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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A problem in Poisson Processes

Let $X_n$ be the interarrival times for a Poisson process $\{N_t; t \geq 0\}$ with rate $\lambda$. Is it possible to calculate the probability $P\{ X_k \leq T \text{ for } k \le n, \sum_{k=1}^{n}{X_k} ...
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Measurability question for martingale (Is $S(X_i, Z_i)- E(S(X_i,Z_i) \mid \mathcal{F}_i)$ $\mathcal{F}_i=\{X_1, \dots, X_i \}$-measurable?)

One condition for a martingale $M_k$ with a general filtration $\mathcal{F}_k$ is that the involved random variables $M_k$ are $\mathcal{F}_k$-measurable. Now I have $M_n=Y_1+\dots +Y_n$ and ...
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Is “a fair coin being tossed $n$ times” the same as “$n$ fair coins being tossed once”?

This is possibly a follow-up question to this one: different probability spaces for $P(X=k)=\binom{n}{k}p^k\big(1-p\big)^{ n-k}$? Consider the two models in the title: a fair coin being tossed $n$ ...
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Lower semicontinuity of the indicator function: stochastic processes

Let $X$ be a Markov process given on a metric space $\mathcal X$ by a transition semigroup $P_t$ acting on $\mathbb B(\mathcal X)$ - the set of all bounded and Borel measurable functions. Such a ...
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Examples: invariant events

In a couple of books I'm reading chapters devoted to the Ergodic theory. As a setting: $(\Omega,\mathcal F,\mathsf P)$ is a probability space, $X:(\Omega,\mathcal F)\to (S,\mathcal S)$ is random ...
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Two identities in probability

I ma reading the book An introduction to stochastic processes in physics by Don Stephen Lemons. I have a question on two identities. One identity is the identity (B.3) in Appendix b page 102. How ...
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475 views

Markov chain with uncountable state space

I'm self-studying probability theory and struggling with understanding Markov chains on uncountable state spaces, notably I would like to solve the following exercise from this book. ...
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1k views

Use stochastic calculus (Ito's lemma) to compute the expectation

Calculate $E[\cos(X)e^X]$, where $X\sim N(0,\sigma^2)$. Use stochastic calculus instead of integrating w.r.t the normal density. During the discussion with friends, we believe that we should use ...
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256 views

Are hitting times of Brownian motion independent?

Suppose that $B_t$ is a standard Brownian motion. And $T_a$, $T_b$ are the hitting time whereas $a<0$, $b>0$. Then are these two random variables independent?
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Observable and unobservable parameters of stochastic processes

Consider the following diffusion process $$ dX_t = \mu\,dt+\sigma(t,X_t)\,dW_t $$ where $X,W$ are 1-dimensional and. Is it true that given a history $(X_s,s\leq t)$ for each $s< t$ one can find ...
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Conditions for Stationary Distributions in Markov Chains?

The book by Durrett "Essentials on Stochastic Processes" states on page 55 that: If the state space S is finite then there is at least on stationary distribution. How can I find the ...
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Distribution after hitting

Let $X$ be a real-valued Markov process, $$ \mathsf P_x\{X_1\in A\} = K(x,A) $$ is its transition kernel. Let $\tau = \inf\{n\geq 0:X_n\geq a\}$ be the first hitting time of the level $a$. I am ...
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488 views

Proving that a process is a Brownian motion

Let $B$ be a Brownian motion with natural filtration $(\mathcal{F}_t)_{t\geq 0}$ and let $\mathcal{H}_t$ be the $\sigma$-algebra generated by $\mathcal{F}_t$ and $B_1$. Define $$A_t = ...
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226 views

Application of Panjer recursion scheme

I'm writing my bachelor (the argument is the Compound Poisson Process applied to insurance) and I need an example to complete it. I need an application of Panjer recursion scheme (for example ...
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404 views

Applications of algebra and/or topology to stochastic (or Markov) processes

Some time back I was reading a PDF about algebra or topology (or algebraic topology, I forget which) and found an extremely enlightening section about an application to stochastic processes. ...
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677 views

How to show Martingale property for sum of $S_k-E(S_k)$-summands where $S_k$ is a function of two RV's

EDIT: new formulation of the question (old version below). In a paper I found the statement that a certain sum $M_n =Y_1+\dotsb Y_n$ is a martingale, $Y_i=f (X_k, Z_k) - E ( f(X_k, Z_k) | X_k)$. (The ...
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Markov processes driven by the noise

Let $\xi_n\in \Xi$ be a sequence of iid random variables with $n \in\mathbb N\cup\{0\}$, which we call a noise process. Construct a process $$ Z_{n+1} = f(Z_n,\xi_n)\quad(\star) $$ with $Z_0\in E$ ...
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Convergence of Brownian integral

Let $B$ be a Brownian motion. I'm trying to show that $$\left(\int_0^te^{B_s}ds\right)^\frac{1}{\sqrt{t}}$$ convergences in distribution as $t \to \infty$. As a hint, we are told to consider the ...
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166 views

Which courses before Stochastics?

I would like to know which maths course I need to take before studying stochastics. Thx for helping, Stephane
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Is $M^2-[M]$ a local martingale when $M$ is a local martingale?

I've learned that for each continuous local martingale $M$, there's a unique continuous adapted non-decreasing process $[M]$ such that $M^2-[M]$ is a continuous local martingale. For a local ...
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Second eigenvalue of a stochastic block matrix

Considering a stochastic block matrix in the form of, \begin{equation} \textbf{$P_{}$} = \left( {\begin{array}{cc} \textbf{$A_{}$} & \textbf{$B_{}$}~; \ \textbf{$B_{}$} & \textbf{$A_{}$} ...
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How can I prove this random process to be Standard Brownian Motion?

$B_t,t\ge 0$ is a standard Brownian Motion. Then define $X(t)=e^{t/2}B_{1-e^{-t}}$ and $Y_t=X_t-\frac{1}{2}\int_0^t X_u du$. The question is to show that $Y_t, t\ge 0$ is a standard Brownian Motion. ...
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580 views

Characterizing previsible processes

I'm having trouble understanding the concept of a previsible process in continuous time, so I'm asking this question to get a better idea of what it means for a process to be previsible. (In what ...
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Method for Constructing Poisson Processes

I'm writing a bachelor thesis about Poisson Processes, and I need a method for the construction of these processes. I know that I can construct them defining inter-arrival times with the exponential ...
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Considering Brownian bridge as conditioned Brownian motion

Let $B$ be a standard Brownian motion. Define a Brownian bridge $b$ by $b_t=B_t-tB_1$. Let $\mathbb{W'}$ be the law of this process. According to Wikipedia, A Brownian bridge is a ...
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403 views

Applications of Compound Poisson Processes

I'm reading the book Non-Life Insurance Mathematics, an introduction with Stochastic Processes by Thomas Mikosch and I'm interested in applications of the Cramer-Lundberg Process to concrete examples ...
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289 views

Brownian motion introduction

I didn't get any answers to my previous question; so I am trying a different tack. I am familiar with a first course in probability theory using measure theory, to the extent of proving the Central ...
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458 views

Process with Markov property but not strong Markov property

I'm trying to find a simple example of a stochastic process with the Markov property, but not the strong Markov property, to give me an intuitive understanding of the distinction between them. All ...
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138 views

Central limit theorems, Almost sure invariance principles and Brownian motion

In a paper I was reading on dynamics, I came across a proof of a central limit theorem in a certain situation using brownian motion and an almost sure invariance principle. I am not very experienced ...
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Classifying a Stochastic Process as transient, null-recurrent or positive-recurrent

For a discrete time Markov chain with state-space the non-negative integers, for $j>0$, $$ p_{j,k} = \begin{cases} p/j & \text{for } k = j+1 \\ 1 - 1/j & \text{for }k=j \\ (1-p)/j & ...
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Hilbert transform of white noise

What is the Hilbert transform of a white noise $\xi(t)$? By the Hilbert transform I mean: http://mathworld.wolfram.com/HilbertTransform.html Thank you.
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Support of family of measures

Let $X$ be a topological space with a Borel $\sigma$-algebra $\mathcal B(X)$. There is a family of probability measure on $X$, which is denoted as $P:X\times \mathcal B(X)\to[0,1]$. I would like to ...
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Reaction-diffusion equations and stochastic processes

The solution to the Fokker-Planck equation can be thought of as a macroscopic description of the dynamics of a diffusion process. Various results make this heuristic more precise - Ito integration, ...
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335 views

$\psi$-irreducibility of Markov Chains

In the book "Markov Chains and Stochastic stability" by Meyn and Tweedie the measurable space $(\mathsf X,\mathcal{B}(\mathsf X))$ is said to be $\varphi$-irreducible for a Markov Chain $X$ if there ...
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817 views

Brownian motion - characteristic function

Let me remind first the construction of Brownian motion. Fix a vector $x \in \mathbb{R}^n$ and define $p(t,x,y) := (2\pi t )^{-\frac{n}{2}} \cdot \exp{\left( - \frac{|x-y|^2}{2t} \right)},$ for $y ...
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Quadratic variation of Brownian motion and almost-sure convergence

Say that $W(t)$ is a Brownian motion. The quadratic variation $[W,W](t)$ is defined in terms of a partition $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = t\}$ by $$ \begin{split} [W,W](t) &= ...
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396 views

Convergence in mean square from expected value/variance

I'm looking for a proof of the following statement: Given a sequence of independent random variables $X_n$ satisfying $$ \lim_{n\to \infty} E[X_n] = T, $$ where T is a constant, then $$ \lim_{n\to ...
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1answer
134 views

Continuous excessive (supermartingale) function

Consider a discrete time Feller Markov process $X$ on $\mathbb R$ with a kernel $K(x,dy) = \xi(x,y)dy$ and the transition operator $$ \mathcal Pf(x) = \int\limits_{\mathbb R}f(y)\xi(x,y)\,dy. $$ ...
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probability terminology for parameter in a Markov process

Suppose $$P(\text{feature present at time} \ t \ \text{and} \ t+\Delta t) = \beta^{2}+\beta(1-\beta) \exp(\Delta t/\tau)$$ where $\tau = 1/(\pi_{01}+\pi_{10})$. What is $\tau$?
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An example of Markov-Feller chain with some properties

Let $X$ be a Polish space and $C(X)$ denote the space of all bounded and continuous functions on $X$. We consider a Markov chain $(\xi_n)_{n\geq 0}$ with transition probability $P:X\times ...
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Different definitions of e-property for Markov-Feller chains

Let $X$ be a Polish space. We consider a stochastic kernel $P:X \times \mathcal{B}_X \to [0,1]$ and the Markov semigroup $(P^{\;n})_{n\geq1}$ of iterations of $P$, which satisfy the Chapman–Kolmogorov ...
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411 views

How to show martingale is bounded in $L^1$?

Fix a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_n)_{n\geq 0},\mathbb{P})$ and an $L^1$-bounded submartingale $X_n$. We can show that, for $n\geq 0$, the sequence $(\mathbb ...
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588 views

Brownian hitting time of a _very_ simple linear boundary

I realize that general results on the hitting times of a curve are practically nonexistant, but I am hoping that someone can string together a sequence of tricks to tell me what $$ \Pr\left( ...
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Question about non-positive Stochastic processes

This question will be a little out-of-character for me. I'm reading an evolutionary game theory book (which isn't for mathematicians), and I'm not sure of the mathematics involved. My definition of a ...
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230 views

dynamic mean: measurement of randomly distributed events

Aim is to estimate an error on a stochastic event rate. I read out the event counter second-wise, every black $1$ is a counted event (new events over time, see the plot below). During the measurement ...
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225 views

Computing quadratic variation for stable Levy flights with $0<\alpha<2$?

The wiki page on semi-martingales states that Every Lévy process is a semimartingale. and that The quadratic variation exists for every semimartingale. Let $X_t$ be a stable Levy process ...
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82 views

Feller continuity of the stochastic kernel: compact set

This question is an extension of Feller continuity of the stochastic kernel. Nate Eldredge provided a nice counterexample, but I failed trying to extend it to the compact set $B$. The setting is the ...
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105 views

Examples when Resonance Overlap fails to predict the onset of Chaos

In a Hamiltonian system Chirikov's resonance overlap criterion approximately predicts the onset of chaotic behavior. Furthermore in a system where resonances overlap, the strengths of the resonances ...
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Feller continuity of the stochastic kernel

Given a metric space $X$ with a Borel sigma-algebra, the stochastic kernel $K(x,B)$ is such that $x\mapsto K(x,B)$ is a measurable function and a $B\mapsto K(x,B)$ is a probability measure on $X$ for ...
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How to model server load?

I want to model the server under load. I'm using following assumptions: The server serves only one request at a time, and all requests take him exactly 100ms to process. All requests that came ...