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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52 views

Markov chain problem 13 [closed]

I have this problem I don't understand, Can you help me, please?
1
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1answer
60 views

Every zero-mean Lévy process has linear variance (wrt $t$)

I'd like to show that every Lévy process with $\mathbb{E}X_t=0, \:\forall t\ge0$ has linear variance, namely $t\mapsto\mathbb{E}X^2_t$ is linear. I showed that indeed the additivity holds, i.e. ...
3
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1answer
33 views

Local maximum of brownian motions

Let $B=(B_t)_{t\geq 0}$ be the standard Brownian motion. I want to show that for every $t_0 \geq 0$ $\mathbb{P}$($B$ has a local maximum in $t_0$)=0. I've already shown that for every ...
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27 views

Stopping and optional times.

Let $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq0},P)$ be a filtered probability space. Put $\mathcal{F}_{t^+}:=\cap_{s>t}\mathcal{F}_s$ and $\{\mathcal{F}_{t^+}\}_{t\geq0}$ be the ...
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1answer
50 views

Concluding from limiting behavior

I've recently seen the following question on the internet: If I wanted to randomly find someone in an amusement park, would my odds of finding them be greater if I stood still or roamed around? ...
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1answer
157 views

Markov process on an Abelian group

Let $E$ be an Abelian group. Let $X$ be a right-continuous process with values in $(E,\mathcal{E})$ (where $\mathcal{E}$ denotes the $\sigma$-algebra on $E$), defined on $\Omega, \mathcal{F}_t,P)$. ...
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26 views

Absolute convergence of $\sum_{n=1}^{\infty} \mathbb{P}(|X|>n)$

I want to show that if $X \in L^1$, where $X$ is a real-valued random variable, the sum $\sum_{n=1}^{\infty} \mathbb{P}(|X|>n)$ converges absolute. My idea was the following: Since $X \in ...
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41 views

How to do integration by parts with brownian motion?

I am not sure how to perform integration by parts in the following expression: $$ \left(1-t\right)\left(B_t - B_s + \int_s^t \frac{r}{1-r} \mathrm{d} B_r \right) $$ Can anyone help me to solve this ...
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9 views

Markov Semigroups worked example

I have been reading this excellent paper on Markov semigroups, in which the assertion is made that a markov semigroup $\mathcal{P: L^1 \longrightarrow L^1}$ is defined by $\frac {d\mu}{dm}$ for some ...
3
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2answers
42 views

Markov chains diagram - what are the numbers above arrows?

Most if not all articles describe the numbers above arrows as probabilities of a transition in that direction, such as this one, or this one. But here, for example, something really weird is ...
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16 views

Expected generation of extinction in branching process with binomial offspring

Consider a branching process with immediate offspring distribution $\xi \sim \operatorname{Bin}(m, p)$, where $m$ is a constant. Let $\phi(s)$ be the generating function of $\xi$, i.e. $\phi(s) = (1 - ...
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17 views

Stochastic integral of local martingales is an extension

I'm trying to prove that the stochastic integral defined for the set of square integrable local martingales is really an extension of ordinary stochastic integral. Define $\mathcal{H}=\{(H_t)_{0\leq ...
4
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1answer
79 views

Application of the Burkholder Davis Gundy inequality

The proof of the Feynman-Kac formula uses a lemma which I need to proof, but I can not figure it out. The lemma is the folllowing: Let $X$ be a weak solution of ...
1
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1answer
13 views

Prove the following r-step transition

Let $X_0, X_1, X_2,...$ be a Markov Chain on state space $S=\{1, 2,..., n\}$ and let $P$ be the Transition Matrix of the above Markov chain Prove that $\Bbb{P}(X_{t+2}=j|X_t=i) = (P^2)_{ij} $ ...
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2answers
31 views

Distribuiton of stochastic integral

If $(W_t)_{t\geq 0}$ is a Wiener process, $X_0=0$ and for all $t$, $t>0$ and $\alpha>0$. $X_t=\int_0^t\frac{u^\alpha}{t}dW_u$. I have want to answer 2 questions: What is the distribution of ...
2
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0answers
23 views

weakly open subset in $M[0,1]$ (the space of finite measures on $[0,1]$)

I came across this question when reading Lynch and Sethuraman (1987): Large deviations for processes with independent increments. Let $M[0,1]$ be the space of finite nonnegative measures on $[0,1]$ ...
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14 views

Intuition behind Stationarity in Delayed Renewal Processes

I was going through excess life and renewal processes in my notes when I came across a proposition in my notes that said that given a delayed renewal process X with independant interarrival times ...
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1answer
30 views

Show that $\omega\mapsto\int_a^bX_t(\omega)\;dt$ is measurable, for a real-valued and continuos stochastic process $X$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $X=(X_t)_{t\ge 0}$ be a real-valued and continuous stochastic process on $(\Omega,\mathcal{A},\operatorname{P})$ $0\le a<b$ I ...
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1answer
32 views

Distribution of particles at infinite time

Let any site of $\mathbb{Z}$ host a number of particles $\eta_0(x)$ which is distributed according to some probability distribution independently and identically for any site $x \in \mathbb{Z}$. At ...
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56 views

Ornstein-Uhlenbeck a Markov process

Consider the Ornstein-Uhlenbeck process defined by $$ X_t = e^{- \alpha t} X_0 + \sigma \int_0^t e^{ \alpha (s-t)} d W_s$$ with $\sigma,\alpha>0$. In many literature I have found they considered ...
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1answer
31 views

Closure of the set of elementary predictable stochastic processes

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}=(\mathcal{F}_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal{A})$ $H=(H_t)_{t\ge 0}$ be a real-valued stochastic ...
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1answer
51 views

Compound Poisson Process Problem

I have the following review problem I've been working through and would appreciate any help towards solving it. Customers enter a store according to a Poisson process of rate $\lambda$ = 5 per hour. ...
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1answer
18 views

Why does differencing an ARMA model induce stationarity?

If I am working with the ARMA(1,3) model $$Y_t=Y_{t-1} + \epsilon_t + 0.9\epsilon_{t-3}$$ Where Y's are the observations and $\epsilon$ is a white noise process. I can decompose and find the lag ...
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1answer
37 views

Markov property when conditioning on future event

I am reading through an introduction to h-transform (available on https://linbaba.wordpress.com/2010/06/02/doob-h-transforms/), and came upon the following equality: $$P\left(X_{t+s}=y;X_T\in ...
3
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1answer
32 views

On random rotational fluctuations in $\mathbb{R}^n$

Imagine first a disk that is mostly stationary, except for random ("thermal" if you like) "rotational fluctuations" around its axis (which is fixed). Something a bit like what's shown in the figure ...
2
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1answer
86 views

A proof which results in Gamma (or Erlang) distribution-From Karlin & Taylor's “A First Course in Stochastic Processes”

The random variables X and Y have the following properties: X is positive, i.e., $P\{X > 0\} = 1$, with continuous density function $f_X(x)$, and $Y\mid X$ has a uniform distribution on $\{0,X\}$. ...
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1answer
30 views

$\sin(W_T)$ and Ito / Martingale Representation Theorem

I've been solving some exercises which require a function to be represented as an adapted stochastic process such that $$ X = \mathbb{E}[X] + \int_0^T \Theta(s)\,dW(s) $$ For example, $X = W(T)$ ...
2
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1answer
42 views

An equality involving the Wiener process

The equality below appears as a step in a proof in a chapter titled "Itô Stochastic Calculus" in Brzeźniak and Zatawniak's textbook "Basic Stochastic Processes", Springer 2005 (in a solution to ...
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28 views

Term Structure and short rates

If I have a term structure/yield curve given by: $$f(t, T) = f(0, T) + σ^2t(T − \frac{t}{2}) + σB_t $$ and want to find the short/spot rate $r_t$, is this simply: $$f(t,t) = f(0,t) + ...
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1answer
16 views

A proof about a Poisson Process

Let the $N(t)$ be a Poisson process with a rate $\lambda >0$. The sequence $T_1, T_2, ...$ is a sequence of interval times between events. The sums $S_k=\sum_{i=1}^{k}T_i$ are the moments in which ...
3
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2answers
51 views

Evolution of a discrete distribution of probability

I am designing a virtual card game and I defined an evolution of probabilities, but I don't have the knowledge on this matter to find out how they will evolve. I hope you help me here, with ...
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0answers
19 views

question about brownian motion and integration

If $X(t)$ is the standard Brownian motion, $0<\alpha<\beta$, and $T$ is the first exit time of $X(t)$ from $[-\beta,\beta]$, then how can I find $E(\int_0^T \mathbb{I}_{(-\alpha,\alpha)} X(t) ...
2
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1answer
56 views

Almost sure convergence of the Poisson process

Let $N = \{N(t) \}_{t\geq 0 }$ be a Poisson process. I already know that $N(t)- \lambda t$ is a martingale where $\mathbb{E} [ N(t) ] = \lambda t$. I want to prove that $$ \frac{N(t)}{t} \rightarrow ...
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0answers
18 views

Power spectral density

Please help can someone help me with these. Any help is accepted. Thanks
2
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0answers
53 views

Absorbed brownian motion is a Markov process

I have been asked to prove that the Brownian motion absorbed at the origin is a Markov process. Formally, let $B_t^x$ be a Brownian motion originating from $x>0$ and let $\tau^x_0 = \inf\{t>0 : ...
2
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2answers
21 views

Product rule with stochastic differentials

I am encountering difficulty in seeing how this relationship holds: with $S_T$ being stock price at time $T$, I want to find the sde for $S_t e^{-rt}$ $$dS_t = rS_tdt + \sigma S_t d\hat B_t$$ Where ...
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0answers
32 views

Transition function for absorbed Brownian motion

I need an help with the following exercise. I've already seen this question Prove that Brownian Motion absorbed at the origin is Markov but I don't understand the answer. Also I would like to prove ...
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1answer
29 views

Maximize value of exponential

How can i go about maximising the value of the following, $$ exp\left( -\alpha e^{rT}x -\alpha\sigma\lambda e^{rT}\int_0^T e^{-rs}\pi_sds+\frac{1}{2}(\alpha\sigma)^2 e^{2rT}\int_0^Te^{-2rs}\pi_s^2ds ...
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1answer
40 views

How to solve Stochastic differential equation?

I do not have a clue on how to solve out this type of question, and how to deal with integration with a combination of brownian motion and linear function. Can anyone help me out please?
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1answer
15 views

Integrate bivariate normal distribution over circular region

Context: Need to compute the probability that a 2D Gaussian random walk falls within distance $ d $ of some point $ p $ on the next step. (Assume the covariance $ \Sigma $ is the identity matrix $ I ...
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0answers
24 views

Infinitesimal generator of a semigroup

I know that if $\{T_t, t>0 \}$ is a conservative Markov semigroup on E, and $f \in D(A)$ has an absolute maximum in x then $Af(x) \le 0$. Where $D(A)$ is the infinitesimal generator of $T_t$. I ...
2
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1answer
71 views

Prove this liminf is a tail events

Let $A_{k}$,$k\geq1$ be [0,$\infty$)-valued random variables on a common probability space. I want to prove the following events are in/not in tail $\sigma$-field T($A_{k}$:$k\geq1$). First, event ...
3
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1answer
63 views

Stochastic continuity

Let $(X_t)_{t \in \mathbb{R}}$ be a square-integrable real-valued process with a continuous mean value function $\mu:\mathbb{R}\rightarrow\mathbb{R}$ and a continuous covariance function ...
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36 views

Differential equation whose solution is Erlang distribution

I am working on a proof (Probability Density Question Involving an Integral Equation (from Karlin & Taylor's A First Course on Stochastic Processes)) and got stuck. Now I would like try ...
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0answers
9 views

Large Deviation Theory

Consider a differential equation of the form: $$dX_0 = f(X_\epsilon) dt$$ and it's perturbed form: $$dX_\epsilon = f(X_\epsilon) dt+ \epsilon dW(t)$$ It's well-known that if one assumes $f$ is ...
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1answer
81 views

Existence of the Brownian Motion using the Kolmogorov extension theorem

Kolmogorov extension theorem: Let $T$ denote some interval (thought of as "time"), and let $n \in \mathbb{N}.$ For each $k \in \mathbb{N}$ and finite sequence of times $t_{1}, \dots, t_{k} \in T$, ...
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0answers
21 views

Reflection principle application

I want to calculate the probability: \begin{equation*} P(W_4>2, \inf_{0\leq t\leq4} W_t >-1) \end{equation*} and $W$ is a Wiener process. I tried: \begin{equation*} P(W_4>2, \inf_{0\leq ...
0
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1answer
35 views

Applying Picard-Lindelöf iteration to a stochastic integral equation

Suppose we have the following stochastic integral equation (we can make it an SDE) where $W$ is a standard Brownian motion $$ X_t = 1 + \int_0^t X_s d W_s. $$ I want to show that by using Picard ...
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1answer
52 views

Application of Ito's formula

I have the following process: \begin{equation*} X_t= \exp \left(\int_{0}^{t}s \, dB_s-\frac{t^3}{6} \right), \end{equation*} where $B$ is a Browinan motion. My textbook asks to write Ito's formula ...
3
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1answer
29 views

Brownian motion proof of Dirichlet problem

I am reading the proof of the Dirichlet theorem stated in the following form: Theorem: Let $D$ be a bounded domain in $\mathbb{R}^d$ such that every boundary point satisfies the Poincare cone ...