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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17 views

Please explain $E[S_{min(n,T)} ]= E [S_{0}]=0$

If $S_{n}$ is a simple random walk i.e $X_{k}= +/- 1$ with prob = 0.5 T = inf {n > = 0 |$S_{n}$ = 1} is a stopping time. T is finite almost surely. .Explain $E[S_{min(n,T)} ]= E [S_{0}]=0$ I know ...
3
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1answer
62 views

Filtrations and Sigma-Algebras

I have been practising a question set by my lecturer and try to verify the answer, unfortunately I am unable to understand the following question and answer. $\textbf{Question:}$ Let ...
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1answer
19 views

Transition rate matrix from transition probability matrix

If I have a $2 \times 2$ continuous time Markov chain transition probability matrix (generated from a financial time series data), is it possible to get the transition rate matrix from this and if ...
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0answers
40 views

About one stoping time definition in Chung's book (A Course in Probability Theory)

In Chung's book, he defines the stopping time $ \alpha^k $ in the following way. $\alpha^1 = \alpha$; $\alpha^{k+1}(\omega) = \alpha^k(\tau^\alpha \omega)$; where $\tau^\alpha$ is the ...
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1answer
16 views

Connection between Expected power and Expected Energy over Frequency - Dirac Delta Squared?

I know math people don't like the Dirac delta, so feel free to answer with your measure theory - I'll try my best to understand. Suppose $x$ is a WSS stochastic process $\{x[n] : n \in ...
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75 views

Generating a stochastic matrix with a given second dominant eigenvalue

I need a procedure (iterative or otherwise) that, given a positive integer $N$ and a (possibly complex) number $\lambda$ such that $0 < \vert \lambda \vert < 1$, will be able to generate an $N ...
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1answer
35 views

Invariance of Brownian motion under orthogonal transformations

Let $\left(B_t\right)_{t \in [0,\infty)}$ be an $n$-dimensional Brownian motion with start at $x \in \mathbb{R}^n$, and let $A$ be an orthogonal $n \times n$ real matrix. I'm trying to show that $AB$ ...
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14 views

Hitting time for a planar diffusion

Let $A$ be an open subset of $\Bbb R^2$, and let us consider a diffusion $\mathrm dX_t = f(X_t)\mathrm dt + g(X_t)\mathrm dW_t$ where $f$ and $g$ are globally Lipschitz continuous maps. Suppose I am ...
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19 views

Show that $ \text{ess-sup}_\Omega g (x+ B_T) = \sup_{y \in \mathbb R ^d }g(y)$

Show that $$ \text{ess-sup}_\Omega g (x+ B_T) = \sup_{y \in \mathbb R ^d }g(y)$$ where $B$ is a d-dimentional brownian motion , $x \in \mathbb R ^d $ and g a Lipschitz bounded function of $\mathbb R ...
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0answers
12 views

Characterizing limit of value functions in a stochastic control problem

Consider a probability space $(\Omega, \mathcal F , \mathbb P)$, $(B_t)_{t\geq0}$ M-dimentional brownian motion adapted to a filtration $(\mathcal F_t)_{t\geq0}$ over $\Omega$. In this context ...
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1answer
78 views

Approximation of stochastic processes in Protter

I'm reading Stochast integration and stochastic differential equation by Protter. In particular I have a question about Theorem 10 in chapter 2.4. Here Protter defines a simple predictable processes ...
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28 views

Continuous time Markov chains, how would this definition be expanded from time-homogeneous to time-inhomogeneous.

Below I have a picture of how we can view a continuous time Markov chain that is time-homogeneous. Now, I am wondering what happens when we have a inhomogeneous continuous Markov chain. I have ...
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0answers
41 views

Expectation of the infimum of a GBM

does somebody know a reference, where I can find the value of the expectation of the running infimum of a geometric Brownian motion, namely: Given a filtered probability space ...
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1answer
34 views

Markov processes and semimartingales

Semimartingales and Markov processes are two fundamental families in probability theory. There are many specific processes that belongs to the intersection of those two families, e.g. Levy processes. ...
2
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1answer
37 views

Three questions about ucp convergence

We say that a sequence of processes $X^n$ converges to a process $X$ uniformly on compacts in probability if for all $\epsilon >0, t>0$ $$P[\sup_{s\le t}|X^n_s-X_s|>\epsilon]\to 0 $$ for ...
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0answers
33 views

Stochastic Differential Equation- When martingale?

Suppose I'd like to check the martingale property for some SDE. What do I have to require for it to be martingale? I know that no drift is one requirement, but what are the others?
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29 views

$\int_t^T 1_C\cdot A\;d\!X=1_C\cdot\int_t^T A\;d\!X$ for $C\in\mathcal F_t$?

Given a semi-martingale $X$ on a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\le\infty},P)$, an integrand $A$ and a set $C\in\mathcal F_t$. Show: $$\int_t^T 1_C\cdot ...
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38 views

If two stochastic integrands are equal on some measurable set, will the stochastic integrals be equal on that set?

Given a $X$ semi-martingale on a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\le\infty},P)$ I am trying to prove: For any $B\in\mathcal F_\infty$ and processes $a_1,a_2$ such that ...
2
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1answer
42 views

The probability of a Brownian motion's tail event is unaffected by the starting point

Consider the measurable space $\left(\mathbf{C}\left[0,\infty\right), \mathcal{B}\left(\mathbf{C}\left[0,\infty\right)\right)\right)$ and the stochastic process $\left(X_t\right)_{t \in ...
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42 views

2-state HMM / ARMA process?

I have issues with this problem: Let $\{X_t, t\in \Bbb N\}$ be a 2-state stationnary Markov chain, with transition $M$ (and $M(1,2)\neq 0 \neq M(2,1)$), let $\{W_t, t\in \Bbb N\}$ be a strong ...
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2answers
71 views

Understanding basic stochastic differential equations

This is from a physics course in economics, the literature provides a bare minimum of mathematical explanations. I am trying to understand how to work with stochastic differential equations given in ...
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1answer
39 views

Interpretation of Power Spectral Density (DTFT of Covariance function)

If we have a deterministic signal $x[n]$ and its transform $$ X(f) = \sum\limits_{n=-\infty}^{\infty}x[n]\exp\left(-2\pi fn\right)$$ I can think of this as containing knowledge of a discrete-time ...
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3answers
46 views

The uniqueness of solution for stochastic differential equation involved with sign function.

When I read a paper about Levy distribution thoerem (http://www.maphysto.dk/publications/MPS-RR/1998/22.pdf). In the first page, the author mentioned the following: There is a unique strong solution ...
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0answers
18 views

The sign of pure jump Levy process

Suppose $(\Omega, \mathcal{F}, P)$ is a probability space. Assume $(X_{t}, P)$ is a Levy process with generating triplet $( 0, 0, \nu)$ with $X_{0}=0$. This means there is no continuous part in ...
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2answers
64 views

Why are these two Poisson-processes independent?

I have two poisson-processes, I have seen a mathematical proof that they are independent, and offcourse they must be independent since the proof is in several textbooks. But logically I can not ...
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198 views

Does the operator $T(f)(t) := f(t) - f(0)$ preserve measurability?

Denote by $\mathcal{B}$ the Borel field on $\mathbb{R}$, denote by $\mathbf{C}_{\left[0,\infty\right)}$ the set of continuous, real-valued functions over the domain $\left[0,\infty\right)$ and denote ...
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1answer
30 views

Markov Chain probabilities

I'm having trouble with this problem from Resnick's Adventures in Stochastic Processes: Consider a Markov Chain on states {0,1,2} with transition matrix $ \left( \begin{array}{ccc} 0.3 & 0.3 ...
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2answers
44 views

Poisson Process and Conditional Probability

Let $X= (X(t); t\ge0)$ be a poisson process with the intensity ($\lambda$ per hour) A) Find the conditional probability of having $m$ events in the first $t$ hours, given that there were $n$ events ...
2
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1answer
53 views

Diffusion processes

I am trying to work out a problem to which I have not found similar solutions on the website. Perhaps you can help me out. Let $X = (X_t)_{t\geq0}$ be a non-negative diffusion process which solves ...
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48 views

Applications of stochastic processes

What are stochastic processes? What are they used for? How can they be applied to real concepts? What is an example of a "stochastic process" problem?
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1answer
40 views

I have trouble understanding the proof of the Wold decomposition theorem

I'm trying to understand the proof of the Wold decomposition theorem in [1, p.187]. I find a few things about it very irritating. The theorem states: Theorem 5.7.1 (The Wold Decomposition). Let ...
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0answers
33 views

Expected value of stopping time of Stochastic Process.

I am trying to solve the following problem: Let $X$ be the strong solution of the following Stochastic Differential Equation: $\mathrm dX_t = sign(X_t)dt + \mathrm dW_t, X_0 = 0$, where $W_t$ is a ...
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0answers
37 views

Convergence a.s. implies convergence in probability

Let $$f,f^n \in L^2([a,b] \times\Omega) $$ such that $$ \int_a^b |f^n(t) - f(t)| dt \rightarrow 0 \;\;\; \text{a.s.}$$ Show that $\int_a^b f^n dB_t \rightarrow \int_a^b f dB_t$ in probability. Ok, ...
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1answer
48 views

Markov Chain: prove that state is positive recurrent by calculating expected # of transitions to return to this state

Given the transitional probabilities below (states: 0,1,2,3), I need to prove that state 3 is positive recurrent by calculating expected # of transitions to return to this state $$P = ...
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0answers
12 views

$N$-point distribution functions of Brownian local time

What is the most explicit formula for $N$-point distribution functions of the local (or occupation, sojourn) time of Brownian motion (Wiener process) with exponentially distributed time duration?
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0answers
26 views

Markov renewal process vs Markov Jump process

The venerable wikipedia for "Markov renewal process" says that: "a Markov renewal process is a random process that generalizes the notion of Markov jump processes". So what's the definition of a ...
2
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1answer
39 views

Distribution of Stopped Brownian motion at hitting time of another Brownian motion.

Suppose $B_t$ and $W_t$ are two independent Brownian motions and $\tau$ is the first hitting time of $B_t$ to some $a >0$. Compute the distribution of $W_{\tau}$. We can try the characteristic ...
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1answer
30 views

A few questions about Stochastic Processes and Numerical Methods

I am having a few problems understanding the Ornstein Uhlenbeck solutions, on wikipedia under solution (http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process) it described using variation of ...
3
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1answer
56 views

When are stable continuous time Markov chains Feller? Always?

This is a question is similar to this 2 year-old one that never got answered (truthfully it's pretty much the same question except that I'm adding a bit more detail and the assumption that the $Q$ ...
4
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0answers
70 views

Determine if this is a Martingale

I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$. We know that $S_t$ is a local martingale because if we stop it ...
2
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0answers
30 views

Local time of fractional Brownian motion

For BM, there is a downcrossing representation of the local time at 0. Namely, $L_t(0)=\lim_2 (b_i-a_i)D(a_i,b_i,t)$, where $D$ is the number of downcrossing between level $b_i$ and $a_i$. I am ...
3
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1answer
58 views

Sum of two Markov processes another Markov process?

Let $dX_{t} = m_1(l_1-X_{t})dt+\sigma_1 dW_{t}$ and $dY_{t} = m_2(l_2-Y_{t})dt+\sigma_2(\rho dW_{t}+\sqrt{1-\rho^2}dW_{t}^{1})$ where the $m_i$'s, $l_i$'s and $\sigma_i$'s are constants, $\rho \in ...
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138 views

One Question about Exponential Variables

This is a problem from Richard Durrett's Essentials of Stochastic Processes. Excited by the recent warm weather Jill and Kelly are doing spring cleaning at their apartment. Jill takes an ...
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1answer
36 views

Quadratic variation - Semimartingale

We know that any Semimartingale has Quadratic variation. I am interested to know if the converse is also true i.e. if a process has quadratic variation then it is semimartingale. Can some one ...
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1answer
69 views

Correlation between two stochastic processes [closed]

Let $$dX_t = k_1 X_t \, dt + \sigma_1 \, dW_t$$ and $$dY_t = k_2 Y_t \, dt + \sigma_2 \left( \rho \, dW_t + \sqrt{1-\rho^{2}} \, dW_t^1\right)$$ where $W_t$ and $W_t^1$ are independent. What is ...
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0answers
33 views

White noise example - but different from a Gaussian white noise signal

I kindly ask for some help in providing an example of white noise series, different from Gaussian white noise. Especially, I would like to know if there is a recipe to generate a series of white noise ...
2
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0answers
53 views

Random walk with $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} < \infty$

Consider a random walk started at $S_0=0$, denoted $S_n = \sum_{k=1}^{n}X_k$, where $X_1$, $X_2$... are the i.i.d increments. If we have $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} ...
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46 views

The difference between Dynamic Optimization, Stochastic Programming, Optimal control and Markov Decision Processes

I've seen the following terms thrown around somewhat interchangeably, and I'm confused. What are the distinctions between them, and what are some representative problems that each deals with? ...
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0answers
15 views

Are the two definitions of branching proceses equivalent?

From Wikipedia The most common formulation of a branching process is that of the Galton–Watson process. Let $Z_n$ denote the state in period $n$ (often interpreted as the size of generation $n$), ...
5
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2answers
80 views

Estimating the maximum of a Brownian motion over the unit interval

Let $\left(B_t\right)_{t \in \left[0,\infty\right)}$ be a standard Brownian motion over the probability space $\left(\Omega, \mathcal{A}, P\right)$. For each $x \in \left(0, \infty\right)$, give an ...