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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Conditional Distributions vs. Stochastic Processes

Is the concept of a version of a stochastic process related to the concept of a version of a conditional distribution? And is a regular version of a stochastic process somehow the same thing as the ...
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15 views

Almost sure convergence of the inverse

If a sequence of non-negative random variables $X_1, X_2, \dots$ converges almost surely to a random variable $X$, that is $X_n \xrightarrow{a.s} X$ or equivalently $P(\lim\limits_{n\to\infty}X_n=X)=1$...
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26 views

Convergence in distribution for two random variables

If $\lim\limits_{n \to \infty} P(X_n\leq T)=P(X\leq T)$ and $\lim\limits_{n \to \infty} P(Y_n\leq T)=P(Y\leq T)$, where $X_1, X_2,\cdots$ and $Y_1, Y_2,\cdots$ are two sequences of random variables ...
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21 views

$X$ is a Right-Continuous process iff $\mathcal{F}^X$ filtration is RC?

I have a doubt on this assertion : $X$ is a right-continuous adapted process $\iff$ $\mathcal{F}^X$ is right-continuous ? I have mainly a doubt on this direction $\Leftarrow$, I do not find a mean ...
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33 views

Wiener process and stochastic int

Let $h:[0,1] \rightarrow \left\{-1,1 \right\}$. How to show that $X_t=(\int_0^th(s)dW_s)_{t \in [0,1]}$ is a Wiener process? I know from the lecture that for every $h$ process $\int h \ dW_s$ is ...
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114 views

Stochastic Integral with respect to Compensated Poisson Process

Proposition: Let $N_t$ be an $\mathcal{F}$-Poisson process and $M_t=N_t-\lambda t$ its compensated process. Then for any $\mathcal{F}$-predictable bounded process $H_t$, the stochastic integral $$(H\...
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49 views

Find $P(B_3>0,B_6>0)$ where $(B_t)$ is a Brownian motion

Suppose that $B_{t}$ is a standard Brownian Motion. What is the probability that both $B_{3}$ and $B_{6}$ take positive values? This is what I've tried but then I get stuck and I'm not sure how to ...
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22 views

Derivation of a property of standard Wiener processes

I am reading A Standard Wiener Process and am struggling to piece together how they arrived at their conclusion. The major properties of any Wiener Process are: $W(t) = 0$ $W(t) - W(s) \sim N(0, t-...
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163 views

Is it Possible to Construct all Proofs in Complex Analysis using Brownian Motion?

(First, I am very aware of the fact that Brownian motion is actually probably more difficult to understand than at least basic complex analysis, so the pedagogical merits of such an approach would be ...
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36 views

Question regarding local martingales.

In the definition of a local martingale I have that for a filtered probasbility space $(\Omega, \mathcal{F},P,\mathbb{F})$. A local martingale is an adapted process M, such that there exists a ...
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29 views

Why an optional process could not be predictable?

We know that a predictable process is also optional (*). Why an optional process could not be predictable ? Why we cannot use the same arguments as the proof for (*) ?
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27 views

What is an example of a second-order markov chain? [closed]

I'd like to see an example of a second-order markov chain. Haven't found one over google or in any of my textbooks
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1answer
32 views

What is the Difference Between a Version and a Modification of a Stochastic Process?

Under what circumstances would one say that: The stochastic process $X$ is a version of the stochastic process $Y$? Background: See here for a related but slightly different question on ...
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81 views

Integral of Wiener Squared process

I don't have a background of stochastic calculus. It is known fact that definite integral of standard Wiener process from $0$ to $t$ results in another Gaussian process with slice distribution that ...
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24 views

Distribution of a certain stochastic process

Consider on a probability space $(\Omega, \cal F, \mathbb P)$ the following stochastic process on $[0, \infty]$, where $W(t)$ is a Wiener process, all the coefficients $\lambda(t), \mu(t)$ and $\sigma(...
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31 views

discretized Brownian motion

These are the definitions I'm working with: A (standard) Brownian motion in $\mathbb{R}$ is a stochastic process $W(t)$ $(t \geq 0)$ such that the following properties hold: $W(0) = 0$ almost ...
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22 views

Algorithmic Efficiency

Suppose there are n extreme points and they are numbered in increasing order of their values. Consider the Markov chain in which $p(1 , 1)=1$ and $p(i , j) = \frac{1}{i-1}$ for $j\lt i$. Use $g(x)= 1 ...
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29 views

Sufficient condition for a measure to be invariant

Given a Polish metric space $H$ and a Borel probability measure $\pi$. Let $\mathcal B_b(H)$ be the set of bounded measurable functions on $H$, and $L^2(H, \pi)$ be the set of square integrable real-...
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11 views

Joint Convergence implication

Assume we have a stochastic process cadlag $X_{t}$ (which may be stochastically continuous if we need). Let $t_{i}$ be an arbitrary fine grid of the time, with $t_{i}\rightarrow \infty$ as $i\...
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1answer
20 views

Maximum of poisson process

Let $X^{(1)}_{t\ge 0},...,X^{(n)}_{t\ge 0}$ independent Poisson Processes with common intensity $\lambda$ Find the distribution of the first time that a)at least one event has ocurred in every ...
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26 views

A stochastic volatility model

An example of stochastic volatility model: $$\begin{cases} \frac{dX_t}{X_t} &= g_t dW_t \\ dg_t &= - k g_t dt + \sigma dZ_t \end{cases} $$ where $Z_t$ and $W_t$ are Brownian motions and $...
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37 views

Marginally Gaussian not Bivariate Gaussian - Ito Integral

Let $(W_t)_{0\leq t\leq 1}$ be a Wiener process defined up to time $1$ on some probability space. Consider the random vector $$\left(W_{1},\int_0^1 \operatorname{sgn}(W_s) \, dW_s\right)=:(W_1,X_1)$$ ...
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25 views

Compute the Mean of the Following Process

Given the following process: $\Delta \ln(St+1)= \mu - (\sigma2/2) + \sigma(\varepsilon(t+1))$ (where both $\mu$ and $\sigma$ squared are of $S$) How does one calculate the mean of $S(t+1)/S(t)$? (...
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Perron-Frobenius Theorem: Markov Chain -> Matrices

I am interested in finding out a way how to transform the stochastic results of perron-frobenius for markov chains to any matrix. I am aware that perron-frobenius was originally proofed with linear ...
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14 views

Spectral Density of an ARMA process.

For an upcoming Stochastic Processes exam, we have had a sudden brief email about Spectral Density as the lecturer had forgotten to mention it in classes. He states, For an ARMA process with $\phi(z)$...
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25 views

Expected location of Brownian motion on the circle

Intuitively it seems likely that the expected whereabouts of Brownian motion on the unit circle would be the origin $\left(0,0\right)$, at least in the limit as $t\to\infty$. Is this right? Are there ...
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26 views

Probability that Brownian motion falls between two piecewise constant functions

I'll first present the problem, and then describe my motivation: Suppose $a_j \in \mathbb{R}$, $b_j \ge 0$, and $0 = t_0 < t_1 < \cdots < t_J$ are time points. Let $W_t$ be a standard ...
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1answer
47 views

show that the process is a martingale [closed]

maybe you will have an idea how to show that : the process $(exp(X_t-\frac{1}{2}Y_t))$ is a martingale? Where $h \in L^2([0,T])$, $T< \infty$, $X_t=\int_0^th(s)dW_s$ and $Y_t=\int_0^th^2(s)ds$ for $...
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22 views

What is the probabilty of event A given event B probability in next T time duration?

Assume: a system S with three component: A, B and C. At any moment any component may fail. System MAY fail due to failure of any one component. Given: From the history, we know how many times ...
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22 views

Transition functions induced by Markov processes

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and denote by $(X_t,\mathcal{F}_t)_{t\geq 0}$ a time-continuous Markov process with values in $(E,\mathcal{E})$. For $s<t\in [0,\infty)$,...
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62 views

Can this approximation result for stochastic processes be modified.(p=1 instead of p=2)?

In McKeans stochastic integrals from 1969 he proves this: You have a filtered probability space $(\Omega,\mathcal{F},P)$, where the filtration is based on a Brownian motion. Assume that $X_t$ is ...
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56 views

process with integral is martingale

How to show that the process $X_t=tW_t - \int_0^t W_s ds $ is a martingale? I guess I have to use the definition of martingale and properties of Wiener process, but I stack with this integral. Please,...
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1answer
34 views

Basic Question on Definition of Brownian Motion

I am quite new to discrete and continuous stochastic processes. It seems there is something I don`t understand about definition of Brownian motion. Let $\Omega, \mathcal{F}, \mathbb{P}$ be a ...
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19 views

Understanding of Second Arcsine law for Brownian motion

Ok I'm trying to understand the second arcsine law which states: Let $g_t:=\sup\{s\leq t:W_s=0\}$, then $$\mathbb{P}(g_t\leq s)=\frac{2}{\pi}\arcsin \left(\sqrt{\frac{s}{t}}\right )$$ This won't be ...
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38 views

Stochastically perturbed fluid flow map ${\rm d}Φ_t(x_0)=u_t(Φ_t(x_0)){\rm d}t+ξ_t(Φ_t(x_0)){\rm d}W_t$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ with finite trace $(W_t)_{t\ge 0}...
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68 views

Discounted price process in Black-Scholes model is a martingale with respect to Q.

I have been presented a proof that the discounted price process in the Black and Scholes formula is a martingale, but there is something important omitted, and I am not able to fill in the gap. I will ...
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12 views

what does a time-reversible Markov matrix look like

I think an unitary symmetric matrix P satisfies the condition for time-reversible Markov chain. Since $P^2=I$ we have $P=P^{-1}$. Let v be the eigenvector with eigrnvalue $\lambda=1$, then we have $Pv=...
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Replicant portfolio with commissions (Jarrow rudd)

I have created a Jarrow Rudd three for a call option that I know how to replicate with a portfolio. A replicating portfolio of a option works this way: At time 0 we form a replicating portfolio ...
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42 views

Feller property of Ito diffusion

Consider the following Ito diffusion $X_t$ satisfying $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x,$$ with Lipschitz coefficients $b,\sigma$. It can be shown that if $g$ is bounded and continuous, ...
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9 views

What does fixed regressor say about our linearity condition?

The linearity condition states that $y_i=(\vec{x}_i)^{T}\vec{\beta}$ for all $i$. Now, if we have fixed regressors, $\{\vec{x}_1,\vec{x}_2,\cdots\}$, our linearity condition only says for those $\vec{...
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1answer
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Definition of Cylindrical Brownian Motion and Spatial Correlation

From Gawarecki and Mandrekar, Stochastic Differential Equations in Infinite Dimensions: We call a family $\{ W_t \}_{t\geq 0}$ defined on a filtered probability space $(\Omega, \mathcal{F}, \{\...
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A random variable is mapping from sample space to real numbers. How about random process?

A random variable is mapping from sample space to real numbers. How about random process? Can we think of the simplest random process as again a mapping from sample space to real numbers, with the ...
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What are the differences in linearity in Non-stochastic and Stochastic Regression?

I have been confused with the differences between stochastic and non-stochastic explanatory variables for a while. I was able to write down some of my understanding and seek approval or comments about ...
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3answers
43 views

Archer Poisson Process problem

An archer wishes to shoot an arrow at a target. The prospective flight path of the arrow is subject to birds flying past at random times, according to a Poisson process with rate $\mu$ per second. To ...
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1answer
24 views

Are these facts about the Poisson process correct?

Before studying theorems one by one, I want to check whether it is right what I know about Poisson process. Let $\left\{N(t)\right\}$ be a Poisson process. Then the number of the event occur during ...
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20 views

Clarification regarding heat kernel for Brownian motion on a manifold

Let $X$ be Brownian motion on a Riemannian manifold $M$ starting at $x\in M$, D a domain and $f$ a bounded continuous function on $D$. Define $\tau_D$ to be the first exit time of $X$ from $D$. $u_f\...
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Mixing convergence

Given a process $X_n \xrightarrow{d} X$ on some probability space $(\Omega,\mathcal{A},P)$. If for every $B \in \mathcal{A}$ it holds, that $$ \lim_{n\rightarrow \infty} P(X_n\in A,B)=P(X\in A)P(B) $$ ...
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Calculation of renewal function $R(t) = \sum{F^n(t)}$?

My textbook defines the renewal function $R(t) = E[N_t] = \sum_{n=0}^\infty F^n(t)$, where $F^n(t)$ appears to be the n-fold convolution of $F$ with itself. $F$ is the distribution of the interrenewal ...
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93 views

Splitting Poisson process formal proof

Let $\{X_t\}_{t\ge 0}$ be a Poisson Process with parameter $\lambda$. Suppose that each event is type 1 with probability $\alpha$ and type 2 with probability $1-\alpha$. Let $\{X^{(1)}_t\}_{t\ge 0}$ ...
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27 views

Why is this a beta distribution?

I'm given a circle with point $A$ defined by $(x,y)$. Then $T=1-d[O,A]$, so $T=1-\sqrt{(x^2+y^2)}$. Asked to find: $P[T<=u]$ $E[T]$ $Var(T)$ Alright, so $d[O,A]$ has the CDF $u^2$. So, for ...