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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Renewal Equation without $[0,t]$ on integral

a renewal process $\{N(t),t \ge 0\}$ with continuous inter-renewal time distribution $W$. Let $A(t)=t-S_{N(t)}$ be the age process. For given constant $x>0$ derive its renewal-type equation for ...
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Poisson process and Heaviside function

Show that Poisson process $p(t)$ of intensity $\lambda$ can be written as $$p(t)=\sum_{t>t_n}\delta(t-t_n),$$ where function $\delta:\mathbb{R}\rightarrow\mathbb{R}$ is Heaviside functions: ...
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Show that $W^2 _t - t$ is a $\mathbb{P}$-martingale.

Claim: $V_t = W^2 _t - t$ is a $\mathbb{P}$-martingale. I have shown via Ito's formula, that $dV_t = 2 W_t \, dW_t$. For reference, I will list this "Proposition": If $X$ is a stochastic process ...
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1answer
29 views

Log normal simulation.

I want to calculate numerically the expectation of a lognormal random variable $Y=e^X$, where $X$ is normally distributed with mean $m$ and variance $V$. The expectation is known as ...
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12 views

Find constants such that transformed simple symmetric random walk is martingale

Let $$S_0 :=0, \quad S_n = X_1 + ... + X_n \quad \forall n \in \mathbb{N}$$ be the simple symmetric random walk on $\mathbb{Z}$, i.e. the $X_i$ are i.i.d. with $$P[X_i = +1] = P[X_i = -1] = 1.$$ Now ...
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22 views

Is the process Markov or not?

Consider the stochastic process with $X_0=0$ and $$ X_t= \begin{cases} 0 & \text{ for } \ \ t<\tau_1 \\ 1 & \text{ for } \ \ \tau_1\leq t < \tau_1+\tau_2 \\ 2 & \text{ for } \ \ ...
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2answers
55 views

Crossing of Brownian Motion Sample Paths

I would like to ask for a more rigorous statement and proof of Lemma on page 5 of this paper. In essence, it states that two distinct sample paths of a Brownian motion does not strictly cross (meaning ...
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22 views

Simulation of a diffusion on $[0,1]$

I have a diffusion process $X=(X_t)_{t \ge 0}$ with the generator $$Af(x)=\frac{1}{2}(a(1-x)-bx)f'(x)+\frac{1}{4}x(1-x)f''(x),$$ where $a,b >0$ are constants. I want to simulate $X$ to a ...
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what does `ensemble average` mean?

I'm studying this paper and somewhere in the conclusion part is written: "Since this rotation of the coherency matrix is carried out based on the ensemble average of polarimetric scattering ...
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1answer
29 views

Ito formula when g(t,x) is an integral

Suppose we have a stochastic process which is written as an Ito process. $$dX_t=\mu_t\ dt +\sigma_t\ dB_t$$. If $Y_t$ is defined as a stochastic process as a function of $X_t$, then we can find $dY_t$ ...
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1answer
30 views

About the expected transitions in Markov Chain

The problem is here: The given answer is here: K = $2+ X_1 + X_2$, where $X_1$ and $X_2$ are independent exponential random variables with parameters $2/3$ and $3/5$. $$ E[K] = 2=2+1/p_1 +1/p_2 = ...
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19 views

A stochastic process $X$ is a martingale $\iff$ $X$ is driftless.

A Collector's Guide to Martingales: If $X$ is a stochastic process with volatility $\sigma _t$ (that is, $dX_t = \sigma _t dW_t + \mu _t dt$) which satisfies $\mathbb{E}[(\int_{0}^{T} \sigma^2 _s \, ...
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1answer
24 views

Do Markov generators form a linear space?

Let $G_1$ and $G_2$ be generators for two distinct continuous-time Markov processes $X^{(1)}$ and $X^{(2)}$ on a common probability space $\Omega$ (with Markov semigroups $S^{(1)}$ and $S^{(2)}$) so ...
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21 views

Book for learning stochastic processes for a beginner

I am a social science student interested in learning stochastic processes. The book used in my university is Grimmett and Stirzaker. I tried to study discrete time markov process from it and was able ...
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58 views

Finding the right $\sigma$-algebra. Question on uncertainty related to the secretary problem.

I'm working on a problem related to the secretary problem. Let me give a short overview on the topic I research: You are supposed to choose the best item presented to you in a row of n items. Any ...
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38 views

what does this integral stand for?

i would really appreciate some advice concerning a paper i'm reading: http://pages.stern.nyu.edu/~dbackus/GE_asset_pricing/disasters/Leland%20port%20ins%20JF%2080.pdf on page 586, there is a problem ...
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2answers
31 views

Why are Optional Stochastic Processes Important?

I understand to some degree why adapted processes, progressive processes, and predictable processes are important. EDIT: I am referring only to the continuous time case, NOT discrete time. But why do ...
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Should I ask about math books? I.e. scan of index page or references page? [on hold]

I mean it's not harmful to anyone if I ask about certain page of certain book? I need page 263 from this book ...
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1answer
25 views

Do Optional and Progressive Processes Have Counterparts in Discrete Time?

We know that predictable $\implies$ optional $\implies$ progressively measurable. Source Predictable processes have obvious/simple counterparts in discrete time. Do optional processes and ...
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40 views

Radon-Nikodym on a Process wrt to filtration

Given a probability space $(\Omega,\mathcal{F},P)$. Let $(X_t)_{t\geq0}$ be a stochastic process defined on it with cadlag paths, lets say on $(\mathcal{X},\mathcal{B}(X))$. Let be $\mathcal{F}_{t}$ ...
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Indistinguishable Processes under local Lipschitz Condition

Let $a,b, \rho, \sigma$ be locally Lipschitz functions on $\mathbb{R}^d$, G an open subset of $\mathbb{R}^d$ and assume that on $G$ we have the equalities $a=b$ and $\rho=\sigma$. Let $\xi \in G$ and ...
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1answer
36 views

What is the probability that a given $ n $ event trains match the beginning of a Poisson process?

Here is my question with which I'm confusing myself: Assume that some event times $ \{\tau_i\}_{i \in \mathbb{N}} $ are a point process with rate $ \mu $ such that number of events that occurred ...
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1answer
26 views

Finding the mean given the probability

I'm doing some work on branching processes and would like to know where the process becomes extinct. If $X$ is the number of offspring of an individual, then the process goes extinct when ...
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1answer
26 views

Expected Square Distance from Origin of Random Walk in $\mathbb{Z}^2$

I'm trying to find the expected value of the squared distance from the origin of a simple symmetric random walk in $\mathbb{Z}^2$ at time $n$. So far, I have calculated that if $(X,Y)$ is the ...
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26 views

How do linear operators acting on paths of Gaussian processes influence the covariance function?

It is well-known that applying a linear transformation $A$ on an $n$-dimensional centered Gaussian distribution with covariance matrix $\Sigma$ results in another centered Gaussian distribution with ...
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17 views

System of SDEs and independence

I am recently reading a paper that seems to claim the following fact without justification: $Y^1_t, \ldots, Y^n_t$ are stochastic processes defined on $\mathbb{R}$. Let $b: \mathbb{R}^2 ...
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1answer
101 views

Stochastic Integral with respect to Compensated Poisson Process

Proposition: Let $N_t$ be an $\mathcal{F}$-Poisson process and $M_t=N_t-\lambda t$ its compensated process. Then for any $\mathcal{F}$-predictable bounded process $H_t$, the stochastic integral ...
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1answer
42 views

Why Are Semimartingales the Largest Possible Class of Stochastic Integrators?

I am trying to understand why semimartingales are the most general possible class of stochastic integrators. (I was hoping that this question would give me my answer, but it didn't.) I thought at ...
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1answer
124 views

One-Dimensional Jump-Diffusion Ito’s Formula

Let, $N_t$ be a Poisson process and let $X_t$ solve the SDE $d{X_t}=a_t dt +J_t dN_t$. Then, what is the correct Ito´s fórmula: i)$df(t,X_t)=(\frac{\partial f}{\partial t} + \frac{\partial ...
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45 views

How to prove that the stochastic integral process is gaussian?

I would like to prove that for a $C^1$-function f and a Wiener process W, the integral process defined by $$ Y_t:= \int_0^t f (s)dW_s := f (t)W_t -\int_0^t W_s f'(s)ds $$ Is a centered gaussian ...
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Functions of a random walk and martingales

Let $\xi_1,\xi_2,\ldots$ be a sequence of iid random variables, such that $$\mathbb{P}(\xi_i=1)=p\ne \frac{1}{2},\,\mathbb{P}(\xi_i=-1)=q=1-p.$$ Consider the corresponding random walk ...
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2answers
23 views

Uncorrelated but not independent uniform distribution

Let $X = (X_1, X_2)$ be uniform distributed on $\{(-1,0), (1,0), (0,-1), (0,1)\}$. First of all I want to show that $X_1$ and $X_2$ are uncorrelated but not independent. Secondly I thought about ...
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38 views

Convergence of a sequence over supremum

Given a cadlag-process $X_{t}$ with stationary independent increments (Levy process) for which $E\left[\sup_{s\in[0,t]}\left|X_s\right|\right]<+\infty$ for all $t>0$. For $n\in \mathbb{N}$ the ...
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28 views

Integral equation with a random variable [on hold]

Suppose we have an integtal of this kind: $$\displaystyle Q(\tau)=\int_0^\tau dt K(w(t)+\nu(t))$$ where $\nu(t)$ is a gaussian noise $\nu(t)=\mathcal{N}(0,\sigma)$ Because $w(t)$ is a known function, ...
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1answer
104 views

Hard Question in Stochastic processes - variance Martingales

I got some hard challenge to solve and I am looking for a small clue/help. My question goes like this: 10 Englishmen are trying to leave a pub in a rainy weather. They do it in the following ...
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1answer
70 views

Why do we need progressive measurability in the approximation of process by simple functions?

I am following Karatzas and Shreve- Brownian Motion and Stochastic Calculus. In the context of Stochastic integration one defines the martingale transform for simple functions: Now we would like to ...
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22 views

Distribution of Double Stochastic Integral

Assume that $f(s)$ is a $C^\infty$ univariate function and that $\{ (W_{1,t}, W_{2,t})\}_{t \geq 0}$ is a two-dimensional, correlated Wiener process. Then, does the random variable $X_T \equiv ...
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32 views

$2D$ random walk stopping time

A $2D$ random walk starts at $(X_0, Y_0) = (k, k)$ where $k>0$ is an integer. At each step $(X_{n+1}, Y_{n+1}) = (X_{n}-1, Y_{n})$ or $(X_{n+1}, Y_{n+1}) = (X_{n}, Y_{n}-1)$ with the same ...
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46 views

Version of the CLT

It is well known, that for a sequence of i.i.d. rv. $X_i$ with $E[X_i]=\mu$ and $Var[X_i]=\sigma^{2}$ that $$ ...
2
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1answer
43 views

Conditional independence of stopping times from i.i.d. stochastic processes

My question is somewhat arbitrary but I was thinking about independence of processes and stopping times. Say that we define two processes $X,Y$ on different probability spaces ...
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1answer
30 views

Poisson Process independent Wiener Process using singular measures

I was reading some stochastic calculus of Jump processes and saw the following result: If $W_t$ is Brownian and $N_t$ is Poisson both adapted to the $W_t$'s natural filtration then these processes ...
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24 views

Compute the Mean of the Following Process

Given the following process: $\Delta \ln(St+1)= \mu - (\sigma2/2) + \sigma(\varepsilon(t+1))$ (where both $\mu$ and $\sigma$ squared are of $S$) How does one calculate the mean of $S(t+1)/S(t)$? ...
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1answer
35 views

What is the difference between an adapted process and a predictable process?

As the footnote on page 1 of this document mentions, even most experts in the field of stochastic processes don't seem to know rigorously what the difference is. However, since I don't have any idea ...
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1answer
32 views

Conditional Probability in Poisson Process

Suppose that $N(t), t\ge 0$ is a Poisson process such the $E[N(9)]=6$. (i) Find the mean and variance of $N(8)$. (ii) Find $P(N(2)\le3)$ (iii) Find $P(N(4)\le5|N(2)\le3)$ - How do I solve this? ...
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1answer
61 views

Indicator Functions - Can someone check my working?

This is a very easy question but since some of my codes aren't coming out properly I thought I should check my theory to see if everything's okay. Say we have two values $K_{1}$ and $K_{2}$ and that ...
3
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2answers
67 views

Integral of Wiener Squared process

I don't have a background of stochastic calculus. It is known fact that definite integral of standard Wiener process from $0$ to $t$ results in another Gaussian process with slice distribution that ...
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1answer
11 views

Riesz theorem and $L^p$ norm in expectation

I am reading a paper that uses the following fact, which claims to be from the Riesz's theorem: For a continuous stochastic process $\{ X_t \}$, let $u_t$ be its density function at each time ...
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Esscher-Transform/ Levy-Process: Measure induced by trajectory

For a Levy-process $X_t$ w.r.t. to a measure P we define $\Theta$ as the set, for which $E[exp(\theta X_t)]$ is defined and finite. Note $\Theta$ is independent of $X_t$. Define ...
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1answer
36 views

To test whether a process is a Martingale (Stochastic calculus)?

If $W_t$ is a standard Brownian motion, I was trying to prove $Y_t = \exp (\int_{0}^{t} s\cdot dW_s)$ is a martingale ! First I started finding $dY_t$ using Ito formula. But I am confused how to ...
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Can this approximation result for stochastic processes be modified.(p=1 instead of p=2)?

In McKeans stochastic integrals from 1969 he proves this: You have a filtered probability space $(\Omega,\mathcal{F},P)$, where the filtration is based on a Brownian motion. Assume that $X_t$ is ...