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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Pathwise integral of $W^{-a}$

Denote by $\tau(x) := \inf \{t \ge 0, W_t=x\},$ where $W_t$ is a Wiener process started at $W_0 = w_0 > 0$ and I would like to show that for any $a>1$ it almost surely holds that ...
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32 views

Transience in a simple Markov chain

Consider the following simple game from a textbook called "Competitive Markov Processes" by Filar & Vrieze (Springer 1996). This is a two player game with two states. In the first state (the ...
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36 views

If a Markov chains converges then the limit is a stationary distribution

Let $p$ be a transition function of a Markov Chain on a countable state $S$ and $i \in S$. Assume for every $j \in S$, $$ \lim_{n\to \infty} p^n(i,j) = \pi(j)$$ Show that $\pi$ is a stationary ...
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41 views

Proof that $p$-th total variation of a brownian motion is $0$ while $p>2$

The p-th total variation is defined as $$|f|_{p,TV}=\sup_{\Pi_n}\lim_{||\Pi_n||\to n}\sum^{n-1}_{i=0}|f(x_{i+1}-f(x_{i})|^p$$ And I know how to calculate the first total variation of the standard ...
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22 views

Does there exist a choice of the parameter $c$ such that the M.C. is time reversible? If so, for what values of $c$? If not, why?

Consider a Markov chain with state space $S = ${$1,2$}. Its transition probabilities are $P_{11} = 0, P_{12} = 1, P_{21} = 1−c, P_{22} = c.$ Does there exist a choice of the parameter $c$ such that ...
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42 views

Why are inter arrival times in the continuous version of discrete-time Markov chains always exponentially distributed?

I am curious whether there exist continuous time Markov processes for which the times between jumping times (which I call inter arrival times) are not exponentially distributed, but have some other ...
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35 views

How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...
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1answer
47 views

Brownian bridge with multiple possible end values

Brownian bridge $Z_t$ is a diffusion process distributed as Brownian motion $B_t$ conditioned on the event $B_1 = 0$. It is rather well-studied, and allows for a Markov-like SDE representation. I ...
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41 views

Why row vectors in stochastic processes?

It seems reasonable to state that column vectors $\mathbf{x}$ are the most frequently seen standard notation, often using $\mathbf{x}^\intercal$ to denote a row vector (transposed column vector). ...
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27 views

Universal localization argument for polynomially bounded functions of a stochastic proceess

As in many mathematical disciplines, many statements about stochastic processes are really easy to prove for bounded objects (sets, function, processes, etc.). This is why many proofs (e.g. for Ito's ...
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34 views

Is there any standard way of analysing this integral?

I have a compound Poisson process $(X_t)$, with jump distribution $F$, which assigns mass only to $(0,\infty)$. In my working I have an expression of the following form: $$ \mathbb{E} \int_0^{\tau} ...
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40 views

Lebesgue Measure of “excursions” of Brownian Motion

I know that the set $S$ where a standard Brownian motion $M:=B[\mathbb{R}]$ attains a strict local minimum is a.s. dense in $\mathbb{R}$. For every point $s \in S$, consider the interval $(s,t)$ such ...
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56 views

Shapely's “Stochastic Games”: An upper bound on each player's gain

In a stochastic game the play proceeds by steps from position to position, according to transition probabilities controlled jointly by the two players. We shall assume a finite number, $N$, of ...
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1answer
15 views

I have a Kolmogorov forward equation with $\lambda_x = x\lambda$ and $\mu_x = x\mu$. How can I prove a particular identity?

This is Problem #12b in Hoel, Port, and Stones' stochastic processes book. I have the forward equation figured out (it's $P'_{xy}(t) = (y-1)\lambda P_{x,y-1}(t) - y(\lambda + \mu)P_{xy}(t) + (y+1)\mu ...
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19 views

Generalization for a Random walk step function on Z?

Consider the random walk on $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$ with transition probabilities $$ p_{i,j}= \begin{cases} p & \text{if } j=i+1,\\ 1-p &\text{if } j=i-1,\\ 0 ...
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19 views

modification process, but with differents paths

Can someone explain me that in every detail pls? It seems obvious, but I don't get it... [Brownian Motion and stochastic calculus, Karatzas (2nd E). Example 1.4] Two processes can be modifications ...
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57 views

Show that $M_n = X_n^2 - n$ is a martingale

Suppose $X_n$ is a symmetric random walk on $\mathbb{Z}$. To show that it is a martingale I need to show $$ \mathbb{E}[M_{n+1}|X_{0:n}] = M_n $$ $$ \begin{align} \mathbb{E}[M_{n+1}|X_{0:n}] &= ...
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19 views

Expectation of a process with stochastic volatility

I would like to compute the conditional expectation of a stochastic process with stochastic volatility. The model is similar to Heston model except here the drift is not constant but an independent ...
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15 views

What is the intuition behind covariation as construct?

I am trying to work through stochastic processes and integrals. It seems covariation as a general concept for functions (let's assume deterministic for now) has the feel of projection. By analogy, ...
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16 views

What are the first two moments of this stochastic process?

The setup. Consider a doubly stochastic Poisson (i.e. Cox) process, which is a Poisson arrival process $X_t$ with stochastic intensity function $\lambda_t$, i.e., a Poisson process whose rate is ...
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45 views

Most General Theory of Stochastic Integration

I've learnt continuous stochastic integration using the classical books: - Revuz & Yor, - Karatzas & Shreve and - Oksendal. Now I want to learn general stochastic integration, i.e. possibly ...
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11 views

What properties of covariance function are violeted for Hurst parameter H>1? and how?

It is known that Hurst parameter H>1 is not possible because properties of covariance function are violated. Exactly what property is violated? What is the detailed proof of this?
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42 views

emails arriving in a Poisson process

Emails arrive according to a Poisson process with rate $λ=2/hour$. You check your inbox (instantly reading all new emails) at time $t=5$ hours and also at some uniformly distributed random time ...
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20 views

Pricing of Binary or Digital Options or Feynman-Kac Equation for $\mathbb E f(X_T)$ with diffusion $X$ and discontinuous function $f$.

I am trying to find references (books, papers, etc.) for calculating $\mathbb E f(X_T)$, where $X_T$ is a diffusion and $f$ is a real function that is not continuous by means of solving a PDE or ...
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19 views

Mean and Variance of an offspring

If I have that the number of offspring of an individual in a population is $0$, $1$, or $2$ with respective probabilities $a>0$, $b>0$ and $c>0$, where $a+b+c=1$, how would I express the mean ...
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1answer
37 views

Ito formula - How to calculate this differential?

Let $W(t)$ be a Brownian motion, how can I calculate the following differential: $$\int_t^T\int_0^t e^{uW(s)}dsdu $$ I do not know how to apply the Ito formula on this problem. Thanks in advance!
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9 views

Proof using Girsanov Theorem

I have parts of proof, but not sure if they are arranged correctly or are sufficient. Is this right? Thing to prove: Given prob space with natural filtration of std Brownian motion, $\hat{W_t} = ...
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1answer
31 views

Electrical components failing in a Poisson process

A machine has infinitely many identical components. They fail according to a Poisson process with rate $\lambda = 4$/hour. A repairman arrives at time $t$ and instantly repairs all of the broken ...
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31 views

Multiclass Markov process

There are two car M/M/1 queues Q1 and Q2. Arrival rate of Red car and Green car in Q1 is $\lambda_{1R}$ and $\lambda_{1G}$ respectively. Similarly arrival rate of red car and green car in Q2 is ...
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19 views

When does the 'expected realization' of a stochastic process exist?

Consider a stochastic process $X_t$ (discrete or continuous time). Under what conditions does there exist a process realization $X^{(\mathbf{t})}=\{X_t\}_{t\in[0,\infty)}$ such that: ...
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67 views

Markov Property Confusion

I feel like I'm being very dense/employing some sort of circular reasoning, but I'm having trouble understanding the Markov Property. According to Durrett (ISBN-10:1461436141), $X_n$ is a Markov chain ...
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55 views

What are some areas of research/industry involving stochastic processes that aren't finance-related?

I've always enjoyed probability and stochastic processes (took two courses in stochastic models in undergrad, and a PhD level intro to stochastic processes course for my master's). Someday I'd like to ...
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35 views

Quadratic Variation and Semimartingales

It is clear that every (I am particularly interested in continuous) semimartingale has a well defined quadratic variation process. However, what can be said about processes that have a well defined ...
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Can Hurst parameter be greater than one? or does nth order fractional brownian motion really exist?

This paper states that there have been some evidences of higher order fractional brownian motion (H>1) http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=917808&tag=1 But in bellow paper it is ...
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1answer
16 views

Shock model conditional expectation

Consider a shock model where $A(t) = \sum _{i=1}^{N(t)}A_i e^{-(\alpha-S_i)}$ where $N(t)$ is distributed as a Poisson, $A_i$ is the amplitude of the shock, distributed as $U(0,5)$, and $S{_i}$ is the ...
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1answer
25 views

4th-moment bound on continuous local martingale

I am struggling with this question: Let $X$ be a continuous local martingale with $X_0=0$, and such that $\mathbb{E} (\langle X \rangle^{p/2}_t) < \infty$, for all $t \geq 0$ and $p \geq 2$. ...
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50 views

Continuity of $t\mapsto \mathbb E X_t$

Is the following statement and its proof correct? Do you know this or related results and where I could read more about such things? Lemma: If $X$ is a right-continuous process, and the collection ...
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75 views

Question about birth and disaster models

We haven't covered birth and death processes yet in class, and my teacher gave us questions to try over spring break. I was hoping to get a head start so I was hoping someone could help me figure ...
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20 views

finite dimensional marginal distribution

I am trying to understand the following theorem: "The distribution of a stochastic process is uniquely determined by the family of all its finite-dimensional marginal distribution (and vice versa)" ...
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Distribution of the increments of a Compound Poisson process

Let $X_t$ be a compound Poisson process defined as $X_t = \sum_{i=1}^{N_t} D_i$, where $D_i$ are i.i.d. and $D_i \sim Exp(\mu)$ and $N_t$ is a Poisson process with parameter $\lambda$. As usual the ...
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1answer
30 views

Help with a simple queueing model question to find transition probabilities?

My prof posted a set of practice questions for our upcoming midterm and I'm trying to work through them (he's not posting solutions for them). He included this question as the most difficult one. A ...
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1answer
25 views

$p$-variation of a continuous local martingale

Here are two interesting statements stated from a book which I do not know how to prove: Let $\{V^{n}_t \}$ be a sequence of processes defined by $$ V^{n}_t = \sum_{k=0}^{\lceil 2^n t \rceil -1} \big| ...
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26 views

Conditional Probability for two independent

Let $X_1$ and $X_2$ be independent geometric random variables having the same parameter $p$. Guess the value of $P\{X_1 = i\mid X_1 + X_2 = n\}$ How do go about making a smart guess for this value? ...
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21 views

Optimum shower occupied retry delay

I cycle to work in an office with a single shower. When the shower is occupied I set my stopwatch for 10 minutes. When the stopwatch expires I trek back to the shower and try again. Sometimes it is ...
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1answer
43 views

Showing that a right-continuous process does not satisfy Kolmogorov's Continuity Criterion

As part of a larger exercise, we've been tasked to show that the following right-continuous stochastic process $X$ does not satisfy the requirements of Kolmogorov's Continuity Criterion. Let $\Omega ...
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59 views

Localisation in the proof of Ito's formula

I am reading Karatza's and Schreve's book "Stochastic Calculus and Brownian Motion" and I don't understand a strange thing as follows: Let $X=X_0 + A +M $ be a semimartingale, where $A$ is a ...
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71 views

Multiple absorbing boundaries

I am interested in the relation between absorbing boundaries and the trajectories of particles (evolving according to a Brownian motion). The probability to hit a boundary at a given time can be ...
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1answer
28 views

Representation of a homogeneous Poisson process

Given a homogeneous Poisson process $\{N(t)\}_{t\ge 0}$ with intensity rate $\lambda>0$. Does someone know how to prove that there is a sequence of i.i.d. $Exp(\lambda)$-distributed random ...
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1answer
37 views

Fubini's theorem for conditional expectations

I need to prove that if $E \int_a^b |X_u|\,du = \int_a^b E|X_u|\,du$ if finite then: $$E\left[\left.\int_a^b X_u\,du \;\right|\; \mathcal{G}\right] = \int_a^b E[X_u \mid \mathcal{G}]\,du.$$ I just ...
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Determinability

If $X$ and $Y$ are random variables taking values in measurable spaces $(E,\mathcal{E})$ and $(D,\mathcal{D})$ respectively, then we say that $X$ determines $Y$ if $Y=f\circ X$ for some measurable ...