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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265 views

Autocovariance of moving average process

Let $\epsilon_t\text{ ~ i.i.d.}(0,1)$, and $X_t=\epsilon_{t}+0.5\epsilon_{t-1}$. I need to find its autocovariance function. I know that $E(X_t)=0$, $E(\epsilon_{t})=0$. Let's say, that $s=t+1$: ...
2
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0answers
91 views

References for basics of Piecewise-Deterministic Markov Processes

I am looking for introductory/pedagogical material to Piecewise-Deterministic Markov Processes (see http://en.wikipedia.org/wiki/Piecewise-deterministic_Markov_process) (For the moment I am interested ...
2
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1answer
329 views

Example of a martingale and a stopping time with $E(T)<\infty$ but $E(X_T) \neq E(X_0)$

Is there an example of a martingale in discrete time $X_0, X_1, X_2,\ldots$ and a stopping time $T$ so that $E(T) <\infty$ but $E(X_T) \neq E(X_0)$? With added assumptions on how $X_n$ behaves, ...
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0answers
94 views

Stopping time inequality Markov process

Let X be a right-continuous Feller-Dynkin process and define the stopping time $$\nu_{r}=\inf\{t\geq 0\mid ||X_{t}-X_{0}||\geq r\}$$ Let $B_{x}(\epsilon)=\{y\mid ||y-x|| \leq \epsilon\}$, for $x$ not ...
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53 views

Distributions representable as Ito diffusions

This is inspired by the following question. Let $X_t$ be an Ito diffusion on the interval $t\in [0,1]$: $$ \mathrm dX_t = a(X_t)\mathrm dt+ b(X_t)\mathrm dW_t $$ where say $a,b$ are Lipschitz ...
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2answers
178 views

why is the expected value of a Wiener Process = 0?

This section of wikipedia says that the expected value of a Wiener Process is equal to 0. Why is that?
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1answer
40 views

how comes this mean recurrect time

I have this transition matrix: $$P=\begin{pmatrix} 0.9 & 0.1 \\ 0.4 & 0.6 \end{pmatrix}$$ I need to find the stationary distribution. I know two formulas for this: $\pi = (\pi_1,\pi_2) = ...
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1answer
82 views

Measure of $\{t:B_t\in E\}$ for some null set $E$.

I am wondering if the following result can be found in any textbook or if you have a proof of it. When $E$ is a null set and $B_t$ is the Brownian motion, we have almost surely : ...
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182 views

Removing deterministic discontinuities from semi-martingales

Let $X:=(X_t)_{0 \le t \le T}$ be a solution of the SDE $$ X_t = X_0 + \int_0^t \sigma(s,X_s) dW_s + \sum_{i=1}^n f_i(X_{t_i^-}) 1_{\{t > t_i\}}$$ where $t_1,\cdots,t_n \in [0,T]$ and $(f_i)_{1 \le ...
2
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1answer
789 views

Applying Geometric Brownian Motion itself into Ito's Lemma

I am new to this forum as well as partial differentiation. I would like to ask the following question. Given a geometric brownian motion: $$dS_t=(\mu S_t)dt+(\sigma S_t)dz_t$$ I would like to apply ...
2
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1answer
112 views

approximating essential supremum

Let $(\Omega,\mathbb{F},P)$ be a filtred probability space. For $t\in [0,T]$, we are given sets $U_t$ of non negative stochastic processes $X=\{X_s;0\le s\le T\}$. We know that for $s\le t$ we have ...
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0answers
50 views

Expectation of the product of a sequence in a simple Markov chain

This is probably simple to those who know stochastic processes, but I am finding it difficult to understand how to solve expectations of a sequence. If $y(t)$ is a simple Markov chain where $y(t) = ...
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1answer
187 views

existence/uniqueness of solution and Ito's formula

Given the Ito SDE $$ dX_t=a(X_t,t)dt + b(X_t,t) dB_t $$ where $a(X_t,t)$ and $ b(X_t,t)$ satisfy the Lipschitz condition for existence and uniqueness of solutions. Given a function $f(X_t,t) ∈ C^2$ ...
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3answers
128 views

why is this Markov Chain aperiodic

I have this Matrix: $$P=\begin{pmatrix} 0 & 1 \\ 0.3 & 0.7 \end{pmatrix}$$ this markov chain is said to be aperiodic, I dont understand how it comes to it. Period $\delta$ is the gcd of ...
2
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1answer
508 views

Kunita Watanabe decomposition

I have a question about generalization of the Kunita Watanabe decomposition. I've learned the following version: Let $M$ be a continuous local martingale. Then every continuous local martingale ...
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142 views

Why can't I use the variance of the sample average in the Central Limit Theorem for the weak-stationary process?

Under mild conditions $\dfrac{\bar{X}-\mu}{\sqrt{\sigma^2/n}}$ approaches the standard normal (where $\sigma^2$ is the process variance, not the marginal variance $\sigma^2_x$). Why is the ...
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1answer
104 views

Absorbing probability and Martingale

This is a problem from the book "Markov chains". Let $(X_n)_{n\ge 0}$ be a Markov chain on $I$ and let $A$ be an absorbing set in $I$. Set $$T=\inf\{n\ge 0 : X_n \in A\}$$ and $$h_i = \mathbb{P}_i(X_n ...
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1answer
76 views

Law of large numbers for a Subordinator.

Let $\left( X_{t}\right) _{t\geq0}$ be a subordinator with the Laplace exponent given by $$ \Phi\left( \lambda\right) =d\lambda+\int_{0}^{\infty}\left( 1-e^{-\lambda x}\right) \nu\left( ...
2
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1answer
69 views

Weird submartingale sequence, $X_n \to -\infty$ but $E(X_n) \to +\infty$.

could you give an example of a sequence $(X_n)$ being a submartingale such that $X_n \to -\infty$ but $E(X_n) \to +\infty$? Thanks a lot!
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2answers
1k views

Conditional distribution in Brownian motion

I need to prove the following: Let $X$ be a Brownian motion with drift $\mu$ and volatility $\sigma$. Pick three time points $s < u < t$. Then, the conditional distribution of $X_u$ given ...
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1answer
77 views

A question about continuity of stochastic processes

Let $X_t$ and $Y_t$ (for $t\geq 0)$ be stochastic processes such that $P(X_t=Y_t)=1$ for every $t \geq 0$. Assume that $X_t$ has a.s. continuous trajectories. If $t_n\rightarrow t$ then clearly ...
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1answer
429 views

Cumulative distribution function (of NHPP inter-arrival time) not tending to 1?

According to this website, for a non-homogeneous Poisson process with mean $m(t) = \int^t_0 \lambda(u) \, dt$, the cumulative distribution function (CDF) for the inter-arrival time to the first event ...
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1answer
51 views

Poisson process (simple question)

Imagine you have two events starting at the same time. The duration time for each event is exponential, with different parameters. Knowing that one of the events is finished (we don't know which) at ...
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1answer
88 views

A equivalent definition of the Feller Process.

I saw this on Liggett's Book (P.95). Let $S=% %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion ,$ and suppose $\left( X_{t}\right) _{t\geq 0}$ is a continuous-time Markov process with ...
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1answer
104 views

A question about Infinitesimal generator of Feller Process

Let $S=% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $, and consider the Feller process $\left( X_{t}\right) _{t\geq 0}$ with state space $S$ such that $X_{t}=t+X_{0}$ for all ...
3
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0answers
127 views

Measurability of number of upcrossing $U_I(\alpha,\beta; X)$ in continuous time

These definitions come from Karatzas and Shreve, Brownian Motion and Stochastic Calculus. We may take for granted that $U_F(\alpha,\beta; X(\omega))$, the number of upcrossings over $[\alpha,\beta]$ ...
3
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1answer
195 views

question about Ito's formula

I'm currently learning about the Ito's lemma / formula In my textbook, a direct application of the formula is to compute quantities like that : (W is a Brownian motion) While trying to prove these ...
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2answers
418 views

How to solve this stochastic integrals?

how can I solve these two stochastic integrals? $$\int_0^T B_t\,dB_t$$ $$\int_0^T f(B_t)\,dB_t$$ where B_t is the BM. Thank you very very much!
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1answer
654 views

Brownian bridge

Let $W = (W_t;F_t)$, $t \leq 0$ be a standard Wiener process, and let $(X_t)_{0 \leq t \leq 1}$ satisfy the stochastic differential equation $$ dX_t =- \frac{X_t}{1-t}dt+dW_t,\quad 0 \leq t \leq ...
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1answer
77 views

Implications between $\mathbb P [\tau < \infty] =1 $ and $\tau \in L_1 (\mathbb P)$

We've got the usual filtered stochastic basis $(\Omega, \mathcal F, (\mathcal F_n). \mathbb P), \space \tau : \Omega \to \mathbb{N}\cup \{\infty\}, [\tau \le n] \in \mathcal F_n$ ($\tau$ is an ...
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1answer
129 views

how do I model probable time until simultaneous availability?

Short question: If several people, all of whom have limited availability, need to meet, how far in the future will I need to schedule the meeting? I was hoping there was a readily available answer ...
2
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1answer
103 views

Lebesgue–Stieltjes integral from 0 to $\infty$ on $\mathbb{R}^+$

In the Stochastic analysis course we encountered the following integral $\int_0^\infty H^2_sd[M,M]_s$, where $H_s$ is a predictable process, $M_s$ is a uniformly integrable martingale in $L^2$, ...
2
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1answer
234 views

When is the following local martingale strict local martingale?

By Section 5.5 of the book [Karatzas and Shreve 1991], the following 1-d SDE has unique weak solution in the form of \begin{equation} d X_{t} = X_{t}^{\gamma} \cdot I_{\{X_{t}\ge 0\}} dW_{t}, \ ...
2
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1answer
159 views

Ito differential equation

Define $$X_t := \left( \begin{matrix} \cos W_t \\ \sin W_t \end{matrix} \right).$$ where $W = \left( W_t,\mathcal F_t \right) _{t\ge0}$ is a standard Wiener process. Find the Ito differential of X ...
2
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1answer
58 views

Proof that stochastic process on infinite graph ends in finite step.

Infinite Graph Let $G$ be an infinite graph that is constructed this way: start with two unconnected nodes $v_1$ and $u_1$. We call this "level 1". Create two more unconnected nodes $v_2$ and $u_2$. ...
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2answers
105 views

stochastic integral equation [closed]

For $0 \leq t \leq T$, define $$Z_t:=\exp {\left\lbrace \int_0^t X_sdW_s - \frac 12 \int_0^t X_s^2ds \right\rbrace }$$ Show that this process satisfies the stochastic integral equation ...
2
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1answer
89 views

Stochastic differential equation problem and applying ito formula

I am given that for $b,a,\sigma >0$ and $x \in (-a,b)$ and $\nu \in \mathbb{R}$, I have the following stochastic differential equation: $$ dZ_t = \nu \,dt + \sigma\, dW_t$$ $$ Z(0) = x$$ and ...
3
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1answer
126 views

Basic doubt about stochastic integrals over general local martingales

Consider $M = (M_t)$ is a continuous square integrable local martingale and $$ \mathbb H ^2(M):= \left \{ \psi =(\psi_t)\ \text{is a real previsible process s.t.,} \forall t\geq 0, \ \mathbb E\left ...
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0answers
58 views

Maximum of a random number of hitting times for a martingale

Let $(M_n)_{n\in \mathbb N}, M_n\geq 0$ be a positive martingale and define $T:=\inf \{n\in \mathbb N \vert M_n=0\}$ as the first hitting time of $0$, knowing that $\mathbb P(T < \infty)=1 $. Let ...
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1answer
178 views

generalized derivative of Wiener process

Defined a standard Wiener process $W = (W_t , \mathcal F_t)_{t≥0}$ and a deterministic, continuously differentiable function $f : [0, ∞) → \mathbb R$. Prove that ...
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1answer
60 views

How to show $t{B^{2}_t}+t^2$ is $\mathcal{F}_t$-addapted process?

How to show $t{B^{2}_t}+t^2$ is $\mathcal{F}_t$-adapted process? Here $B_t$ is Brownian Process. Please Help
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0answers
278 views

Good books on “advanced” stochastic analysis

Any good books suggestion for studding advanced features of stochastic analysis ? Thank's in advance
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1answer
93 views

Tricky question about Ito's stochastic integral and continal law

Consider $B=(B_t)_{t\geq 0}$ real $\mathcal F_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. Then, consider a new real ...
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1answer
45 views

Prove that $Z_k := \int_{T_{2k+1}}^{T_{2k+2}}f^2(B_s) ~ds$ are independent and have same law

Consider $B=(B_t)_{t\geq 0}$ a real $\mathcal F_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. Then, consider for $a,\ b \ ...
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1answer
66 views

Process' mean, covariance and stationarity

Let $X_t=Yt+Zt^2$ be random process, where $Y$,$Z$ are uncorrelated random variables, with characteristics: $EY=3$, $EZ=0.5$, $DY=1$, $DZ=0.05$. Find $X_t$ mean and covariance and prove whether ...
2
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1answer
149 views

Proving that $T_t := S_t -\left| x \right| -\frac {n-1}{2} \int _0 ^t \frac {1}{S_u}~du$ is a brownian motion

Consider $B=(B_t)_{t\geq 0}$ $\mathcal F_t$ - brownian motion in $\mathbb R ^n, \ (n\geq 2)$ starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. ...
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1answer
98 views

Proving that brownian motion pass almost ever by zero in $]0,t[$

Consider $B=(B_0)_{t\geq 0}$ a real $\mathcal F_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. Then, consider $$ \Phi_t(x) ...
4
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1answer
235 views

Tricky a.e. limit question, studying the $ \text{a.e.-lim}_{t \rightarrow \infty} \frac {1}{t} \int_0 ^ t W_s ~ds$

Could someone give some advice in order to study $$\underset{t\to\infty}{\operatorname{a.e.-lim}} \frac {1}{t} \int_0 ^ t W_s ~ds,$$ where $(W_t)_{t\geq 0}$ is a standard brownian motion starting at ...
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0answers
126 views

Poisson point process convergence

Let Π be a Poisson point process on [0,∞) with intensity measure $\mu$. Assume $μ([0,t])<∞$ for all $t<∞$ and $μ([0,∞))=∞$. Also assume $μ({x})=0$ for all x. Prove ...
1
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1answer
163 views

Random walk probability non-symmetric steps

I currently have a probability class tutorial question that I have no idea where to begin. At first instinct, I thought it may have been a CTMC question or branching question, but now I have no idea, ...