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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Deriving the process of successfully consumed requests from the process of request-producers and the process of request-consumers

The title is not very straightforward I understand, but you will soon realize it was not so simple to describe in few words this problem. The problem Consider a system consisting of: A process of ...
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1answer
350 views

Wiener process with a random mean [closed]

I have found this kind of stochastic process $$ dX=dW-{\rm sgn}(dW)dt. $$ What would the probability distribution be for $X$ assuming that the distribution for ${\rm sgn}(dW)$ is a Bernoulli with ...
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131 views

A different Markov property definition

In Shreve's Stochastic Calculus in Finance, the Markov property is defined as Definition 2.3.6. Let $(\Omega,\mathcal F,P)$ be a probability space, let $T$ be a fixed positive number, and let ...
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1answer
138 views

Application of Fokker-Planck equation in Ito calculus

In http://markov.uc3m.es/2009/02/ito-calculus-for-the-rest-of-us/, is derived. But I don't get this: after all, the process is defined as - which means that $f(X,t)$ in this context is zero (or am I ...
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99 views

Basic partial derivative calculus and Ito Calculus

In http://markov.uc3m.es/2009/02/ito-calculus-for-the-rest-of-us/, after some statements about processes, it says that Now I am not getting how this is resulted. Can anyone explain this? This seems ...
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269 views

When is a stochastic process defined via a SDE Markovian?

I was wondering when a stochastic process defined via a SDE is Markovian? The SDE may involved Ito integral, Lebesgue integral, jump component, and any other things. The reason I ask this question is ...
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1answer
96 views

Calculating the average of $\sin^2$ of a stochastic process

I have a random process $\phi_t$ which evolves according to the SDE $$d \phi_t = \mu dt+ \sigma \sin \phi_t \,dW_t$$ with $\mu$ and $\sigma$ constants and $W_t$ a Wiener process. The initial condition ...
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1answer
227 views

Distribution of the integral of a diffusion process

Suppose $X(t)$ is a diffusion process with $E[X(t)]=0$ and variances $\sigma^2_t$ concave in time. If $X$ is also a Brownian motion, then the distribution of $\int_0^T X(t) dt$ is known to be ...
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0answers
360 views

Chapman-Kolmogorov equation for conditional probabilities?

From Wikipedia (note that I have modified it from for a Markov process to for a general stochastic process): the conditional probability density $p_{i;j}(f_i\mid f_j)$ is the transition ...
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1answer
131 views

functional analytic interpretation of the (co)variation and the doob decompostion

I have a question concerning the covariation of two time-discrete stochastic processes. Let $(\mathcal{F}_i)_{i\in T}$ be a filtration. We call a time-discrete, real-valued, adapted process $X$ ...
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1answer
88 views

How to prove that for Brownian motion in $(a, b)$ $\mathbb{E}^x[\min(H_a, H_b)] = (x-a)(b-x)$?

i'm wondering if anyone can help me with proving the fact that for BM in the interval $(a,b)$ and with $$H_y = \inf\{t>0: X_t = y\},$$ the following is true: $$\mathbb{E}^x[\min(H_a, H_b)] = ...
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1answer
120 views

Finding the exact solution of a difference equation

We know a particle moves two units to the right with probability $p$, or $1$ unit to the left with probability $q$, hence $(p+q=1)$. $$q_k=P\left(S_n=0\mid S_0=k\right)$$ We are asked to find the ...
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78 views

lower bound product of correlated gaussian random variables

I am looking for the lower bound of $$P(XY>u)$$ where $X$,$Y$ are correlated centred Gaussian random variables.
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122 views

Upper bound concerning Snell envelope

Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the habitual ...
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1answer
134 views

State Space and Sigma Algebra for stochastic Process

Let $X_t$ be a random variable on $(\Omega,\mathcal{F} ,\rho)$ for all non-negative $t$. From what I understand, the state-space for the stochastic process is the infinite tuple $(\omega_1, ...
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1answer
406 views

Markovian and the Chapman–Kolmogorov equation

From Wikipedia In a Markov process, one assumes that $i_1 < \cdots < i_n$. Then, because of the Markov property, $$ p_{i_1,\ldots,i_n}(f_1,\ldots,f_n)=p_{i_1}(f_1)p_{i_2;i_1}(f_2\mid ...
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78 views

Stochastic Exponential: $dZ=-\lambda Z dM + dL$ to $dZ=-\lambda Z dM + Zd\tilde{L}$ while $\tilde{L}$ is still orthogonal to $M$

I have a question concerning the paper http://www.researchgate.net/publication/228648002_No_arbitrage_and_the_growth_optimal_portfolio, Lemma 6.3, which is based on ...
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61 views

Question on Conditional Probabilities

This one is from Introduction to Probability Models by Sheldon Ross. Not homework, was just trying to solve through some exercises again. Let $X$ and $Y$ be independent exponential random variables ...
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2answers
533 views

Stochastic process and unit variance

What does it mean when in stochastic process, we say that the process has unit variance? What is its exact definition?
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67 views

Is independence preserved in this special setting under a change of measure?

This is a question due to the answer of Did in this post Independent increments of $X_t:=\int_0^t\phi(s) dW_s$. Precisely, we assume that the dynamics of a stock prices follows ...
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178 views

Mean Duration of Stochastic/Markov Game

An urn contains five red and three green balls. The balls are chosen at random, one by one, from the urn. If a red ball is chosen, it is removed. Any green ball that is chosen is returned to the urn. ...
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324 views

Markov Chains Probability

A Markov chain $X_0$, $X_1$, $X_2$, ... has the transition probability matrix $$ P = \left[ \matrix { 0.3&0.2&0.5 \\ 0.5&0.1&0.4 \\ 0&0&1 } \right] $$ and is known to ...
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839 views

Recursion for Finding Expectation (Somewhat Lengthy)

Preface: Ever since I read the brilliant answer by Mike Spivey I have been on a mission for re-solving all my probability questions with it when possible. I tried solving the Coupon Collector problem ...
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173 views

Wiener process and joint distribution of $M_t$ and $W_t$

Why is $f_{M_t,W_t}(m,w) = \frac{2 ( 2 m - w)}{t\sqrt{2 \pi t}} e^{-\frac{(2m-w)^2}{2t}}, m \ge 0, w \leq m$ ? I now know what running maximum is, but unsure why joint distribution goes as above ...
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1answer
542 views

Running maximum of Wiener process

The joint distribution of the running maximum $ M_t = \max_{0 \leq s \leq t} W_s $ and $W_t$ is $f_{M_t,W_t}(m,w) = \frac{2 ( 2 m - w)}{t\sqrt{2 \pi t}}e^{-\frac{(2m-w)^2}{2t}}, m ...
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279 views

Does an independent-increment Gaussian process necessarily have Gaussian increments?

Suppose a stochastic process is both independent-increment and Gaussian. Are all its increments Gaussian distributed? Thanks!
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101 views

Alternative definitions of stochastic processes?

A stochastic process is defined as a family of random variables $\{X_t: \Omega \to S, t \in T\}$ , where $\Omega$ is a probability space, $T$ is a set, and $S$ is a measurable space. Equivalently, a ...
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1answer
686 views

Multidimensional infinitesimal generator of a jump-diffusion

Let $X=\{X_t\}_{t\geq0}$ be an $n$-dimensional Markov process, defined by the SDE $$dX_t = \mu(t, X_t) \, dt + \sigma(t,X_t) \, dB_t+\beta(t-,X_{t-}) \, dN_t,$$ where $\mu, \sigma$ and $\beta$ are ...
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53 views

Stochastic Processes Question

Give an example of a stochastic process $X_{n}$ that is not a Markov chain, such that $P_{y}(N(y)=\infty)=0$ but $E_{y}N(y)=\infty$
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255 views

Stochastic process as an Ito integral with time-dependent integrand

Will the following process $$r(t)=\int_0^ta(s,t)dW(s)$$ be adapted to the Brownian motion $W(s)$? Will $r(t)$ be an Ito process? Edit: Maybe I should rephrase it a bit. The question is: does ...
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87 views

Integral function being measurable or not

Consider a conditional measure $m(\cdot | \cdot)$ such that 1) $x \mapsto m(Y\mid x)$ is measurable on $\mathbb{R}^n$ for all Borel sets $Y \subseteq \mathbb{R}^m$; 2) $m(\cdot \mid x)$ is ...
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1answer
104 views

Stochastic Process measurability

I've come across a statement that i cannot grasp. When $T=[0,\infty)$, $E=\mathbb{R}$ and $\xi=B(\mathbb{R})$ then the collection of all continuous E-valued functions is not $\xi^T$-measurable. Here ...
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640 views

Can we prove directly that $M_t$ is a martingale

Suppose we define the stochastic process $$M_t:=e^{\int_0^t\phi_s dW_s -\frac{1}{2}\int_0^t\phi_s^2ds}$$ where $\phi\in L^2[0,T]$, $t\in [0,T]$. Note that $M_t$ is just the stochastic exponential of ...
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1answer
62 views

Sample continuity and continuity at every index value

From Wikipedia Let $(Ω, Σ, P)$ be a probability space, let $T$ be some interval of time, and let $X : T × Ω → S$ be a stochastic process. For simplicity, the rest of this article will take the ...
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74 views

Does a process satisfying this condition for jumping between states necessarily have exponential waiting time?

This is a property of Poisson process. But I will ask about it for a more general process. For a stochastic process $X$ with continuous time and discrete state space, if $\forall i$ in the state ...
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4k views

What's a good intro book to stochastic processes?

I'm taking stochastic processes now (I'm an undergrad math major), and feel the book we're using is inadequate and lacks detail. Here is a link to the book: ...
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104 views

Variance for the distance between two Brownian particles vs. a Brownian particle and a stationary particle

I have two Brownian particles, $B_1$ and $B_2$ (with diffusion coefficients $D_1$ and $D_2$), at coordinates $P_1$ and $P_2$ in a three-dimensional fluid. I let the system evolve for $t$ seconds. ...
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1answer
40 views

Motion of the centroid of $k$ Brownian particles?

Imagine we have $k$ Brownian particles diffusing in a three-dimensional solution, where each particle has the same diffusion coefficient $D$ (measured in $\mu^2/sec$). Now imagine that we have a ...
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1answer
55 views

Breaking median records

$\newcommand{\median}{\operatorname{median}}$ Suppose $X_1,X_2,X_3,\ldots$ are independent identically distributed real random variables. The process $Y_n=\max\{X_1,\ldots,X_n\},\quad n=1,2,3,\ldots$ ...
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1answer
563 views

Classify the states of a markov chain

a) P =$\begin{bmatrix} {1-2p} & 2p & {0} \cr {p} & {1-2p} & {p} \cr {0} & 2p & {1-2p} \cr \end{bmatrix}$ b) P = $\begin{bmatrix} 0 & p & 0 & 1-p \cr 1-p & 0 ...
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1answer
112 views

Conditional independence of components of an $n$-dimensional Hawkes process

Let's say we are given a $n$-dimensional Hawkes point process. To be more precise, let $N = (N_1,\dots,N_n)$ be a point process and $\mathcal{H}_t=\sigma\{N_1(s),\dots,N_n(s):0\leq s\leq t\}$ be the ...
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1answer
126 views

Branching Process

Consider a branching process $X=\{X_n, n=0,1,\dotsc\}$ where $X_n=\sum\nolimits_{i = 1}^{{X}_{n-1}}{Z_i }$ , $X_0=1$, and let $Z_i$ be such that $P[{Z_i=0]}=1/2$, $P[Z_i=1]=1/4$, $P[Z_i=2]=1/4$. ...
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1answer
359 views

How to show that a stochastic process is a measurable function-valued mapping?

Given a probability space $\Omega$, a measurable space $S$ and a set $T$, a stochastic process $X: \Omega \times T \to S$ is defined as a $T$-indexed family of random variables $\{X_t: \Omega \to S, t ...
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134 views

How to compute conditional probability for a homogeneous Poisson process?

Let $N$ be a homogeneous Poisson process with intensity $\lambda$. How do I compute the following probability: $$P[N(5)=2 \, | \, N(2)=1,N(10)=3]?$$
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120 views

Order Statistics : What's the deal with $n!$?

Assuming I have $n$ IID Random Variables $X_1,...,X_n$ the Order Statistics of this set $:= (Y_1,...Y_n)$ has a distribution of $f_Y(y_1,...,y_n)=n!f_X(x_1,...,x_n)$ I have read a couple of ...
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1answer
131 views

Find E[N]. (Absorbing States MC)

Assume we toss a dice until odd number appears four consecutive times, and in such a case we stop the game. Let $N$ be the total number of times we will toss the dice. Find $E[N]$. Lets say for ...
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1answer
125 views

Convergence to Brownian motion integral

Let $X_i$ be i.i.d with $\mathbb{E}(X_i) = 0$ and $Var(X_i) =1, \, S_n = \sum_{i=1}^n X_i$. I would like to show that $\sum_{i=1}^n \frac{f(S_i/\sqrt{n})}{n}$ converges to $\int_0^1 f(B_t)dt$ in ...
4
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205 views

Brownian motion integral

Let $(B_t)$ be a standard Brownian motion, $f$ a continuous function and $X_t = \int_0^t f(s)B_s ds$. I was able to prove that $(X_t)$ is a Gaussian process with zero mean and trying to find the ...
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49 views

Is a stochastic process being Markovian or Martingale completely determined by its law?

Suppose a stochastic process is Markovian. Let $L$ be its law on its sample path space (note that here I assume its initial distribution is known, not just conditional distributions). If there is ...
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141 views

A question about discrete and continuous-time Markov Chains

I have a test tomorrow about Stochastics Process and I couldn't solve the following questions: A gambler starts with 500\$ and plays till he runs out of money. In each round the probability to win ...