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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A question about the proof that right-continuous modifications are indistinguishable.

I'm trying to understand the proof that if $X,Y$ are modifications $(P\{X_t = Y_t\}=1\,\, \forall t \in T$) of each other and are right continuous, that they are then indistinguishable ($P\{X_t = ...
2
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1answer
1k views

Different versions of Girsanov theorems?

I am reading two different versions of Girsanov theorem regarding change of measure to preserve Brownian motion. Wikipedia has the following Girsanov theorem: If $X$ is a continuous process and ...
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1answer
55 views

What does an unaugmented sigma field mean?

What does an unaugmented sigma field mean in Wikipedia's Girsanov's theorem? Then for each $t$ the measure $Q$ restricted to the unaugmented sigma fields $\mathcal{F}^W_t$ is equivalent to $P$ ...
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1answer
62 views

Existence Brownian Motion

I'm reading through a proof of the existence of a Brownian motion and at some point they state that for $0\leq t_{0}<t_{1}...<t_{n}$ there exist multivariate normal distributions with covariance ...
2
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388 views

Ito integral almost sure and $L^2$ limit

why does one define the Ito integral as the $L^2$ limit, although it can be shown by Doob's martingale inequality and Borel-Cantelli lemma that there exists a t continuous version, which is ...
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159 views

Is the following a martingale?

Let $X_{n}$ be a martingale with respect to a filtration $\mathbb{P}_{n}$. Define: $Y_{n}$ := $X_{n}^{3}$ Is $Y_{n}$ a martingale? Supermartingale?
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62 views

Asymptotic behaviour of sums of covariances of RVs with LRD

Our assumptions are: $X_t$ is a stationary sequence of standard normal random variables such that $\gamma _X (k)\sim L_{\gamma}(k)k^{2d-1}$ with $d \in (0,1/2)$, where $L_\gamma (k)$ is a slowly ...
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3answers
1k views

Finding the transition probability matrix, two switches either on or off..

Each of two switches is either on or off during a day. On day n, each switch will independently be on with probability [1+number of on switches during day n-1]/4 For instance, if both switches are on ...
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2answers
231 views

How to describe discretization to a novice?

While going through some C++ code about stochastic processes, I came across this concept of discretization repeatedly. I have checked the Wikipedia link but description goes into deeper details too ...
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70 views

Is there a canonical probability measure on smooth curves?

For continuous curves, we have Brownian motion giving the most natural probability measure. However, the sample paths of Brownian motion are almost surely terribly behaved (not of bounded variation, ...
4
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1answer
176 views

Example of a martingale which is not jointly measurable

Suppose we have a measurable space $(\Omega,\mathcal{F})$ and an $\mathbb{R}$-valued continuous-time (but not necessarily continuous) stochastic process $X$. $X$ is jointly measurable if it is ...
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147 views

Different meanings of $\int_0^T X(t) dt$, and its meaning in Ito isometry?

Given a stochastic process $X: [0,T] \times \Omega \to \mathbb R$, I realized there are different meanings of $\int_0^T X(t) dt$. $\int_0^T X(t, \omega) dt$, $\forall \omega \in \Omega$ or a.e., ...
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265 views

Infinite Doubly stochastic matrix questions

I have the following question about a Markov chain ${(X_n)}_{n \geq 0}$ with infinite irreducible doubly stochastic matrix $P$. We have the state space $\{1,2,...\}$ . Determine the stationary ...
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51 views

reference on SDE driven by Levy processes

I am reading on the Shephard Nielson model. I am just wondering if there are books on processes driven by jump processes? Can anyone give suggest a reference book for me on this?
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1answer
26 views

Proof involving different distributions in a discrete time Markov chain

Prove that if the initial distribution $a_0$ equals the stationary distribution $\pi$, then the transient distribution $a_n$ equals $\pi$ for all $n$.
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107 views

Superposition of simple birth process

Theorem Suppose $X$ and $Y$ are independent simple birth processes with birth rates $\lambda n$ with the same $\lambda$. Then $X+Y$ is a simple birth process rate $\lambda$ Proof let $Z=X+Y$ ...
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248 views

Proof of thinning theorem

Thinning theorem If $N= (N_t)_{t\geq0} $ is a poisson process rate $\lambda$ and it is thinned by removing incidents with probability p independently of each other and the poisson process, then what ...
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359 views

Integrability in Ito isometry

Itō isometry from Wikipedia: Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical real-valued Wiener process defined up to time $T > 0$, and let $X : [0, T] \times \Omega \to ...
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1answer
56 views

Numerically solve SDE

I am not really into solving stochastic differential equations, but I was trying to numerically solve an OED given by: $\frac{dy}{dt} = f(t,y,p) + N(0,\sigma^2)$ where normal noise with 0 mean and ...
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36 views

What does “s” represent in the solution of this stochastic problem?

http://i.stack.imgur.com/LZPmN.png My concern is what does s represent on the equation i circled?Is it the amount of Penalty he received or the number of ticket Harry get in order to achive 50 ...
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345 views

Canonical processes of a stochastic process

From a old handwritten note without references cited, the first canonical process of a stochastic process $\Omega \times T \to S$ is defined as the identity mapping on $S^T$. I was wondering if there ...
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1answer
335 views

Is an orthogonal-increment process submartingale?

If I remember correctly, a stochastic process is said to be orthogonal-increment, if it is a second-order process, and the increments over disjoint intervals are uncorrelated. I wonder if an ...
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1answer
236 views

Strong Markov property explained.

I have got 2 theorems, Theorem 1 The increment $ (N_{t+u} - N_t)_{u\geq 0} $ of a Poisson process rate $\lambda$ is again a Poisson process rate $\lambda$ and is independent of $(N_s)_{0\leq s \leq ...
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1answer
457 views

Books at similar levels as Kallenberg' Foundation of Modern Probability?

Thanks to many people who have mentioned it to me and others on this site before. I was just able to peek into Kallenberg' Foundation of Modern Probability. It is more comprehensive, deep and thorough ...
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60 views

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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341 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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127 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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130 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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1answer
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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3answers
247 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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220 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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349 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 ...
4
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1answer
129 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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86 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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77 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.
3
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1answer
117 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
132 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, ...
2
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1answer
393 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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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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1answer
59 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
506 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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65 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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177 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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320 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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2answers
802 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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1answer
162 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
513 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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261 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!