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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83 views

Are $L$-diffusions unique in law?

I've been trying to understand diffusions. We can show they exist by noting they solve particular SDEs, but are they unique? More precisely: Fix a filtered probability space satisfying the usual ...
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2answers
279 views

Question about an exercise in Revuz/Yor

I'm solving exercise 2.28 in Revuz/Yor. I was able to prove 1). Unfortunately at 2) I got stuck. I have to show: Let $B$ be a d-dimensional Brownian motion and $A\in \mathcal{A}:=\cap_t ...
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1answer
72 views

Computing the joint distribution of a 1-dependent series

Let $F(u,v)$ be a continuous bivariate cdf, $F(u|v)$ the conditional distribution of U given V=v and $F^{-1}(s|v)$ the corresponding inverse conditional distribution. Let $G$ be a continuous cdf and ...
2
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2answers
238 views

Convergence of $\frac{a_n}{n}$ where $a_0=1$ and $a_n=a_{\frac{n}{2}}+a_{\frac{n}{3}}+a_{\frac{n}{6}}$

Given $a_0=1$ and:$$a_n=a_{\frac{n}{2}}+a_{\frac{n}{3}}+a_{\frac{n}{6}}$$Find convergence or divergence of $\frac{a_n}{n}$. Such a weird problem; I don't know how to attack it. I'm also fairly ...
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1k views

On hitting time of Brownian motion and Ito's lemma

I have two possibly related questions. Let $\tau:=\min\{t\geq0:B_t=1\}$, where $B_t$ is a standard Brownian motion. I am supposed to derive the fact that $\mathbf{E}\tau=\infty$ by applying some ...
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0answers
72 views

stationary process is invariant under sliding time

$(\Omega,\Im,P) $ is a probability space and $\xi(t)\equiv\xi(w,t)$ is a stochastic process which is defined on $\Omega\times T$ ,$T=[0,\infty)$. If for every ...
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2answers
418 views

Markov process and transition semigroup

I read in my lecture notes an introduction chapter about Markov process. Here is the setting: Let $(\Omega,\mathcal{F},P)$ a probability space, $(S,\mathcal{S})$ a measurable space and $X=(X_t)$ a ...
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1answer
161 views

Simulation of diffusion processes on the canonical space $C([0,t],\mathbb{R})$

I'm currently reading the article "Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes" by Beskos, Papaspiliopoulos, Roberts and Fearnhead. I'm ...
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1answer
497 views

Stopping theorem for continuous martingale

I've a question about a proof in my lecture notes. We want to prove the following theorem. $M=(M_t)_{t\ge 0}$ be a $(P,F)$-martingale, where $P$ is a probability measure and $F=(\mathcal{F}_t)$ a ...
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1answer
732 views

Independent increments of Brownian Motion

Suppose we have the $(W_t)$ Brownian Motion and the filtration $F=(\mathcal{F}_t)$, where $\mathcal{F}_t:=\sigma(W_s;s\le t)$. I know that for any $n\in \mathbb{N}$ and $0\le ...
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1answer
308 views

Convergence of quadratic variation of Ito processes

I need to find an example of an Ito process $X=\{X_t:t\in[0,T]\}$ with non-zero Ito integral part and a sequence of Ito processes $\{X_n\}$ such that $X_n$ converges uniformly to $X$, as ...
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0answers
483 views

Martingale transform question

I was reading my notes and I was having trouble understanding theorem 4.3 below. I understand essentially what it is saying, but to me its simplying stating something rather intuitive? That given ...
1
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1answer
82 views

Is this question related to Poisson process?

Consider a computer system which employs two copies $A$ and $B$ of some chip. A chip $C$ on reserve is used to replace either $A$ or $B$ whichever fails first. What is the probability that $A$ is ...
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1answer
175 views

How to find the z-transform of a given quantity?

The arrival of messages to a communications channel is modeled as a Poisson process, with rate $\lambda$ messages / unit time. Let $\{N(t), t \geq 0\}$ denote that process. Each message contains a ...
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3answers
502 views

If conditional expectation $E[Y|X]$ is constant.

It is well-known fact, that if $X,Y$ are independent, integrable random variables then $E[Y|X]=E[Y]$. Next assume that $Y$ is centered and $E[Y|X]=0$. What reasonable conclusions can be made about the ...
3
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2answers
899 views

How to prove this property in a Poisson process?

For a Poisson process show, for $s < t$, that $P(N(s)=k|N(t)=n) = \binom{n}{k} (\frac{s}{t})^k (1-\frac{s}{t})^{n-k}$
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148 views

Uniqueness of a local martingale problem

First, some notation: let $X = (X_{t})_{t\geq 0}$ be some strong Markov process in $E = \mathbb R^n$ with cadlag paths. Let us denote by $P_t$ the transition semigroup of $X$ and by $\mathbb B$ the ...
5
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1answer
328 views

Right continuous version of a martingale

This is an exercise in chapter 2 of the book "Continuous Martingales and Brownian Motion" by Revuz and Yor: Consider the probability space $([0,1], \mathcal{B}([0,1]), dx)$, where $dx$ denotes ...
2
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2answers
235 views

Martingales, finite expectation

I have some uncertainties about one of the requirements for martingale, i.e. showing that $\mathbb{E}|X_n|<\infty,n=0,1,\dots$ when $(X_n,n\geq 0)$ is some stochastic process. In particularly, in ...
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0answers
919 views

Monotone class theorem

I have some question about the Monotone Class Theorem and its application. First I state the Theorem: Let $\mathcal{M}:=\{f_\alpha; \alpha \in J\}$ be a set of bounded functions, such that ...
4
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1answer
406 views

How to make this heuristic extension of Itô-Tanaka's formula valid

Here is my story, I have the following function : $$ g(x)=(1+x)\cdot\exp\left(-\frac{(\log(x+a)+c)^2}{2\sigma^2}\right)1[x\ge y]=f(x)\cdot1[x\ge y] $$ with $a,c,\sigma$ being "good" reals so that ...
4
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2answers
195 views

Stratonovich SDE coefficient selection

Is it possible to find a strictly positive function $\sigma:\mathbb{R}\to\mathbb{R}$, such that a solution $X_t$ to an SDE $$dX_t=-X_tdt+\sigma(X_t)\circ dB_t,$$ with $X_0$ being arbitrary, is a ...
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1answer
242 views

Differential form of “random walk with reset” based on Wiener process

Assume such a "random walk with reset" X(t) is defined based on Wiener process (GBM) ...
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228 views

Random walk with reset?

Is there such a random walk, that "good times" it just looks like a random walk, while when "bad moment" comes, it will reset => jump to zero, afterwards continue doing random walk again? Thanks!
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372 views

Martingale convergence almost surely

Let $(X_i)$ be a sequence of r.v. and define $Y_i^n:=X_i\mathbf1\{|X_i|\le n\}$ and $Z_i:=T^i(Y_i^i)$, where $T^i$ is defined as $$T^i(X) :=\sum_{l=-\infty}^\infty ...
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1answer
205 views

Alternating renewal process

A machine breaks down repeatedly and after each breakdown it takes a length of Y_n to repair the machine. It then runs for a period of Z_n before breaking down again. If N(t) is a renewal process ...
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80 views

flipping $k$ out of $n$ coins - $\sigma$-algebras

I'm stuck with a part of a homework problem and need some clarification. Let $\Omega$ be the sample space for flipping $n$ fair coins, i.e. the set of all $n$-tuples of $E$ and $N$, eagle and number, ...
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1answer
394 views

Renewal process with geometric interarrival times

How would I go about determining the renewal function, $m(n)$, for a general $n$, if the interarrival times, $X_i$ are geometrically distributed with $P(X_i = k) = p \cdot (1-p)^{k-1}$. I believe I ...
2
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1answer
349 views

Understanding the Kesten Multiplicative Process

I read in D. Sornette's Critical Phenomena in Natural Sciences about the Kesten Multiplicative process: $$X_{n+1} = a_n X_n + b_n$$ Where $a_n$ and $b_n$ are stochastic variables drawn from the pdfs ...
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342 views

Problem of cumulants

I take the problem of cumulants to be this: given a sequence $(\kappa_1,\kappa_2,\kappa_3,\ldots)$, is it the sequence of cumulants of some probability distribution? In one sense, this is trivially ...
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1answer
206 views

Poisson Process (Renewal) Question

I am having difficulty with the following problem. I tried conditioning on T_n but I am unsure how to proceed with that conditional expectation. Thanks for the help!
2
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1answer
297 views

Renewal process (excess / current life)

I am self-studying renewal processes and I came across an interesting problem. How would one go about finding the expectation of excess life given that current life is equal to x? I am assuming in ...
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2answers
1k views

Random walk on n-cycle

For a graph G, let W be the (random) vertex occupied at the first time the random walk has visited every vertex. That is, W is the last new vertex to be visited by the random walk. Prove the following ...
3
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1answer
124 views

stochastic analysis problem

Suppose $X$ and $Y$ are Ito processes, $X_t=x+\int^t_0Y_sdB_s$ and $Y_t=y-\int^t_0X_sdB_s,\ t\geq 0$, here $B$ is a standard Brownian motion. I need to prove that ...
2
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1answer
464 views

Expected time of mouse's survival (stochastic matrix)

In the following wikipedia page explaining stochastic matrices, there is an example with 5 boxes and a cat and a mouse where they jump to a left or right box at every turn and it explains how to ...
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2answers
949 views

Brownian bridge expression for a Brownian motion

Let $B_t$ be a standard Brownian motion in $\mathbb R$, then the Brownian bridge on $[0,1]$ is defined as $$ Y_t = a(1-t)+bt+(1-t)\int\limits_0^t\frac{\mathrm dB_s}{1-s} $$ for $0\leq t<1$. Here ...
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0answers
82 views

Scale invariance and $1/f^2$ power spectrum

In the paper Occlusion Models for Natural Images : A Statistical Study of a Scale-Invariant Dead Leaves Model; Lee, A. B. Mumford, D. B. Huang, J.; International Journal of Computer Vision I read ...
4
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1answer
171 views

Simple stochastic integral

Let $(B_1,B_2)$ be a two-dimensional Brownian motion. Let $$ X_t = \int\limits_0^t B_1(s)\mathrm \; dB_2(s). $$ Is there a closed form for $X$ or the integral above is all one can get?
2
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1answer
112 views

MA and AR process stationarity intuition

$y_{MA}$ = $ε_t$ + $ε_{t-1}$ <- stationary $y_{AR}$ = $ε_t$ + $y_{AR_{t-1}}$ $y_{AR_{t-1}}$ = $ε_t$ + $ε_{t-1}$ + $y_{AR_{t-2}}$ $y_{AR_{t-2}}$ = $ε_t$ + $ε_{t-1}$ + $ε_{t-2}$ + ...
2
votes
1answer
130 views

question with stopping times

Suppose we define for $A\in \mathcal{B}(\mathbb{R^n})$ the first hitting time $$ T_A:= \inf\{t\ge 0;X_t(\omega)\in A\}$$ where $X=(X_t)$ is a stochastic process, adapted to a Filtration and with ...
2
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1answer
241 views

M/G/1/K - evalutate birth and death rates

Within a queue with capacity = K and exponential interarrival times, death rate is μ and birth rate λ. A packet is discarded when the queue is full. When the source is active there's a probability ...
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1answer
220 views

Pointwise convergence almost surely of an Approximation sequence

Unfortunately I've trouble to see the following: If you work with stochastic process $X$ you often want to approximate this in the following sense, define: $$ X^{(n)}(s,\omega) = ...
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1answer
209 views

Reference request for Optimal Stopping (Stochastic Analysis)

I would like to start and get into the habit of reading some publications in different areas of mathematics, to get used to the writing style / mathematical sophistication etc. that is expected. In ...
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5answers
7k views

What's the difference between stochastic and random?

What's the difference between stochastic and random? I've read in the portuguese wikipedia that there's a difference, but I still didn't see this point on english wikipedia.
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314 views

Distribution of Maximum of Sum of Sum of Gaussians

Let $X_i$ be a sequence of i.i.d. standard normal random variables. Let $Y_i=\sum_{k=1}^iX_k$ and $Z_i=\sum_{k=1}^iY_k$. I am interested in upper and lower bounds for $P(\sup_{1\leq i\leq m}|X_i|\leq ...
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1answer
208 views

Growth condition for Ito diffusions

Given a one-dimensional SDE $$ \begin{cases} dX_t &= b(t,X_t)dt+\sigma(t,X_t)dB_t, \\ X_0 &= Z \end{cases} $$ for $t\in[0,T]$ where $Z$ is square integrable: $\mathsf E[Z^2]<\infty$ ...
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1answer
444 views

Quadratic Variation of Sum of Local Martingales

I have a question about calculating covariances of local martingales. Suppose $M_1$ and $M_2$ are local martingales. Put $M = M_1+M_2$. Is there a nice way to calculation $[M]$ in terms of $[M_1]$ and ...
4
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1answer
4k views

Martingale that is not a Markov process

1. On the internet, it is suggested that $$ X_t=\left(\int_0^t X_s \;ds\right)\;dW_{t} $$ is a martingale, but not a Markov process. I understand that the process $$ I_t(C)=\int_0^t C_s \; dW_s$$ ...
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2answers
648 views

Proof of Markov property for Ehrenfest urn

[the question got downvoted on MO with the recommendation to ask here] In many books Ehrenfest Urn is used as an example of a homogeneous Markov chain, where entries in transition probabilities ...
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vote
1answer
886 views

Proving a process is Markov chain

Could anyone give me an example of a problem where it is requested to prove rather than assume that a stochastic process forms a Markov chain. I can think of something like this: if $X_{n+1} = X_{n} + ...