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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How can we easily compute $\mathbb{E} [ \left|W_t\right| ^\alpha]$?

How can we easily compute $\mathbb{E} [ \left|W_t\right| ^\alpha]$, where $\alpha \in \mathbb R^*_+ $ and $W = (W_t)_{t \geq 0}$ is the one dimensional standard Brownian motion (or wiener process)?
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46 views

Time Series: existence of moments $\Rightarrow$ existence of distribution?

This might come to you as a bit silly, because normally we are used to the vice-versa question. But here is what I have: a nonlinear time-series model, for which I can derive by infinite backwards ...
1
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1answer
393 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
6
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1answer
942 views

Expectation of an integral w.r.t. Brownian Motion

I know the following statement: if $f$ is a deterministic function and continuous, i.e. $f\in C^0([0,T],\mathbb{R})$, then $\int f(s)dW_s$ is normally distributed with mean zero and variance $\int ...
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1answer
213 views

Brownian Motion Expectation-Like Integral

How much is $$\int_0^T tB_t \, dt$$ where $B_t$ is Brownian motion and $T$ an universal constant?
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3answers
248 views

Singular covariance matrix

I am looking into the process $\{X_t, t\in\mathbb{Z}\}$, $X_t=A\cos(\lambda t)+B\sin(\lambda t)$, here $\lambda\in(0,\pi)$ is fixed, $A$ and $B$ are uncorrelated random variables with $EA=EB=0$, ...
2
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0answers
140 views

Independence of Brownian Motion with respect to a stopping time

Let $B_t$ be a brownian motion, $B_0=0$, and $\gamma \in \mathbb{R}$. Now, let's build the following stopping time: \begin{equation} T = \inf \{ t \geq 0 : |B_t + \gamma t| = 1 \}. \end{equation} If ...
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1answer
245 views

two question about poisson processes

I'm solving an exercise from a last year exam. Suppose we have an Poisson process $(N_t)$ with parameter $\lambda=\frac{1}{3}$ given with respect to a filtration $(\mathcal{F}_t)$. The first ...
2
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1answer
100 views

Fractional Brownian motion as integral, mean zero

Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $$X(t)={1\over ...
2
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1answer
843 views

Passing a limit into expectation

While reading about random walks, I started thinking about this and got a headache: Given a random process $\{X_n\}_{n\in \mathbb{Z}^+}$ with a real state space (i.e., $X_n$ takes on real numbers), ...
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1answer
106 views

Computation of Conditional Expectation

Let the stochastic process $\{X_n\}$ be constructed inductively as follows: $X_0=0$, and for $n\ge 1$, and conditionally on $\mathcal{F}_{n-1}=\sigma(X_0,\ldots,X_{n-1})$, we set $\;\,\,\, ...
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0answers
78 views

Is the following interpretation for the stationary distribution of a Markov process correct?

Imagine I have some Markov process with stationary distribution $\pi$ and a mixing time of $\tau$ after which $|Prob[x=s_i] - \pi(s_i)| \leq \epsilon$. Can I assume the following: A state $(x=s_i)$ ...
2
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2answers
205 views

How do I derive the Gaussian Mixture distribution of an Ito Integral?

I have a question about the distribution of an Ito Integral. Consider the integral $$ \int_0^1 B_1(r) \mathrm{d}B_2(r), $$ where $B_1$ and $B_2$ are two independent standard Brownian motions. I am ...
2
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1answer
178 views

Translational invariance of Brownian motion

Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space, $(X_t,\mathcal{F}_t)_{t \geq 0}$ a time-homogeneous Markov process. A paper I read defines a probability measure $\mathbb{P}^x$ by ...
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1answer
111 views

Supermartingale with vanishing drift

Is a continuous supermartingale with vanishing drift already a martingale? In my concrete problem, I have a continuous nonnegative local martingale $ (X_t) $ on $ \left[0, T\right] $ which is bounded ...
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1answer
136 views

Brownian motion interesting question

I found this interesting question on the internet, but unfortunately I could not solve it. What is probability that Brownian motion (starting at origin) has value 1 before having value -2?
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21 views

Problem with the uniform transience

Let $X$ be a Borel space and let us consider a Markov Chain $(\Phi_n)_{n\geq 0}$ on this space given by the stochastic kernel $$ P(x,\mathrm dy) = p(x,y)\mu(\mathrm dy) $$ where the density $p$ is ...
3
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0answers
119 views

Failure criteria for a collection of independent evolving discrete random variables

I have built a computer model that contains a collection of independent discrete random variables. They each have values of between $0$ and $k$ where $k$ is between $4$ and $15$ and is constant for ...
2
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1answer
60 views

Is there a standard procedure for conditioning a stochastic process?

I've got a two-dimensional Markov stochastic process $(X_t, Y_t)$ that runs on time interval $[0, t_f]$. I know the transition function (or the infintesimal generator, if you like) of this process. ...
2
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1answer
889 views

Is continuous L2 bounded local martingale a true martingale?

I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...) I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
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74 views

Find a density function for the endpoint of this stochastic process

$(X_t, Y_t, Z_t)$ is a three-dimensional stochastic process described as follows: $X_t$ is a Brownian Motion. $Y_t = \int_0^t X_s ds$ $Z_t = \inf_{s \in [0, t]} X_s$ I would like to find a density ...
3
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1answer
192 views

Find the transition function of this stochastic process

Let $(X_t, Y_t)$ be a two-dimensional Markov stochastic process that runs on time interval $[t_0, t_f]$. Its infintesimal generator is described by the functions $\mu_X, \mu_Y, \sigma_X, \sigma_Y$. I ...
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59 views

Right continuity of mean value function from a submartingale

This may be a stupid question but: Let $(X_t,\mathcal{F}_t)_{t\geq}$ be a submartingale and define $$m_X(t+):=\lim_{s\downarrow t} E[X_s]$$ and assume it exists (I know it always does). Why is then ...
3
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1answer
712 views

What's the difference between expected values in binomial distributions and hypergeometric distributions?

The formula for the expected value in a binomial distribution is: $$E(X) = nP(s)$$ where $n$ is the number of trials and $P(s)$ is the probability of success. The formula for the expected value in a ...
3
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1answer
382 views

Biased Random Walk Converging to a Brownian Motion with drift (Donsker's Theorem)

Fix $N$ and suppose $\{X_n\}_{k=1}^{N}$ are i.i.d steps that are $\pm 1$ with equal probability. Then $S_n = \sum_{k\leq n} X_k $ is a simple random walk, and (with the right scaling) we know that the ...
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437 views

Generated sigma algebra from Brownian Motion

Suppose that we have a Brownian motion and we define the P-augmented filtration by $$\mathcal{F}^W_t:=\sigma(\mathcal{F}^0_t \cup \mathcal{N})$$ where $\mathcal{F}_t^0:=\sigma(W_s;s\le t)$ and ...
2
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2answers
292 views

Basic stochastic integral

I am new to this stuff. Can some one explain how I could compute the stochastic integral of the form $\int_0^t W_sds$, where $W_t$ is Brownian process? Thanks!
2
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1answer
207 views

How to show that $X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$ (“inverse brownian”) is a martingale?

Consider $$X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$$ where $ \left(B_{t }\right)_{t \geq 0}$ is a $ \mathcal F_t$- brownian motion in $\mathbb R ^3$, null at ...
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1answer
52 views

Covariation of a gaussian process $ G_t = \int_0 ^t \frac{B_u}{u}du$

Consider $$ G_t = \int_0 ^t \frac{B_u}{u}du$$ where $\left(B_{t} \right)_{t\geq0}$ is $\mathcal F _t $ - brownian motian in $\mathbb R$, null at the origin. It's simple to show that $\left(G_{t} ...
2
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1answer
103 views

Show that $M_t = \int_0 ^t \exp{((B_2(s)^2)} dB_1(s)$ is not a continuous square integrable martingale

Consider the following $\mathcal F_t$- (continouous) local martingale $$M_t = \int_0 ^t \exp{((B_2(s)^2)} dB_1(s)$$ where $\left(B_t\right)_{t\geq0} =\left(B_1(t),B_2(t)\right)_{t\geq0}$ is ...
3
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0answers
67 views

Is this a valid method for time-integrating a stochastic process?

I have a stochastic process $X_t$, and I have a function $a(x | t)$ that reflects my beliefs about the value of $X_t$ ($a$ is a density function in its first parameter). I am studying the properties ...
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0answers
45 views

How do you convert an infintesimal generator of a Markov process to a transition function?

Suppose a continuous-time continuous-step Markov stochastic process $X_t$ has infinitesimal generator $\mu(x, t)$, $\sigma(x, t)$ ($\mu$, $\sigma$, and $X_0$ are known). How can we use this ...
3
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1answer
314 views

Some basic questions about Stochastic Calculus

I have a transition function for a Markov process $X_t$. I want to find a density function for the stochastic process $Y_t := \int_0^t X_s \,ds$. Some questions about this: Is this the same as the ...
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1k views

How to prove the martingale?

How to prove that the integral $\int_{0}^{+\infty}\upsilon e^{-ru}S_{u}dW_{u}^{Q}$ is a martingale under Q where $S_{t}$ is a martingale under Q and $\mathbb{E}^{Q}[\int_{0}^{+\infty}|\upsilon ...
5
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0answers
91 views

Confusion in the proof of properties for $\psi$-irreducibility

Let $P$ be a stochastic kernel on a measurable space $(\mathsf X,\mathfrak B(\mathsf X))$. The kernel $P$ is called $\varphi$-irreducible if for a positive measure $\varphi$ and for all measurable ...
4
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1answer
314 views

Funny problem about stochastic integrals and Ito' s lemma

Consider a probability filtred space $ (\Omega, \mathcal F, \mathcal F_ t, \mathbb P)$ and a continuous $\mathcal F _t$-martingal starting from $0$, $ M = (M_t)_{t \geq 0}$, such that $\left \langle ...
2
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2answers
597 views

Explicit solution of a linear SDE

I'd like an explicit formula as a function of $W_t$ (standard Brownian motion) and $\lambda >0$ for the solution of the following SDE: $$\mathrm dX_t = \mathrm dW_t - \lambda X_t \,\mathrm dt$$ ...
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1answer
139 views

Upper bound for the $\sup$ of a martingale defined as a stochastic integral of a general continuous martingale

Consider a probability filtred space $ (\Omega, \mathcal F, \mathcal F_ t, \mathbb P)$ and a continuous $\mathcal F _t$-martingal starting from $0$, $ M = (M_t)_{t \geq 0}$, such that $\left \langle ...
1
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1answer
101 views

Explicit solution of a SDE envolving a probability measure changing

Let's consider the probability space $ (\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb{P})$ and a $\mathcal F_t$-Brownian motion under $\mathbb{P}$, $(W^{\mathbb P}_t, t\geq 0) $ with ...
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1answer
126 views

Itô's lemma to solve the SDE

Given $dG_{t}=\alpha S_{t}dt+\upsilon S_{t}dW_{t}$ and $dS(t)={dG_{t}}-\epsilon_{t}dt$. How can I have ...
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2answers
918 views

Kolmogorov Extension Theorem vs. Caratheodory Extension Theorem

I noticed that CET together with monotone-class arguments is commonly used in theory of discrete-time stochastic processes to construct a joint probability measure from finite-dimensional ...
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3answers
442 views

Probability of rejecting faulty Bearings

A ball bearing diameter is $3.00 \pm 0.01$. The Mean and Standard Deviation are given (you can assume them to be any value). Using normal distribution, find the probability of a faulty bearing.
3
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1answer
151 views

Autocorrelation of wrapped Wiener process

Let $\phi(t)$ be a Brownian Walk (Wiener Process), where $\phi\in[0,2\pi)$. As such we work with the variable $z(t)=e^{i\phi(t)}$. I would like to calculate $$E(z(t)z(t+\tau)).$$ This is equal to ...
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1answer
115 views

Compound Poisson process: calculate $E\left( \sum_{k=1}^{N_t}X_k e^{t-T_k} \right)$, $X_k$ i.i.d., $T_k$ arrival time

Let $N_t$ be a Poisson process with rate $\lambda$. $T_k$ the inter arrival times of $N_t$. $\{X_k\}$ a collection of i.i.d. random variables with mean $\mu$. $X_k$ is independent of $N_t$. Calculate ...
7
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1answer
253 views

Integral of the positive part of a Brownian motion

Let $X(t)$ be the standard Brownian motion, I need to find the distribution of $S=\int_{0}^T(X(t))^+dt$, where $(x)^+=\max\{0,x\}$. I want to use the distribution to get a concentration bound for ...
3
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1answer
1k views

The law of absolute value of a standard Brownian motion

How can we easily compute $\mathbb{E} [ \left|W_t\right|]$, where $W = (W_t)_{t \geq 0}$ is the one dimensional standard Brownian motion (or wiener process)?
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vote
2answers
145 views

Using random walks to predict behavior rather than matrix decomposition

I want to create a model that tries to predict a user's behavior based on the random walks of similar users. The problem is similar to Netflix's recommendation challenge. One of the popular solutions ...
4
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1answer
210 views

Quadratic Covariation

I am not sure about the answer to this question. For a Brownian motion $B_t$ and a process $M$ defined by $M_t=B_{t-s}$ if $t>s$ and 0 else, what is the Quadratic Covariation $[B,M]_t$ ? I find ...
5
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2answers
278 views

Brownian Motion Covariance: max instead of min

It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion. Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that ...
8
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1answer
3k views

Markov process vs. markov chain vs. random process vs. stochastic process vs. collection of random variables

I'm trying to understand each of the above terms, and I'm having a lot of trouble deciphering the difference between them (note, my mathematics training isn't very strong - so please go easy on the ...