am a bit rusty on this. So let us consider the following two dimensional standard Brownian motion issued from zero defined on the probability space $(\Omega, \mathcal{F},\mathbb{P})$ (note that, in ...
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65 views

2-D exponential functional brownian motion

I'm looking for the distribution of $X = \int_0^T e^{-W_t} dt \int_0^T e^{W_t}dt$ and $Y = \frac{\int_0^T e^{-W_t} dt}{ \int_0^T e^{W_t}dt}$ (where $W_t$ is a standard brownian motion) On most ...
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1k views

Markov chains: is “aperiodic + irreducible” equivalent to “regular”?

I have two books on stochastic processes. In one book, it says that the limiting matrix is possible to find if the matrix is regular, that is, if for some $n$ $P^n$ has only positive values. The ...
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2answers
401 views

Is wide sense stationary iff second order stationary?

Wide sense stationary (WSS) process is defined by covariance function being independent of time $E[X(t)X(t+\tau)] = g(\tau)$ and mean is a constant $E[X(t)]=\mu$ where $\mu$ is a constant and $g()$ is ...
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316 views

Probability for asymmetric random walk

How to express this in equation form(in terms of position(x) and time(N)), like the one for symmetric random walk, $\displaystyle P(x,N) = \frac{N!}{(\frac{N+x}{2})! ...
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715 views

Prove the scaling property of a Brownian motion.

I have to prove that $X_t:=c^{-1/2}W_{ct}$, $t\ge0$, where $c>0$ is a constant is a Wiener process. My attempt: 1) $X_0=c^{-1/2}W_0=0$ 2) We know that $(W_t)$ has continuous trajectories. It ...
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73 views

Introduction to stochastic control [closed]

I'm looking for an introductory text on stochastic control. Any suggestions?
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475 views

Wald equality, expectation of a stopping time

Let $(X_n)$ be a sequence of iid random variables such that: $$\mathbb{P}(X_k=-1)=q \\ \mathbb{P}(X_k=1)=p=1-q$$ (two points distribution) Let $\tau$ be the first moment when number of successes ...
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1answer
104 views

How to show that two random proceses have the same family of finite dimensional distributions?

I got two random processes: $$y_t=e_t-\frac{1}{3}e_{t-1},\ e_t\sim\mathcal{N}(0,9)\ \text{i.i.d.}$$ $$y_t=e_t-3e_{t-1},\ e_t\sim\mathcal{N}(0,1)\ \text{i.i.d.}$$ I want to show that both have the ...
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93 views

Is there a version of Slutsky theorem for stochastic process?

To be more specific, if a stochastic process $X_n(t)$ converges weakly to a tight Gaussian process $G(t)$, and another stochastic process $Y_n(t)$ converges uniformly to a deterministic function ...
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1answer
47 views

Is the non-negativity of a potential preserved in the limit?

If $\{ Y_t, 0 \leq t < \infty \}$ is a non-negative supermartingale such that $\lim_{t\rightarrow \infty} E(Y_t) = 0$, is it true that $ Y_{\infty} \geq 0$ a.s.? Note that $\lim_{t \rightarrow ...
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97 views

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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106 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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711 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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310 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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73 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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454 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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483 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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54 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 ...
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68 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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731 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 $\sigma$? $X_t$ ...
3
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1answer
283 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 ...
3
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1answer
190 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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2answers
632 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 ...
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88 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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43 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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56 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
316 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
296 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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445 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. ...
3
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1answer
315 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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167 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 ...
3
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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 ...
2
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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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149 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 ...
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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 ...
2
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0answers
167 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
597 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
119 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 ...
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71 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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82 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 ...
3
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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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140 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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60 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" ...