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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Locally square integrable (local) martingales

I'm reading Protter and sometimes he says "locally square integrable martingale", and sometimes he says "locally square integrable local martingale", and I wonder if these two are the same. Protter's ...
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16 views

Formula for contingent claim similar to European call option but with two dates for option to buy

So in a normal European call option with one maturity date, you'd buy a share of a stock if the price of the stock at the maturity date was higher than the exercise price. How would you come up with a ...
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1answer
46 views

Random walk on $\mathbb Z/m\mathbb Z$ converges to uniform distribution

Let $(X_t)_t$ be the standard continuous time random walk on $\mathbb Z/m\mathbb Z$ with $X(0)=[0]$ almost surely, then I want to show that for $t \rightarrow \infty$, $\lim_{t \rightarrow \infty} ...
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31 views

heuristic for expected number of visits random walk

What is the heuristic argument that explains why, on $\mathbb{Z}^d$, $d \geq 3$, the expected number of visits of a random walk starting from the origin at $x$ is of order $$ O(|x|^{2-d})? $$
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48 views

Prove that the limit in probability of normally distributed random variables is normally distributed, too [duplicate]

Let $X_n\sim\mathcal N_{\mu_n,\sigma_n^2}$ for some $(\mu_n,\sigma_n^2)\in\mathbb R\times(0,\infty)$ and $X$ be a real-valued random variable with $$X_n\stackrel{\text{in probability}}\to X\;.\tag 1$$ ...
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33 views

Convex closure of the support of a Levy Process

Given a Levy Process $X_{t}$ on a filtered Probability Space $(\Omega,\mathcal{F},\mathcal{F}_{t},P)$ with distribution-function $F_{t}$ for $X_{t}$. We look now for the cumulant transform $\phi_{1}$ ...
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30 views

How can we prove that the generalized stochastic process induced by a real-valued Brownian motion is Gaussian?

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle ...
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13 views

Covariance functional of a generalized real-valued Brownian motion

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle ...
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32 views

Identity for return times in continuous Markov chain

I need help with this problem about return times in continuous time Markov chains: We are given a continuous time Markov chain $(X_t)$. Define $T_i=\inf{\{t>0; X_t=i, X_{t-} \neq i\}}$, which ...
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15 views

How to show a hitting time is finite almost surely?

A one-dimensional symmetric simple random walk starts at $S_0 = 1$. How to show with probability one it passes $x = 0$ (or I guess equivalently, the stopping time of hitting $x = 0$ at the first time ...
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27 views

Distribution of “range” of a process

Let $X_t$ be a stochastic process, for example a brownian motion (i.e. $X_{t+h} - X_t \sim \mathcal{N}(0,\sqrt{h}^2)$). The difference between now's value $X_t$ and a past value $X_{t-100}$ is $$M_t ...
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28 views

First return times and continuous markov chains.

We are given a generator matrix $Q$ (Q-matrix) for a continuous time Markov chain $(X_t)_t. We want to calculate the probabilities of: returning to State 3 before State 1, while starting at State 3: ...
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1answer
20 views

Does such a Markov chain exist?

Suppose it has finite state space $S$, and $\lim\limits_{n\to \infty}p_{ij}^{(n)}=0$ for all $i,j\in S$. But guess is there isn't, since for a finite transition matrix, it is unlikely to have ...
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22 views

stationary distribution of outputs in Markov chain

consider a hidden Markov model with two states, with following transition/observation matrices: $T = \left( \begin{array}{cc} 0.9 & 0.1 \\ 0.1 & 0.9 \end{array} \right), O = \left( ...
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1answer
19 views

show that $X_n$ is a supermartingale $\Rightarrow$ $Y_n=\text{min}(X_n,x)$ is a supermartingale

I have to show that: if $X_n$ is a supermartingale then $Y_n=\text{min}(X_n,x)$ is a supermartingale; ($x\in R$) This what I did: since we can write : $\text{min}(X_n,x)=X_n{1}_{X_n \leq x} + ...
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40 views

Why do people all the time exploiting almost sure properties of a stochastic process as if they were sure properties?

All the time, I see people working with a given Brownian motion $(B_t)_{t\ge 0}$ on a fixed probability space $(\Omega,\mathcal A,\operatorname P)$ and suddenly exploiting its almost sure properties ...
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1answer
20 views

A one-sided continuous stochastic process is product measurable. Does the same hold true for almost surely continuous processes?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $(X_t)_{t\ge 0}$ be a real-valued almost surely continuous stochastic process on $(\Omega,\mathcal A,\operatorname P)$. Let ...
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1answer
27 views

Expectation of a generalized real-valued Brownian motion

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $\mathbb R$ and $$\langle ...
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1answer
25 views

Is $\phi B(\omega,\;\cdot\;)$ Lebesgue integrable over $[0,\infty)$ for a real-valued Brownian motion $B$ and $\phi\in C_c^\infty(\mathbb R)$?

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$ and $\lambda$ be the Lebesgue measure on $\mathbb R$. Is ...
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39 views

Distance between Brownian Motion and scaled Gaussian random walk

I'm currently reading this paper: http://user.math.uzh.ch/barbour/pub/Barbour/SteinDiffusion.pdf and in equation (2.26) the author uses the following fact: If $Z(t)$ is a standard Brownian Motion and ...
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62 views

Will the generated sigma algebra have this property?

Lets say you have a measurable space $(\Omega, \mathcal{A})$. And a measurable function $X: (\Omega, \mathcal{A})\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$. We then know that for the sigma ...
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1answer
22 views

how to show that definition for stochastic process in continuous time applies to stock prices

I know that the formal definition of a stochastic process is: {$X(t,\omega)\,\,t\ge0$} is a stochastic process if: For any fixed $t\ge0$, $X(t,\omega)$ is a random variable For any fixed $\omega$ ...
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21 views

part of proof about stochastic processess

I need help proving a part of a proof. The exercise in the book(exercise 3.14 in Bernt Øksendals: Stochastic Differential equations) is given as to prove something, but I will only talk about the ...
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18 views

Probabilities of events involving random variables with random indices

Let, on some probability space $(\Omega,\mathscr F,\mathbb P)$, $(Z_t)_{t\geq0}$ be a real-valued (measurable) stochastic process. Moreover, let $X:\Omega\to\mathbb R$ and $Y:\Omega\to\mathbb R$ be ...
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1answer
48 views

Why is the drift of an Itō process considered to be a Riemann integral even when it's not even Riemann integrable?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(B_t)_{t\ge 0}$ be a real-valued Brownian motion with respect to $\mathcal ...
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24 views

Prove the probability that X takes on an even value for a Poisson R.V.? [duplicate]

We have a random variable $X$ that is distributed according to the Poisson distribution with $ \lambda$, such that $P[X=k]=e^{-\lambda}\frac{ \lambda^k}{k!}$ for $k=0,1,\dots$. We want to show that ...
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1answer
59 views

Prove that the stochastic process can not have continuous paths.

This problem is about stochastic processes, but what I really need help with is using the dominated convergence theorem in the end: What I need to prove is that a stochastic process having these ...
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1answer
42 views

Rigorous argument of the Markov property used in discrete-time Markov chains

I am reading an example related to discrete-time Markov chains which I do not really understand rigorously. Suppose that $\{ X_n \}_{n \in \mathbb{N} }$ is a time-homogeneous discrete-time Markov ...
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19 views

Symmetric random walk ergodic [closed]

Consider a symmetric random walk on $\mathbb{Z}/m \mathbb{Z},$ i.e. we start in some state $[k]$ and then propagate with equal rates either to $[k+1]$ or $[k-1]$ and so on. How do I show that this ...
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21 views

Weak convergence time-continuous random walk

I was wondering whether the time-continuous random walk on $\mathbb{Z}$ that I want to denote by $X:[0,\infty) \rightarrow \mathbb{Z}$ with $X(0)=0$ a.s. and transition rates(NOT probabilities) ...
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1answer
38 views

Are stopping times the same?

In the context of stochastic integration, we showed how it's possible to define the stochastic integral $\int H dM$ for $H \in L^2(M)$ and $M \in \mathcal M^2_0$ (martingales null at $0$ such that ...
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1answer
30 views

linear combination of infinitely divisible random variables

If $X$ and $Y$ are real valued random variables with infinitely divisible distributions, does $aX + bY$ also have an infinitely distribution ($a, b \in \mathbb{R}$). I've seen this stated in several ...
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17 views

How to identify a process via its Karhunen-Loeve expansion?

Suppose that you are given the following Karhunen-Loève expansion of a real-valued continuous Gaussian stochastic process, $x$. $$x(t) = \sum_{k=1}^{\infty}z_{k}\cdot \frac{\sqrt{2}\sin((k-0.5)\pi ...
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1answer
34 views

Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E[\langle\xi,φ\rangle]$ an element of the dual space of $C_c^∞(G)$?

Let $G\subseteq\mathbb R^d$ and $$\mathcal D:=C_c^\infty(G)$$ be equipped with some topology $\tau$ $\mathcal D'$ be the dual space of $\mathcal D$ and $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the ...
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1answer
24 views

What is a generalized stochastic process? I've found two different definitions. Are they equivalent?

Let $\mathcal D:=C_c^\infty(\mathbb R^d)$ and $\mathcal D'$ be the dual space of $\mathcal D$. What is a generalized stochastic process? I've found two different definitions in some textbooks: ...
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50 views

Transition Probability Matrix of Tossing Three coins

Three fair coins are tossed, and we let $X1$ denote the number of heads that appear. Those coins that were heads on the first trial (there were $X1$ of them) we pick up and toss again, and now we let ...
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34 views

how to derive exponential growth equation from stochastic growth?

consider an exponential growth process of a population starting that has initial size $N_0$ and grows at rate $r$: $$\frac{dN}{dt} = rN$$ assuming deterministic and constant growth, the population ...
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8 views

Bounded Stochastic discrete process

I just came across this stochastic process (link): $dY_t = (a-bY_t)dt + c \sqrt{Y_t(1-Y_t)}dW_t$, where $dW_t$ is a Wiener Process. According to the author under certain conditions this process is ...
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84 views

Probability mass function of the sum of the function of the sum of iid random variables

How can I get an expression of the probability mass function of: \begin{equation} Y_i=\sum_{k=1}^i f\left(\sum_{n=1}^{k} X_n\right) \end{equation} being $X_n, n=1,2,...$ iid random variables and ...
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1answer
41 views

Question on Gambler's Ruin

Adam and Bob bet on the outcomes of coin tosses. $P(H)=p$ and $P(T)=1-p=q.$ Adam wins \$1 from Bob if it shows heads and Bob wins \$1 from Adam if it shows tails. Adam begins with \$ $k$ and Bob \$ ...
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14 views

Deriving Time of extintion of a Small neural Network

I'm trying to derive the Expected Value of the Time of Extintion $\tau_{ext}$ of a small Neural Stochastic Network with the following dynamics, where I consider $\tau_{ext}$ to be the time of the last ...
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17 views

How to arrive the following results?

I am reading the book "stochastic differential equations and diffusion processes" written by Ikeda and Watanabe. In the chapter IV about uniqueness of stochastic differential equation, there is a ...
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2answers
30 views

Expectation equation with wiener process [closed]

Can somebody help me with working out $E((W_{t}^2-t)(W_{s}^2-s)$ where $W_{t}$ denotes a Brownian motion. I tried it with rewriting $W_{t}=W_{t}-W_{s}+W_{s}$ but it doens't work yet. Many thanks!
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8 views

What does Karhunen-Loève expansion have to do with cosine-sine basis expansion?

According to my research, Karhunen-Loève(KL) expansion is a version of Fourier series for stochastic processes and states that under some conditions, a stochastic process $X\left(\omega, t\right)$ can ...
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1answer
37 views

Showing equality Wiener process [closed]

Let $M_t=\max W_s$ over $0 \leq s \leq t$ with $W_s$ a Wiener process. Can somebody help me with showing out: $P(M_t>a, W_t<b)=P(M_t>a,W_t>2a-b)$ with use of the reflection principle.
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39 views

Measurability of the set where sample path is continuous

Let $(\Omega,\mathscr F, \mathbb P)$ be a probability space and let $(X_t)_{t>0}$ be a collection of random variables such that $X_t:\Omega\to\mathbb R$ is $\mathscr F$-to-Borel measurable. Fix ...
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1answer
28 views

Cont Time Markov Chains. Stationary Probability

A barber finishes haircuts at rate $3$, measured in hours, so on average it takes him 20 minutes to cut a person’s hair. Customers arrive at rate 2. There is, however, only a two chair waiting ...
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1answer
39 views

What's the distributional derivative of a Banach space valued almost surely continuous stochastic process?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $\lambda$ be the Lebesgue measure on $[0,\infty)$ $(H,\left\|\;\cdot\;\right\|)$ be a Banach space over the field $\mathbb ...
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1answer
28 views

Find a Martingale in a game of exchanging hats

$n$ people play a game of exchanging hats, with the following two rules: --They throw their hats in to a pile and everyone chooses one uniformly at random, those who got back their own hat are out of ...
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21 views

Is there any definition of entropy of a stochastic process?

Entropy of finite random variables is defined in Wiki https://en.wikipedia.org/wiki/Entropy_(information_theory) Entropy rate of a stochastic process is defined in Wiki ...