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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Where is the assumption of right continuity used in the following proof?

Lemma:If $X$ be a right-continuous positive local martingale then , $X$ is a generalized super martingale Proof: $\forall s<t$ $$E[X_t\mid F_s]=E[\lim_{n\to\infty} X_{t \wedge\tau_n}\mid F_s] \leq ...
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43 views

Birth immigration process

I'm having some problem with this question. A model for the distribution of the number of goals scored in soccer matches suggests that if n goals have already been scored by time t, then the ...
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21 views

If a stochastic process follows Geometric Brownian Motion, does it imply that it is Log-normally distributed and vice-versa?

This might be a naive question, but it doesn't stop haunting me. Wiki page for GBM writes the SDE for GBM process and shows it follows log-normal distribution. Is it true every time or are there any ...
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1answer
36 views

A die whose score cannot be as before (Markov chains)

A die is "fixed" so that each time it is rolled the score cannot be the same as the preceding score, all other scores having probability $1/5$. Given that the first score is 6, what is the probability ...
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20 views

Translated stochastic process

Let $M$ be a (compact) Riemannian manifold and let $L$ be some second-order elliptic operator on $M$. Now for a vector field $v$, I can consider the flow $\Psi_t$ of $v$ and consider the following ...
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93 views

Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let ...
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1answer
33 views

Cellular automata (Random walk)

Here is the context of my question below. I cite from "Some Rigorous Results for the Greenberg-Hastings Model" by Richard Durrett- Consider the following cellular automaton known as the ...
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59 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
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1answer
27 views

Marginal probability density function of Stochastic process

I was solving the following question and I derived the Auto correlation function and proved that it is a WSS process. However, I am not sure how to go about finding the Marginal probability density ...
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6 views

Stochastic Process with mean reverting property

Here I am seeking for a definition of what kind of stochastic processes are called mean reverting stochastic process. That is, what are the properties that a stochastic process should obey in order to ...
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23 views

Usual augmentation filtration? (Sigma algebra generated by a descreasing family of sets)?

My aime is to understand the usual augmentation filtration. More pricesely, I want to understand the last identity in this PDF file. http://onlinelibrary.wiley.com/doi/10.1002/0470863617.app1/pdf ...
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1answer
28 views

Finding mean and variance of stochastic process

If I'm given a Stochastic Process Xt that satisfies a stochastic diff. equation, let's say fXt, what is the formula to find the mean and variance of Xt? I think it's: $mean = dE(X_t) = dX_0e^t$ ...
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1answer
32 views

Algebra and partions of a set

My book in mathematical finance introduces algebras and partitions of a set, in order to explain how information is modeled to the investor. But there is one thing I don't get. They say that for every ...
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2answers
49 views

quadratic variations of Brownian motion squared

I'm trying to refresh my memories about stochastic processes. We know that Brownian motion has as quadratic variation equals to t. What is the quadratic variation of the Brownian motion squared ? ...
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1answer
31 views

I want to find Sigma-field generated by $X=5I_{\{1,3,7\}}+6I_{\{4,5\}}+7I_{\{8,9\}}$. with follow condition. [closed]

Let $\Omega=\{1,\dots,10\}$ and $\mathcal F= 2^\Omega$ and we have $$P(\{1\})=P(\{3\})=P(\{7\})=\frac{1}{15}$$ $$P(\{2\})=P(\{6\})=P(\{10\})=\frac{1}{6}$$ $$P(\{4\})=P(\{5\})=\frac{1}{20}$$ ...
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28 views

Monte Carlo Markov Chain simulation

I am going to post the python code logic we used however I want someone to look at the number that are printing out. The Markov chain is uniformly distributed across all $50x50$ matrices with entries ...
3
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1answer
53 views

Recurrence/Transience of random walk with +2/-1 steps

Consider the Markov chain with state space $S=(0,1,2,...)$ and transition probabilities: $p(x,x+2)=p$ , $p(x,x-1)=1-p$, $\forall$ $x>0$. $p(0,2)=p$ , $p(0,0)=1-p$. For which values of $p$ is this ...
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3answers
47 views

an exercise about mean and probability

Let $(\Omega, \mathcal{F}, P)$ be a probability space, $X : \Omega \rightarrow \mathbb{R}$ be a discrete random variable and $$\phi : [0, \infty) \rightarrow (0, \infty)$$ an increasing function (so ...
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1answer
35 views

$E[1_{\lbrace P_T-P_{\tau_n}=0\rbrace}\int_{\tau_n}^T h(s)dN_s]=0?$

If $P_t$ is a standard Poisson process, and $N_t=P_t-t$ the associated martingale then $\int_0^t h(s)dN_s$ is a martingale (assuming that h satisfies the neccessary hypothesis). Thus, considering ...
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1answer
44 views

Inequality for the expected values of norm of stochastic processes

Let $\underline{X}=(x_1, x_2, x_3), \; x_i \sim \mathcal{N(0,1)}$ i.i.d. For any fixed $t>0$ and $\underline{X}_0$ prove that the following holds ($\Vert\cdot\Vert$ is the Euclidean norm): ...
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39 views

Optional Sampling Theorem Application

Let x, y > 0. Define the first passage time of a Brownian motion $W_t$ as $\tau_a$ = min{t $\ge$ 0: $W_t$ = a}. I need to show that $$E[e^{-u\tau_x}1_{\tau_x < \tau_{-y}}] = ...
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17 views

equality involving the arrival times of a poisson process

Let $P_t$ be a Poisson process with arrival times $\tau_1,\tau_2,\dots$ and $h$ a bounded function, $F$ a square integrable function of the arrival times of $P_t$ until the time T . I am wondering if ...
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1answer
28 views

distribution of $\sup\limits_{0\le t \le 1}|W(t)|$

My prof on class told us that distribution of $S=\sup\limits_{0\le t \le 1}|W(t)|$ has been well studied, where $W$ is a Wiener process, but I need a table to find $c$ such that $P(S < c) = 0.95$. ...
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31 views

Bivariate stopped processes

Take two dependent Levy processes $L_1(t)$ and $L_2(t)$ with law $\mathcal{L}(L_1(1),L_2(1)$. If we stop the first process at a general time $t=s_1$ and stop the second process at another general time ...
2
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1answer
24 views

Showing that if $B_t$ is a Brownian motion then $t B_{1/t}$ is Gaussian

I want to show that if $B_t$ is a Brownian motion then $t B_{1/t}$ is a Gaussian process, i.e. that it has increments which have the normal distribution. It seems like a trivial fact, since the ...
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1answer
46 views

Stochastic differential of Bessel process [closed]

Let $ \underline{B}_{t}=(B_1(t), \dots, B_d(t))$ be a $d$-dimensional Brownian motion. How to calculate the stochastic differential of $ \Vert{\underline{B}_t}\Vert$? $\Vert . \Vert$ denotes the ...
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1answer
41 views

mean of random variables

Let $(\Omega,\mathcal{F}, \mathbb{P})$ be a probability space, $X : \Omega \rightarrow \mathbb{R}$ a discrete random variable and $g : \mathbb{R} \rightarrow \mathbb{R}$ a random variable. I can't ...
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0answers
20 views

Stochastic scheduling to maximize the expected number of customers arrived at the root of a Jackson tree

In a Jackson network, organized as a tree rooted at queue r, several customers are queued at time 0 and there is no new customer arrival. The service time of each customer in queue i is geometrically ...
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1answer
27 views

Probability of Wiener process hitting a particular point at an independent stopping time

Assume we have a stopping time $T$ that is independent of a Wiener process $W$. If $T$ were taking discrete values (let's say in $\mathbb{N}_0$), one can easily show (using the independence and the ...
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21 views

integral involving wiener process

Suppose $W_t$ is standard Brownian motion and define $$ R(x,y) = \int_{0}^{T} W_{t+x}\,W_{t+y}\,dt, $$ which is sort of the sample covariance function. What is the distribution of $R(x,y)$?
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20 views

Level sets of a Wiener process

Assume we have a Wiener process $W$ starting at $W_0=0$. What can one tell about the Lebesgue measure of "level sets" $A_y = \{t>0; W_t=y\}, y \in \mathbb{R}$? I actually need to estimate these ...
3
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1answer
54 views

Martingale regularization with right continuous filtration

The standard textbook presentation of the Doob Regularization Theorem for a martingale $(X_t, \mathcal{F}_t)$ assumes that the filtration satisfies the usual conditions. It is clear that the ...
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46 views

Is $0$ transient, positive recurrent or null recurrent?

Suppose that you have a Markov chain with state space $E$ containing $0$. Assume that $$ p_{00}^{(2n)}=\binom{2n}{n}\left(\frac{1}{2}\right)^{2n-1}~~~\text{ and ...
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1answer
29 views

Stochastic intensity poisson process

I am wondering if, considering $N_t$- an $\mathcal{F}_t$ poisson process with stochastic intensity $\lambda_t$ on $(\Omega,P)$ and $\tilde{N}_t$- an $\mathcal{G}_t$ poisson process with stochastic ...
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34 views

How to calculate probability of an event in a stochastic setting?

Let $\left(\, B_{t}\,\right)_{t\ \geq\ 0}$ be a Brownian motion. Calculate the probability of the event: $$ E\equiv\left\{\,\exists\ \epsilon > 0 : \forall\ 0 < h < \epsilon, \max_{t\ \in\ ...
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1answer
29 views

A question on proving the existence of a martingle which has a deterministic square bracket

Let $g:\mathbb{R^+} \to \mathbb{R^+}$ be a non decreasing and continuous function . Show that there exists a continuous martingale M such that its square bracket $<M>_t=g(t)-g(0)?)$ I have ...
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2answers
57 views

On the definition of Markov chains

A Markov chain with discrete time dependence and stationary transition probabilities is defined as follows. Let $S$ be a countable set, $p_{ij}$ be a nonnegative number for each $i,j\in S$ and assume ...
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2answers
63 views

Jumps of Lévy process

Let $X:=(X_t)_{t\geq0}$ be a Lévy process with triple $(b,A,\nu)$. Is there any known relation between the "distribution" of its jumps and the Lévy measure $\nu$? E.g. can we express something like ...
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1answer
51 views

Applying Ito's formula

This is probably an easy question but I am getting aquanted with Ito's formula and stuck on an exercise in my textbook. Let $X_{t}=W_{t}-a t/2$ where $a$ is a real number and $W_{t}$ is brownian ...
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1answer
37 views

How to connect the deterministic and probabilistic descriptions of the SIR model

I am a 17 year old student and I was reading up on epidemic modelling for a math project, specifically the SIR model and I came across this: "This" refers to the assumptions to which the Markov Chain ...
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32 views

Exercise on quadratic variation

I am faced with the following exercise: Let $X_{1},X_{2},...$ be independent random variables satisfying $\mathbb{E}(X_{n}^{2})<\infty$ and $\mathbb{E}(X_{n})=0$ for all $n\in\mathbb{N}_{0}$. ...
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45 views

A basic measure theory question on Stochastic Process

Let $(Ω, F, P)$ be a probability space, $T$ some index set, and $(S, Σ)$ a measurable space. $X : T × Ω → S$ is a stochastic process, so it is measurable map. Let $S^T$ be the collection of all ...
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60 views

Let $X_n \sim U[0,1]$. Let $A_n$ count the number of local maxima of the sequence unto $n$. Prove a suitable central limit theorem for $A_n$.

Let $X_n $ be uniformly distributed on $[0,1]$. We say $X_k$ is a local maximum if $X_k> X_{k\pm 1}$. Let $A_n$ count the number of local maxima of the sequence unto and including $n$. Find $a_n, ...
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1answer
22 views

How to transform a process into a Markov Chain?

This problem is in the book Introduction to Probability. The question goes this way. Consider the process {$ X_n, n = 0,1,...$ } with values 0,1 or 2. If P{$X_{n+1} = j | X_n = i, X_{n-1} = ...
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1answer
29 views

Random Process, how do I understand this?

I think I have little difficulty in understanding the "Random Process". Here is a definition taken from Oppenheim's book. In Section 7.3 we defined a random variable X as a function that maps ...
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1answer
27 views

Combining different rates in Poisson process?

I have the following problem: A machine has two critically important parts and is subject to three different kinds of shocks. Shocks of type $i$ occur as a Poisson process with rate $\lambda_i$. ...
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19 views

Unconditional variance of approximately simple oscillator

I am considering the Ornstein-Uhlenbeck like system $$ dx_t = -\sinh(x_t) dt + \cosh(x_t) \sqrt{1 + 2\sinh(x_t)} dW_t $$ and wish to compute the unconditional mean $E[ x_t sinh(x_t)]$ amongst other ...
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50 views

Method to find composition of stochastic processes?

Consider the following problem: You have 2 distributors offering the exact same product, and there are multiple producers. At each time you know the total number of units each distributor has ...
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124 views

6-digit password - a special decoding method

Consider the situation of decoding a 6-digit password that consists of the symbols A to Z and 0 to 9, where all possible combinations are tried randomly and uniformly. Consider the ...
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1answer
70 views

Martingale Transforms and quadratic variation

Let $M$ be a martingale with $\mathbb{E}M_{n}^2<\infty$ for all $n$. Let $C$ be a bounded predictable process and set $X=M\cdot C$. Show that $\mathbb{E}X_{n}^2<\infty$ for all $n$ and that ...