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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Brownian Motion Conditional Expectation Question

I have a real number $x$, and $W$ is a standard Brownian motion. Let $0 < s < t$. How to find $$ \mathsf E[W_s | W_t = x] $$ Please provide me with a step by step answer as I want to ...
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6k views

Stochastic Process Examples

I was wondering if people could give me examples of how stochastic processes are seen and used in research in real life.
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421 views

Solving Stochastic Differential Equations

Can anyone help me with the following SDE? Solve the following stochastic differential equation: $$dY_t=aY_tdt+(b(t)+cY_t)dB_t$$ with $Y_0=0$. Hint: Try a solution of the form $Z_tH_t$ where $Z_t = ...
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1answer
146 views

Lack of right-continuity of the filtration adapted to Brownian motion

Let us consider the standard Brownian motion and the natural filtration $(\mathcal{F}_t^B)$. It is known that $(\mathcal{F}_t^B)$ is not right-continuous at $t=0$. But what about $t>0$? Is it true ...
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60 views

stationary Markov chain without starting in stationary distribution?

What would be a concrete example (i.e. the transition Matrix $P$) for a discrete time stationary Markov chain, i.e. $(X(t_{1}+t),t_{2}+t),...,t_{n}+t))$ does not depend on $t$, $\forall n\geq 1, ...
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316 views

Application of central limit theorem for triangular arrays

A (1-dim) Brownian motion $(B_t)_{t \geq 0}$ satisfies the following properties: (B0): $B_0=0$ a.s. (B1): $(B_t)_t$ has independent increments (B2): $(B_t)_t$ has stationary increments, ...
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79 views

Determining the spectral density?

Suppose you have a process $X_{t} = 0.5X_{t-1} + w_{t}$ where $w_{t}$ is $WN(0,\sigma^{2})$. How does one determine the spectral density of the process? Do you first find the ACF of the process and ...
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1answer
435 views

Poisson Process - Finding conditional expectation and conditional variance

Consider a Poisson process $X={X(t);t\ge0}$ of rate $λ=5$. Here $X(t)$ is the number of customers arrived up to the time$=t$. Suppose that $X(1)=5$ (so 5 customers arrived by the end of the first ...
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2answers
776 views

Poisson Processes

Shocks occur to a system according to a Poisson process of rate $\lambda$. Suppose the system survives each shock with probability $a$, independently of other shocks, so that its probability of ...
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1answer
134 views

Interpolation result for Brownian Motion in Donskers Theorem

Suppose we have an increasing sequence of stopping times $\{\tau_n\}$ such that $\tau_n-\tau_{n-1}$ are iid. Furthermore let $B$ be a Brownian Motion and we define $S_n:=B(\tau_n)$ which gives a ...
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5k views

How to read transition probability matrix for Markov chain

Suppose that whether or not it rains today depends on previous weather conditions through the last two days. So if $RR$ (rained yesterday and today), then it will rain tomorrow with probability $0.7$. ...
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90 views

Disease sprease in a population

Unknown to public health officials, a person with a highly contagious disease enters the population. During each period he either infects a new person which occurs with probability $p$, or his ...
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1answer
241 views

Brownian Motion and the Functional CLT

Suppose we have a time series $(x_t\mid t\in \mathbb{Z})$ for which the partial sum process $X_T$ defined on the unit interval by $$ X_T(\xi)=\omega_T^{-1}\sum_{t=1}^{[T\xi]} ...
3
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0answers
99 views

Negative moments of a functional of Wiener process

At the moment I am reading D. Nualart's The Malliavin Calculus and Related Topics. The problem I am trying to solve is the following: Show that the random variable $\int_0^1 s^2\arctan W_s\, ds$ ...
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241 views

Very basic doubt about Itô's lemma

While trying obtain the dynamics of $X_t = \exp( \int_t ^T \phi_s ds)$, where $\phi$ is an Ito process following $$ d\phi_t = \mu dt+ \sigma dW_t \ ,$$ I had some doubt concerning the application of ...
2
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1answer
103 views

Markov chain property

Suppose $\{Y_{n}, n \ge 0\}$ is a Markov chain consisting of $N$ states. Suppose that $i$ and $j$ are states of this Markov chain and that $i \hookrightarrow j$, i.e state $j$ can be reached from ...
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Example of a random process which is strictly stationary but not iid

I understand $IID\subseteq SSS\subseteq WSS$. What could be an example of a stochastic process which is not iid but is strict sense stationary? I will appreciate examples for $SSS\setminus IID$ and ...
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72 views

Recurrence criterion for a specific Markov chain

Let $(X_n)$ be a Markov chain on $\mathbb N_0$ defined by $(\alpha \geq 0)$ $$ p(0,1) = 1 \\ p(x,x+1) = 1-\frac{1}{(1+x)^\alpha} \\ p(x,0)= \frac{1}{(1+x)^\alpha}$$ Define the shifted moments for ...
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322 views

Random Variable on a Sphere

Not sure where to start with this problem: For any $d\geq 1$, we admit that there is only one probability measure $\mu$ on $\mathcal S_d$, (the $(d-1)-th$ dimensional sphere embedded in $\mathbb ...
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3answers
2k views

Why is stopping time defined as a random variable?

I've been given a crash course in stochastic processes and martingales for the purposes of a semester project on them. The guy I'm working with has been, I feel, a little vague in the definition of ...
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1answer
116 views

How to calculate $x_t$ from $ARIMA(1,2,0)$ (second difference $AR(1)$ process)?

It sounds so simple but I'v struggled with this problem for a quite long time now. I have come to this: $$X_t = \phi(x_{t-1} - 2x_{t-2} + x_{t-3}) + 2x_{t-1} - x_{t-2} + \epsilon_t$$ it just doesn't ...
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2answers
312 views

hitting time of one of two barriers

Let's consider a one-dimensional Random Walk. At each time the walker moves of one step to the right with probability $p$ and to the left with probability $q$, with $p+q=1$. The walk is not symmetric, ...
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95 views

Discontinuities of stochastically continuous Gaussian process

Can a stochastically continuous Gaussian process have essential discontinuities (non-jump)? Thanks!
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73 views

Separable processes

If $X_t$ is a separable (Gaussian) process, is the process defined by $Y_0=0, Y_h= \frac{\lvert X_{t+h}-X_t \rvert}{|h|^\gamma} $ for $h>0$, and $-t<h<0$ and some fixed $\gamma>0$ also ...
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2answers
762 views

Expected value of stochastic process

I have the following problem: $X_1,X_2,...$ are positive identically distributed random variables with the distribution function $F(x) :=P(X_n \leq x)$ and we assume that $F(0)<1$ for all $n$. Let ...
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1answer
348 views

Covariance of Autocorrelated Error Terms

I have the following multiple regression model with autocorrelated error terms: $Y_t=\beta_0+\beta_1X_{t1}+\beta_2X_{t2}+...+\beta_{p-1}X_{t,p-1}+\epsilon_t$ ...
4
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1answer
111 views

Dice game modelling: Lose everything on “3”, double everything on “1” or “6”

I was recently playing a quite easy dice game: You trow a fair dice: if you get a "3" the next player continues, if you get something else it is up to you to continue. If you continue and you throw ...
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0answers
102 views

Poisson Processes and “Cooldown” Effects

This question is a spin on the traditional Poisson Process. Let's assume that I have a machine that has the ability to turn on a light. It does so by following a Poisson distribution with some ...
0
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1answer
207 views

Distribution of integral with respect to Brownian motion

Let $B$ be a Brownian motion and define the complex sequence $(X(n))_{n\in \mathbb Z}$ as $$ X(n) := \int^\pi_{-\pi} e^{inx} dB(\pi + x)$$ What is the distribution of $X(n), n\in \mathbb Z$?
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198 views

Specific use of reflection principle for Brownian motion

Let $B$ be a Brownian motion with $B(0)=0$, $x,y>0$ and $B^*$ the Brownian motion reflected at $-x$. I came across the following: $$ \mathbb P_0(\inf_{s\in [0,t]}B(s)<-x, B(t)\geq y-x) = ...
3
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1answer
6k views

Poisson Process Conditional Probability Question

I was hoping to get verification that I am on the right track and doing/ thinking about the following problem correctly: Customers arrive at a service facility according to a Poisson process of rate ...
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104 views

Conditional distributions of (higher-order) autoregressive Markov processes

If we specify an $p$-th order autoregressive process in discrete time by its transition distribution $F_{t|t-1,\ldots,t-p}$, what can be said about lower order conditional distribution where we ...
2
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1answer
285 views

Doob's stopping time theorem with unbounded stopping time

Let $(X_t)_{t\geq0}$ be Brownan motion on $\mathbb R$, and $\tau$ is a stopping time adapted with the natural filtration generated by the Brownian motion. If $X_0=0$, $E(e^{\tau/2})<+\infty$. ...
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49 views

What are $C_b^2 (\mathbb R)$ and $C^{2,1} (\mathbb R × \mathbb R^+ )$?

From a note, for a diffusion process with its transition probability $P(, t|x, s)$, Theorem 1. (Kolmogorov) Let $f (x) ∈ C_b (\mathbb R)$ and assume that $$ u(x, s) := ∫ f (y)P (dy, t|x, s) ∈ ...
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1answer
46 views

How to show a function belongs to $H^2$

How do you show that $\{\exp(B_t(\omega))\}_{0 \le t \le T} \in H^2$ where $B_t$ is a standard wiener process $H^2=\{f\in L^2(P\times m):f~~\text{adapted}\}$ and $P\times m : {\cal F} \times {\cal ...
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1answer
129 views

correlation between two different variables

I am studying stochastic processes and found the next problem: Let $A$ and $\Phi $ be two independent random variables such that $E(A) = 0$, $E(A^2) < \infty$, and $\Phi$ is uniformly distributed ...
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0answers
113 views

Understanding two definitions of diffusion processes

From Wikipedia: A diffusion process is a Markov process with continuous sample paths for which the Kolmogorov forward equation is the Fokker-Planck equation. I was wondering if "for which the ...
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1answer
398 views

Stochastic processes for beginers (good links and books)

I've a syllabus like that.. Markov chains with finite and countable state space. Classification of states. Limiting behavior of n state transition probabilities. Stationary distribution. Branching ...
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2answers
90 views

Identity for exponential of Brownian motion using scaling relation

Let $B$ be a Brownian motion and $s\wedge t := \min\{s,t\}$, $s\vee t := \max\{s,t\}$. I stumbled over the following identity: $$ \mathbb E[\exp(B(s\wedge t) + B(s\vee t))] \\=\mathbb ...
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1answer
423 views

Stopping time and Brownian motion (specific example)

Let $B$ be a Brownian motion. I want to show that $$ \inf\{t\geq0 \mid B(t)=\max_{x\in [0,1]}B(s)\} $$ is not a stopping time w.r.t. the standard filtration. How can one intuitively see that this ...
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1answer
60 views

What is $1_{\{\tau_n>0\}}X^{\tau_n}$ process saying?

As title says, what is $1_{\{\tau_n>0\}}X^{\tau_n}$ process? I do have understanding of what stochastic processes are, but not sure what is this specific process saying.
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1answer
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what is F-previsible process? And what would be F?

What is F-previsible process? I tried to search in the Internet but I couldn't find it... Also what is F here? context: http://en.m.wikipedia.org/wiki/Martingale_representation_theorem#section_2
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246 views

Lévy measure of a pure jump process

Let ($\xi_t)_{t \geq 0}$ an infinititely divisible cadlag process on $[0,\infty)$ and denote by $p$ its jump measure. Define a measure $p_t$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ by $$p_t(B) := ...
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1answer
166 views

Extinction Probability

Suppose that lions have a geometric offspring distribution. $p_{k}=(1-\theta)\theta^{k}$, $k\geq 0$ with mean 1.5. If the current world population of lions is 10, what is the probability they will go ...
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1answer
110 views

skorokhod embedding from Rogers Williams

I am reading Rogers and Williams "Diffusions, Markov Processes and Martingales" and have a question about the proof of the Skorohod embeding. What we want to proof: Let $X$ be a zero-mean finite ...
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1answer
162 views

What is some books at the level which including this inequality and its proof?

I always wanting to looking into harder random variable/probability/stochastic process/statistics books that are harder than the intro one and have multiple random variable but easy enough to have ...
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1answer
167 views

Calculate the determinant of the matrix ad hence prove

By computing the determinant of $\lambda I-L$ where $L$ is the Leslie matrix, derive the Euler Lotka equation. $$L= \begin{bmatrix} b_{1} & b_{2} & \ldots & b_{w-1} & b_{w}\\ s_{1} ...
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1answer
385 views

M/M/1 queues And finding equilibrium probability that the shop is empty

Customers arrive at a barbers shop at the incidents of a Poisson process of rate λ. Each person is served in order of arrival (by the single barber), and takes an exponential, rate μ service time. ...
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1answer
147 views

Birth processes with immigration and catastrophe

On the volcanic island of Montserrat the number of species increases(by immigration from neighbouring islands) at rate α. However, at rate η the volcano explodes, and all life is wiped out, although ...
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1answer
739 views

Linear birth death process, probability of extinction by time t

I have a linear birth death process with birth rates $\lambda n$ and death rates $\mu n$ . Let r(t) be the probability of extinction by time t. If there is 1 individual alive at time 0 explain why ...