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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469 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
votes
2answers
310 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
219 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 ...
1
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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
108 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
68 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
46 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
325 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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1answer
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 ...
6
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1answer
98 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
321 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
votes
2answers
700 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: $$\mathrm dX_t = \mathrm dW_t - \lambda X_t \,\mathrm dt$$ ...
1
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1answer
142 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
vote
1answer
104 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 ...
1
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1answer
128 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 ...
8
votes
2answers
989 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
473 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
votes
1answer
161 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
118 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
votes
1answer
259 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
votes
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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2answers
151 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
votes
1answer
224 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
votes
2answers
288 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
votes
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 ...
2
votes
1answer
80 views

probability equation multiplication understanding

I want to understand this probability equation. I'll be grateful, if someone can help. $$P(\text{Birth})P(\text{Death}) + (1-P(\text{Birth}))(1-P(\text{Death}))$$ The above is the transition ...
4
votes
1answer
488 views

Covariance of Brownian-motion-like processes

We know that $\operatorname{Cov}(B_s,B_t)=\min(s,t)$ if $B_t$ is Brownian motion. What is $\operatorname{Cov}(B_{f(s)},B_{f(t)})$ for some injective $f$? How can I write $B_{f(t)}$ in an Ito ...
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1answer
131 views

Equivalence between detailed balance and time reversal

I need help in proving the following popular claim A continuous time and stationary Markov jump process obeys the detailed balance equations $$ P(x)q(x,x') = P(x')q(x',x) $$ where $q(\cdot,\cdot)$ is ...
2
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0answers
270 views

In finite-state Markov chain state $i$ is transient

Can you help me please with proof of this question: Prove, that in finite-state Markov chain state $i$ is transient if and only if is exist state $k$ such that $i\rightarrow k$ but k $\nrightarrow ...
3
votes
1answer
193 views

Example of stationary 2D Gaussian process with non-symmetric auto-covariance function

Let $(X_t, Y_t)$ be a stationary 2D Gaussian process, therefore $\mathbb{E}\left(X_t\right) = \mathbb{E}(Y_t) = 0$. I am looking for an explicit example of a valid auto-covariance matrix, i.e: $$ ...
3
votes
2answers
1k views

Expectation of Brownian motion Integral

I want to calculate $\mathbb{E} \left[\left(\int_0^tB_s\text{d}B_s\right)^3\right]$ where $B_t$ is a standard Brownian motion. Using Ito's formula for $f:\mathbb{R}\rightarrow\mathbb{R}$ with ...
2
votes
2answers
142 views

Relation between $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$

Let $B_t$ be a standard Wiener motion. What can we say about $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$? Is there a relation?
2
votes
1answer
180 views

What is the conditional distribution of $B(s)\mid B(t_1)=x_1,B(t_2)=x_2$ for $0<t_1<s<t_2$?

Given that $\{B_t,t\ge0\}$ is a standard Brownian process. What is the conditional distribution of $B(s)$ given $B(t_1)=x_1$ and $B(t_2)=x_2$, for $0<t_1<s<t_2$? My try: First i tried to ...
1
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1answer
99 views

calculate $E(e^{B(t)}|e^{B(u)},0\le u\le s )$

Suppose $\{B_t,t\ge0\}$ be a standard brownian motion and suppose $0\le u\le s<t$, calculate $E(e^{B(t)}|e^{B(u)},0\le u\le s )$. Attempts: $E(e^{B(t)}|e^{B(u)},0\le u\le s ...
1
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2answers
174 views

What's the probability of a gambler losing \$10 in this dice game? What about making \$5? Is there a third possibility?

Can you please help me with this question: In a gambling game, each turn a player throws 2 fair dice. If the sum of numbers on the dice is 2 or 7, the player wins a dollar. If the sum is 3 or 8, ...
2
votes
0answers
58 views

Find distribution of time instants where a random signal assumes fixed value

I have a stationary and ergodic stochastic process $N(t,\omega)$. For a fixed $t^*$ I know the distribution of random variable $N(t^*,\omega)$. Is there any way to know the distribution of time ...
4
votes
1answer
2k views

Expectation of Stopping Time w.r.t a Brownian Motion

How do you take the expectation of a stopping time with respect to a Brownian motion? The specific question is: $$ \tau = \inf\{ t \ge 0: B(t) \in \{-a, b\}\} $$ I understand the optional stopping ...
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votes
1answer
50 views

What is the expectation of $E(\min_{1\le s\le 2}B_S)$?

Let $\{B_t:t\ge0\}$ be a standard brownian process. What is the expectation of $E(\min_{1\le s\le 2}B_S)$? I think the problem is i am not sure how $\min_{1\le s\le 2}B_S$ is distributed. I try that ...
0
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0answers
205 views

Explain the existence of limit in Persistence Excitation — mostly zero and non-existent?

Definitions Persistence Excitation on page 121 here or shortly here and here. A signal is PE if this limit exists $$r_u(\tau)=\lim_{N\rightarrow\infty}\frac 1 N \sum_{t=1}^{N} ...
3
votes
2answers
684 views

Show that this process is a martingale

Let $B_t$ be a Brownian motion and $M_t=\max_{0\leq s\leq t}B_s$. Show that: $$(M_t-B_t)^4-6t(M_t-B_t)^2+3t^2$$ is a martingale for $t\geq0$.
2
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1answer
44 views

Quasimartingale is Quasi-Dirichlet process

a paper I read states, that a Quasimartingale (an process $(X_t)_{t\in [0,T] }$ with $\mathbb E[|X_t|]<\infty$ for all $t\in [0,T]$, which suffices $$\sup_\Delta \sum^{n-1}_{j=0} \left\|\mathbb ...
0
votes
2answers
557 views

Using the Memoryless Property to Explain the Expected Value of the Maximum of iid Exponential RVs

Let $T$ and $V$ be independent random variables that are exponentially distributed with rates $\lambda$ and $\mu$. Consider their maximum, $$W = \max(T,V)$$ From the answer to a previous post, I ...
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1answer
434 views

Mean Square Differentiability

If I want to show that a stochastic process is not mean square differentiable, is it enough to show, that the process $a.s.$ does not have differentiable sample paths?
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146 views

Convergence in distribution of Wiener Process

I am considering this in the sense that I know according to the central limit theorem, for an i.i.d. process $X_n$ (with mean $m$ and variance $σ^2$), the corresponding normalized sum process is: $$ ...
4
votes
1answer
951 views

Why does the Central Limit Theorem break down for the Poisson Process

I am considering this in the sense that I know according to the central limit theorem, for an i.i.d. process $X_n$ (with mean $m$ and variance $σ^2$), the corresponding normalized sum process is: $$ ...
0
votes
1answer
765 views

Combinatorics question, ways to distribute 20 cookies among 7 children

You have 20 identical cookies that you want to distribute amoung 7 children, without breaking the cookies, how many ways can this be done... a) if every child gets at least one cookie? b) if not every ...
3
votes
1answer
95 views

Cheeger's inequality: Markov chain version is a special case of graph version?

For a Markov chain the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite-state, discrete-time, reversible stationary Markov chain. It can be seen as a ...
2
votes
1answer
396 views

Let X be a geometric random variable with parameter p compute E[X^3]

Let X be a geometric random variable with parameter $p$ compute $E[X^3]$. How would I approach this and how would I simplify the series? Can I use a moment generating function? I am able to write ...
1
vote
1answer
417 views

$X$ is a Geometric random variable find the expectation of $1/X$

Let $X$ be a geometric random variable with parameter $p$, find the expectation of $E[1/X]$. I need help simplifying the series.
1
vote
1answer
220 views

Example of a reversible Markov chain which has a stationary but non-reversible distribution?

For a Markov chain, I define a reversible distribution to be a distribution wrt which the MC is reversible to. A stationary distribution is defined as a distribution that once reached will stay. A ...