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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51 views

The sum of two random processes with independent increment also has independent increment?

$X_n$ and $Y_n$ are random processes. Both of them have independent increment. Does $X_n + Y_n$ has that property?
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104 views

Show that $\mathbb{P}(B_t<x,B_s<x)\geq\mathbb{P}(B_t<x)\mathbb{P}(B_s<x)$ for $0<s<t$, $x>0$, and $B$ Brownian motion

Let $B$ be a standard Brownian motion. How can we show that $$\mathbb{P}(B_t<x,B_s<x)\geq\mathbb{P}(B_t<x)\mathbb{P}(B_s<x)$$ for $0<s<t$ and $x>0$ without actually computing the ...
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95 views

Time series (stochastic process) estimating parameters using characteristic function

I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector ...
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157 views

Martingale property of a counting process subtracting its compensator

I shall show that for a point process (counting process) $\Phi((0,t])=\sum_{n \geq 1} \mathbf{1}_{\lbrace T_n \leq t \rbrace}$, \begin{align*} M_t = \Phi((0,t]) - \int_{0}^{t} \mathbf{1}_{\lbrace s &...
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76 views

Examples of decreasing-in-some-time-interval variance of a time homogeneous Markovian process

Let $x_t$ be a zero mean, time homogeneous Markovian process over time $t$ starting from $x_0=0$. What are the examples of $x_t$ where the variance at $t$ decrease over some interval of $t$? The ...
2
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1answer
96 views

Distribuation Max - Min of Brownian motion

I'm looking for the distribuation of $M_X(t) - m_X(t)$ of the brownian motion and not the joint distribuation. where $m_X(t) = \min\limits_{0\leq s\leq t}X(s)$ and $M_X(t) = \max\limits_{0\leq s\leq ...
2
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0answers
74 views

mean displacement inequality for random walk with drift away from zero

Suppose $X_n$ is a nearest neighbor random walk on the integers with transition probabilities biased towards moving away from zero but with the bias asymptotically vanishing as you move away from zero....
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77 views

Infinite discounted sum of weakly dependent Normal random variables

Say I have the expected value of a sum of weakly dependent Normal random variables of the form $\mathbb{E}\left[\sum_{n=1}^\infty a^n X_n\right]$, where $0<a<1$. I was wondering under what ...
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2answers
109 views

Ito's Isometry for three factors

Ito's Isometry states the following: If $\{W_t\}_{t\ge0}$ is a Brownian motion and $\{\phi_t\}_{t\ge0},\{\psi_t\}_{t\ge0}$ are two non-anticipative piecewise-continous processes with $\mathbb E[\int\...
2
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1answer
517 views

Kolmogorov Backward Equation for Itô diffusion

Let $(X_t)_{t\ge 0}$ be the solution of the SDE $$ X_t = X_0 + \int_0^t \mu(s,X_s) \,ds + \int_0^t \sigma(s,X_s) \,dB_s, \quad t\ge 0 $$ where $\mu(s,x)$ and $\sigma(s,x) $ are Lipschitz continuous ...
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1answer
336 views

Showing that two Gaussian processes are independent

Say that $Z_t = (X_t, Y_t)$ is a 2-dimensional Gaussian process (by definition, it means that the random vector $(X_{t_1},Y_{t_1},...,X_{t_n},Y_{t_n})$ is a Gaussian random vector for all $t_1 ,...,...
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1answer
173 views

Verifying a standard Brownian Motion? [closed]

Let $\{X_t, t\ge 0\}$ be a standard Brownian motion process. For a fixed positive number s and all $t\ge 0$, we define $Y_t = X_{t+s} - X_s$. Is $\{Y_t, t\ge0\}$ a standard Brownian motion? Attempt: ...
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147 views

Computing Replacement rate in a renewal process

I found this problem on some slides that I'm studying as I prepare for a final: "A machine in use is replaced by a new machine either when it fails, or when it reaches the age of $T$ years. If the ...
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3answers
54 views

Get one of the two random variables's distribution function from limitation [duplicate]

This is a very fundamental problem. In the Stochastic Processes textbook, it says that: The Continuity Theorem of Probability allows us to conclude that $$F_X(x)=\lim_{y \to \infty}F_{XY}(x,y)$$ ...
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1answer
540 views

Background for studying and understanding Stochastic differential equations

Assume I have back ground of the following knowledge based on the textbook as : ODE : ODE by Tenenbaum Baby probability : Ross 's baby probability Baby real anlysis : Bartle's introduction to real ...
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641 views

$\int_0^tB_s^2\ dB_s$ - Gaussian Process and independent increments?

For $(B_t)_{t\ge0}$ a standard Brownian motion (Wiener process) define the stochastic process $X_t:=\int_0^tB_s^2\ dB_s$. I am currently trying to assess if $(X_t)_{t\ge0}$ is a Gaussian process and ...
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61 views

Merton's Problem Stochastic Differential Equation

Solve the following numerical case of Merton's optimal portfolio selection problem: find an optimal policy function $(s, y) \mapsto u(s, y)$ such that for the Ito diusion determined by $dX_t =X_t(u(t, ...
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1answer
31 views

A basic doubt on Poisson

For a Poisson process the event "arrival at time $t$" = ${N(t+h)-N(t) =1}$ when $h->0$. Is this correct ? How ?
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1answer
134 views

time sampling of a Poisson process

In Sheldon Ross, one paragraph has a heading "time sampling of a Poission process" and it describes that each arrival we toss a biased coin with $p(t)$ being bias prob.Then the process generated is a ...
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1answer
449 views

Blumenthal 0-1 law

Let $(B_t)$ be a Brownian motion. Consider the event : $B(n)>a \sqrt n $occuring infinitely often. I want to prove that this event has probability 1. we can see that, by rescaling property, $$\...
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2answers
146 views

Solving the SDE $dX(t) = (c(t) + d(t)X(t))dt + (e(t) + f(t)X(t))dW(t)$

How to solve $dX(t) = (c(t) + d(t)X(t))dt + (e(t) + f(t)X(t))dW(t)$ together with the initial condition $X(0) = X_0$.
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1answer
240 views

differentiability of brownian motion

for a fixed $t \in [0, \infty)$, I have to show that $ \mathbb{P} (D^+W_t = + \infty$ and $D_+W_t = -\infty )$, where $D^+$ (and $D_+$) denotes the upper right-hand derivative (and respectively the ...
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1answer
205 views

Time-invariance and spatial-invariance of a stochastic process

Many stochastic processes have independent and stationary increments, i.e. let $(X_t)_{t\ge 0}$ be a stochastic process on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, then $X_t - X_s \...
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2answers
189 views

change of sign for brownian motion

for a fixed $\epsilon$ $> 0$, I want to show that almost surely (ie with probability 1), a standard brownian motion $W_t$ would change sign over [o,$\epsilon$ ]. I thought about defining a random ...
2
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1answer
45 views

Ant problem with discrete combinatorical background.

an ant can move along a grid in $\mathbb{Z}^2$. But the ant can only go upwards and to the right(with equal probability). The ant starts in the point $(0,0)$, but there is an electrical wire from $(0,...
2
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1answer
135 views

Law of Large numbers using Brownian limit

Given a standard Brownian motion $\{B_t;0 \leq t < \infty \}$, we know that $\lim_{t \to \infty}\frac{B_t}{t} = 0$ a.s. I am interested to know if we can prove Strong Law of Large Numbers for any ...
3
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1answer
297 views

What is the probability a random walk hits x before it hits y?

This problem was motivated by my bitcoin trading and recalling some of my math education back in the day. I thought I'd ask people who know this much better than I... Suppose there is a continuous, ...
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1answer
1k views

How to prove difference between two independent poisson process is not a poisson process?

It will come under properties of poisson process in some books. The sum of two independent poisson process can be proved as a poisson process using its memoryless property but how to prove difference ...
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78 views

Ergodic and space-time differences

I read a resolution of St. Petersburg paradox, which says that the game is not ergodic. Yet, if you play n games in row, you average income will be the same as if you play n parallel games, This ...
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1answer
263 views

Inequality for Euclidean norm

Let:| | be Euclidean norm on $\mathbb{R}^{n}$ and $b : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n}$ and $\sigma : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n\times m}$ two continuous functions. ...
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1answer
103 views

Reducing a double summation with infinite limits

I've been solving a Renewal theory problem and I end up with this function $m(t)=e^{-4t}\sum_{k=1}^{\infty}\sum_{i=2k}^{\infty}\frac{(4t)^i}{i!}$. How do I solve or reduce the double summation? Is it ...
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1answer
268 views

Law of large numbers for Brownian Motion

Let $\{B_t: 0 \leq t < \infty\}$ be standard Brownian motion and let $T_n$ be an increasing sequence of finite stopping times converging to infinity a.s. Does the following property hold? $$\lim_{...
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26 views

Renewal function equivalence

I know that $m(t)=E[N(t)]$ for a Renewal process that has $N(t)$ arrivals. I've seen at my class that there is a way of calculating $m(t)$ by doing $m(t)=\sum_{i=1}^\infty P(N(t)\geq i)$. But I also ...
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1answer
176 views

I want to calculate $\int B(t)^2 dB(t)$ where $B(t)$ is Brownian motion

Let $B(t)$ be Brownian motion. I want to calculate $\int B(t)^2 dB(t)$. definition.A process $\{X(t),0\le t \le T \}$ is called a simple adapted process if there exist times $0=t_{0}<t_{1}<t_{2}...
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0answers
316 views

Essential supremum of a conditional expectation

Given the function \begin{equation} P(x,t) := \sup\limits_{t \le \tau \le T} E\left( g(X^{t,x}_{\tau}) \right) \end{equation} where $X^{t,x}$ is the unique solution to the SDE \begin{equation} X_u ...
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1answer
50 views

I want to show $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. [closed]

Let $B(t)$ be Brownian motion. Show that $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. Find its mean and covariance functions. thanks .
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1answer
313 views

how to prove $\frac{B(t)+W(t)}{2}$ is a Brownian motion where $B(t)$ and $W(t)$ be two independent Brownian motions.

Let $B(t)$ and $W(t)$ be two independent Brownian motions. Show that $\frac{B(t)+W(t)}{2}$ is also a Brownian motion. Find correlation between $B(t)$ and $X(t)$. thanks for any help
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708 views

Stationary Increments of a Poisson process

Let $\{N(t),t\geq0\}$ be a Poisson process, i.e. for $t\geq0$ and $n\geq0$, $P(N(t)=n) = \dfrac{e^{-\lambda t}(\lambda t)^n}{n!}$, with $\lambda>0$ a constant. Prove that $$P(N(t+s)-N(t) = n) = P(N(...
2
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1answer
58 views

I cannot see what happens to $P_{ij}(s)=p_{ij}^0+\sum^\infty_{n=1}\sum^{n-1}_{k=0}s^{n-k}f_{ij}^{n-k}s^kp_{jj}^k$ to get result

This is self learning and it is stats. $P_{ij}(s)=\sum^\infty_{n=0}p_{ij}^ns^n$ (which you'll probably recognise is a generating function) and $F_{ij}(s)=\sum^\infty_{n=1}f_{ij}^ns^n$ (note n=1 ...
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1answer
79 views

Does uniqueness always hold for the Kolmogorov's Extension Theorem?

Kolmogorov's Extension Theorem (KET) implies the existence and uniqueness of a product measure given its finite-dimensional distributions (FDDs), provided that the latter are consistent. KET puts some ...
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194 views

Please rate this stochastic processes course. What should my next course be?

I am currently doing a course in Stochastic Processes that uses the book "Adventures in Stochastic Processes" by Sidney I. Resnick. The topics covered in the book are as follows: Discrete Index Sets/ ...
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2answers
106 views

How to show that $\mathbb{E}(\int_0^T t\mathrm \, dW_t) = 0 $?

I just want to know why $\mathbb{E}\left(\int_0^T t \,\mathrm dW_t\right)=0$. I know it's got something to do with the Gaussian distribution but I don't really know what.
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1answer
27 views

How to generate a series of points in a convex set?

For an arbitrary convex set, may be a polyhedron or an ellipsoid how can I generate N uniformly distributed points inside?
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1answer
233 views

Ruin time for a two-input “risk only” slot machine

Imagine a "risk only" slot machine that takes 'coins' corresponding to some real number fraction of a dollar $p$, returns the coin with probability $p$, and eats the coin with probability $(1-p)$. ...
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2answers
110 views

Expectation Geometric Brownian Motion

Can someone help show me a simple way to show: $$\mathbb{E}(S_t)= S_0e^{\mu t}$$ for $$ S_t = S_0\exp\left( \left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t\right) $$ from this page: http://...
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0answers
63 views

Is a deterministic path a Markov process

I have a question about what classifies as a Markov chain and what does not. Consider a system with state space $\left\{ 1,\ldots,n \right\}$ and a trajectory for the system defined by the following ...
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1answer
85 views

State space reduction of a CTMC

I have a CTMC with six states $\{0,1,\ldots,5\}$. It turns out that states 3 and 4 are equivalent and so are states 1, 2 and 5. I would love to clump equivalent states into one. $$Q_1=\matrix {& ...
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2answers
119 views

Expectation for a Poisson processes

This is a very elementary question I am asking where I want to make sure my procedure is right. I am asked to calculate: $V=E[X_2\mid N(1)=1,N(2)=2,N(3)=3]$, where $N(t)$ is the counting function of ...
2
votes
1answer
59 views

What is the hitting time distribution for white noise?

What is the distribution of the hitting time for a stochastic process $(W_t)_{t\in [0,T]}$, where $W_t$ are i.i.d. Gaussian random variables? How about in cases, in which $W_t$ are i.i.d. with a ...
0
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1answer
137 views

Random Process not mean square continuous

Does a WSS (Wide Sense Stationary) process exist which is not mean square continuous? If so, can you give me an example. Note: A WSS process is mean square continuous iff the autocorrelation ...