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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Continuity theorem in Itô integral explanation

What is the continuity theorem used here in the explanation of the Itô integral? I cannot seem to find anything that would be exactly useful in my measure and integration text.
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Wasserstein metric: conditions for the existence of minimizer and duality

Let $(X,d)$ be a metric space and let $\mathcal P(X)$ be the set of all Borel probability measures on $(X,d)$. The Wasserstein distance on $\mathcal P(X)$ is given by $$ W_d(\mu,\bar\mu):=\inf_{M\in ...
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39 views

Property of Wiener process sample path

What is a mean of time, when the trajectory of wiener process $W_t$ is over the line $y=t$? We need to find $\mathbb{E}\tau$, where $\tau=\sum\limits_{a,b:\forall t\in(a,b) ; ...
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16 views

Expectations of certain Brownian motion equations

$B_t$ is Brownian motion. It is assumed that motion starts at $0$. I do not understand how the highlighted equalities hold true. Is the first one equivalent to ...
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Why are stochastic processes with decreasing expected value called supermartingales?

I am curious to know why a process which has decreasing expected value is called a supermartingale. From a beginners perspective it would seem reasonable to have the following picture: ...
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54 views

Filling of a tank - recurrence relation

Suppose a tank has a maximum limit of 100 units. Each day 2,1 and 0 units are added to the water level with probability p,r and q. Any excess water would overflow and if it reaches the minimum level ...
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25 views

Two Reflecting Barriers

A chain with stats 1,2,....,n has a matrix whose first and last rows are (q,p,0,...,0) and (0,...,0,q,p). In all other rows Pk,k+1 = p, Pk,k-1 = q. Find the stationary distribution. I am ...
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39 views

Card Shuffling and Convergence in Probability

There are $4n$ cards, and we denote the set of cards with number $4k,k \in \{1,2,\ldots,n\}$ as $S$. The we shuffle the whole cards randomly, which means that each permutation will happen with the ...
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Higher powers of transition matrix of the Markov chain $X_n =\max\{Z_1, …, Z_n\}$

I have a question I am working on: Suppose $\{Z_n, n ≥ 1\}$ are iid outcomes of successive throws of a fair die. Then, let $X_n = \max\{Z_1, ..., Z_n\}$. It is easy to show that $X_n$ is ...
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37 views

First hitting time Geometric Brownian motion

I have the following problem: My Process underlies the SDE $ d W_t = \mu W_t dt + \sigma W_t d B_t $ with $B_t$ being a standard Brownian motion, $\mu,\sigma >0$, i.e. $W_t = S_0 \exp\Big( ...
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40 views

Covariance between real and imaginary parts of Fourier transform of a stationary time series

Since Fourier transform of a random stationary time series(in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary parts of ...
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71 views

How to make sense out of this: Ergodic theorem for Markov chains

We had the ergodic theorem for Markov chains, stating that: For a state space $S \subset \mathbb{N}$ and all functions $f \in L^1$ (meaning that $\sum_{s \in S} |f(s)|\pi(s) < \infty$) and an ...
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33 views

law of iterated logarithm

Wikipedia claims see this link that the law of the iterated logarithm marks exactly the point, where convergence in probability and convergence almost sure become different. It is apparent from the ...
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45 views

Does irregularly transformed stationary process preserves stationarity?

I would like to apply the following theorem in a probably unusual way. Let $Z_t=f(Z_{t-1},Z_{t-2},\dots,Z_{t-M})+\varepsilon_t, t=1,2,\dots$ be a stationary and ergodic Markov chain as well as ...
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35 views

How to prove that convergence in MGF implies Convergence in Distribution?

I know that if the moment generating function of two distribution converges to the same function then the two distribution converges in CDF. But how can we prove this thing explicitly ?
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22 views

Return time Markov chain

I have been wondering about this for quite a while now that I found in a textbook in the proof that an irreducible positive recurrent markov chain $(X_n)$ has a stationary distribution Let $t_i$ ...
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56 views

Need help with a basic exercise about Markov chains

Suppose $\left\{ X_{n}\right\} _{n=1}^{\infty}$ is a Markov Chain taking real values. Are the following Markov Chains? $$Y_{n}=\sum_{i=1}^{n}X_{i} , Z_{n}=\left(X_{n},X_{n-1}\right)$$ Edit1 I ...
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Branching Brownian Motion and KPP equation

I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...
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30 views

Long run behavior of a absorbing markov chain

$$A=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0.2&0&0.6&0.2\\0&0.2&0.2&0.6\end{pmatrix}.$$ In the above matrix how do I calculate the probability that in the ...
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386 views

Zombie outbreak on a $k$-regular graph

Suppose we have a zombie outbreak on a connected $k$-regular graph of order $n$. There are $n_0$ initially infected zombie nodes, and each turn, each zombie infects its neighbors with probability ...
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39 views

Can ergodic Markov chains be periodic?

I found a statement in one of my notes which said If a state is persistent, aperiodic and not null the it is said to be ergodic Is it necessary that it should be aperiodic? This statement ...
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Markov chains: An issue in classification of states

I recently came across a lemma which goes as follows. Suppose a Markov chain has N states. Let i and j be pair of states. Then j can be reached from i iff there is an integer $ 0 ≤n< N$ such ...
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Matlab code for Simulation of SDE [duplicate]

I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated.. Q: "Simulate ...
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47 views

What is the distribution of this random variable? [closed]

Find the distribution of this random variable: $$X_t=\exp\left(t \int_0^t sdW_s\right)$$ knowing that $W$ is a Brownian motion in the filtered space $(\Omega, \mathcal{F},P,(\mathcal{F}_t)_{t\geq0} ...
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59 views

Proving a property of hitting times of a simple random walk on $\mathbb{Z}$

I'm reading the course notes of a probability course about martingales currently and I'm trying to solve some of the exercises, however I'm very much stuck with the following exercise: Let $\left\{ ...
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18 views

Random process with Cauchy distribution

The problem is as follows. Let $X(t)$ be a stochastic process such that $X(t) = V + 2t, t \ge 0$, and $V$ has the Cauchy distribution $x_0 = 0, \gamma = 1$. Find the probability that $X(t) = 0$ for ...
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Expected number of steps to absorbtion - Markov chain

I want to calculate the expected number of steps (transitions) needed for absorbtion. So the transition probability matrix $P$ has exactly one (lets say it is the first one) column with a $1$ and the ...
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Solution to linear system around the topic of Markov-chains

Let $(X_n)_{n\geq 0}$ be a Markov-chain with the state space $S$ and transition matrix $P=(p_{xy})_{x, y \in S}$. For $A\subset S$ be $H^A:=\inf\{n = 0, 1, \dots | X_n \in A\}$ the first visit time ...
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23 views

Are random walk variations Markov-Chains?

Let $S_{n}:= S_0 + \sum_{i=1}^{n}X_i$ be a simple random walk, $X_i$ are independent random variables with $P[X_i=1] = p, P[X_i = -1] = 1-p$. Let $M_n:=\max\{S_0, \dots, S_n\}$. The task at hand is ...
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Is squared Brownian Motion a gaussian process?

I am working at the following SP, given by $(X_t)_{t\geq0} = \alpha W_t^2+\beta t$ where $W_t$ is Brownian motion and $\alpha,\beta$ real. I managed to calculate mean and covariance function and now I ...
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186 views

Properties of Markov chains

We covered Markov chains in class and after going through the details, I still have a few questions. (I encourage you to give short answers to the question, as this may become very cumbersome ...
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166 views

Expectation of stopping time $\inf\left\{t: W_t>t\right\}$ [closed]

Let $W_t$ be a Wiener process, $\tau=\inf\left\{t>0: W_t>t\right\}$. How to find $\mathbb{E}\tau$? Update: Useful statements the theorem of Bachelier: $\sup\left\{W_t: 0\leq t\leq ...
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43 views

Stochastic Differential equations with $\sin(x^2)$ as drift.

Can somebody help me how to solve the following SDE analytically or suggest me to go through some literature to understand this or can give me a little bit hint to work by myself. Thanks in advance. ...
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39 views

Levy processes, vanilla option and Fourier Transform

The context to this problem is mathematical finance, although the answer does not need specific knowledge of the area. I am trying to work out the expression for the price of a call option using Levy ...
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25 views

Convergence to a finite random variable

The Martingale Convergence Theorem is typically stated that we have convergence to a (finite) random variable $X$ as $X_n \to X$ a.s when the conditions of the theorem are satisfied. What is meant ...
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Stability of simulation of brownian noise

As I understand, Brownian noise can be simulated by the process $$x_{n+1}=x_n+R_n$$ where $R\sim U[-a,a]$. The expected value for $x_n$ is then $x_0$. But $\text{Var} x_n\to\infty$ as $n\to\infty$ ...
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Brownian Motion calculation

I am reading the following statement and cannot justify it. Any help would be greatly appreciated. Thanks. $$ E\left[\exp\left(2\int_{0}^{T}B_{s}^{2}\,{\rm d}s\right)\right] < \infty \quad\mbox{if ...
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32 views

Convergence of probability density functions

Assume that a sequence of random variables, $(X_t)_{t\geq 0}$, converges in distribution to a random variable $X_0$, as $t\to 0$. Also assume that $X_t$ and $X_0$ have $C^{\infty}$-probability density ...
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Brownian motion starts fresh variant

It is a standard result that if $W_t$ is a Brownian Motion and $S$ is a stopping time of the standard filtration $F_t$ then we have that $B_t = W_{S+t} - W_S$ is a Brownian Motion. I quote the ...
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26 views

Chances of winning if only a certain amount of people can participate

So we got 2 million (2_000_000) cornflakes boxes. Hidden in 6 of those cornflakes boxes is a gold nugget. Due to a wrong printing, 500_000 cornflakes boxes don't get into the market. So there are ...
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48 views

What is a.e. a.s

I am reading a paper which uses almost everywhere almost surely (a.e.,a.s.) simultaneously, I am not quite sure what it means then. To be specific, they consider a stochastic process $\{X_t\}$ such ...
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A question in a textbook about Blumenthal 0-1 law for a general Markov process

This question came up as a result of reading this question . Here is the Blumenthal 0-1 law in the book Stochastic Processes by Richard F. Bass. Proposition 20.8 Let $(X_t , \Bbb{P}^x)$ be a ...
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Independence Lemma, is it non-trivial?

I'm reading Steven E. Shreve's "Stochastic Calculus for Finance II, Continuous-Time models", and a bit confused on the Independence Lemma (Lemma 2.3.4). The lemma says: Lemma 2.3.4 (Independence). ...
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Why does a Gaussian process have a gradient whose determinant is Gaussian?

I'm trying to understand something in Adler and Taylor's book, Random Fields and Geometry. Let $T \subset \mathbb{R}^N$ be a compact parameter set (for simplicity, suppose it is a closed hypercube) ...
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Integral of different types of noise

I have been learning about the Wiener process and read this on Wikipedia; "the Wiener process is used to represent the integral of a Gaussian white noise process" It got me wondering what the ...
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An exponential martingale

Let $H_{t}$ be a bounded continuous and $\textbf{F}^{B}_{t}$ an adapted process. $B$ Brownian motion. Show that $M_{t}= \exp\left(-\int^{t}_{0}H_{s}dB_{s} -\frac{1}{2}\int^{t}_{0}H^{2}_{s}ds\right)$ ...
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Writing down the transition matrix of a discrete Markov chain

Please consider the following scenario: One person is walking along a discrete circle induced by $\mathbb{Z}/n\mathbb{Z}$ In each round we roll a dice with $w\in\left\{2,\ldots n\right\}$ sides If ...
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44 views

Geometric BM tends to zero but is strictly positive a.s.?

The process $\{S_t\}_{t\ge0}$ following $dS_t = \sigma S_tdW_t$ with $S_0>0$ has the solution $$S_t=S_0 e^{-\frac12\sigma^2t+\sigma W_t}$$ Now for any $\epsilon>0$ we have $$\mathbb ...
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Multipe Ito Integrals

Im working on a Lemma 10.8 in the Book "Numerical Solution of Stochastic Differential Equations by Kloeden And Platen" I have been stuck on one point. Can somebody help me to understand how he moved ...
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Fractional Brownian motion---construction via Hilbert space?

The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms: Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, ...