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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54 views

Quadratic variation of semi-martingale

$X_t = e^{B_t-\frac{1}{2}t^2}$ I need to find $[X]_t$, the quadratic variation process. I have tried to solve the problem and my main question is whether this approach is correct or not. ...
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17 views

An application of Ito's formula

I am reading a proof in which I don't understand how to use Ito's rule to derive the following: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space such that $M^{(i)}$ and $M^{(k)}$ are ...
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25 views

Brownian motion and sup of a Brownian motion

I am stuck with the following problem: let $B_t$ be a standard Brownian motion and let $S_{t}:=\sup_{0 \leq s \leq t} B_s$. Prove that for every $\lambda \geq 0$ and $\mu \leq \lambda$, ...
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25 views

Proof finite stopping time and Wiener process bounded

Let $T_{-a,b}=\inf\{t\geq 0: W_{t} \notin [-a,b]\}, a,b>0$. I want to show that this is a finite stopping time ($P(T_{-a,b}<\infty)=1$) and that $|W_{\min(T_{-a,b},t)}|$ is bounded by a ...
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34 views

Connection of expectation and PDE; help understanding step in proof

The following is taken from a set of lecture notes available online by Nizar Touzi. Some points before reading the proposition: $\mathcal{A}$ is the generator of a stochastic process $X_s^{t,x}$. The ...
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69 views

Theorem 4.14 Brownian Motion and Stochastic Calculus

I have been reading the proof of Theorem 4.14 of Karatzas' book. I wonder whether there is a typo in the description of the process $\eta^{(n)}_{t}$ as $\xi^{(n)}_{t+}-\min({\lambda,A_{t} })$ ...
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24 views

Elementary renewal process theorem proof

Let $N_t$ denote the renewal process associated with independent, identically distributed r.v.s $T_1,T_2,\dots$ with mean $\mu$. I am going to derive the elementary renewal process theorem ...
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1answer
24 views

Why is the little o(h) in the Poisson Process dissimilar to other field's?

I learned about little oh notation in Algorithm class last year. $$ \mbox{if } \lim_{n\to\infty}\frac{f(n)}{g(n)}=0 \mbox{, then } f(n)\in o\big(g(n)\big) \mbox{ or } f(n)=o\big(g(n)\big) \mbox{.} $$ ...
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69 views

Linear non-homogenous SDE

I'm struggling to understand how to resolve the following SDE: $$dX(t)=(\sin(t)-2X(t)) dt + (1+X(t))dB(t)$$ I understand that I should use the Ito formula but I have no idea how the $F(X(t),t)$ should ...
3
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1answer
52 views

Measurable projection theorem proof reference

I'm beginning to study about stochastic processes, and currently focusing on stopping times and hitting times. The textbook I'm using is "Stochastic Integration Theory" by Medvegyev (and Karatzas ...
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37 views

A simple symmetric random walk is adapted

$\newcommand{\ee}{\mathbb{E}}$The fact that for all $n$ we have $\ee[S_n \mid \mathcal{F_{n-1}}]=S_{n-1} ~\text{a.s.}$ and $\ee[ |S_n|]<\infty $ is usually shown explicitly when showing something ...
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39 views

Distribution of the random time for queuing system to change from full to empty.

Question: Find the distribution for the (random) time it takes an $M/M/1/2$ queuing system with $\lambda = \mu = 1$ to change its state from being full to being empty. ($\lambda, \mu$, arrival rate ...
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1answer
35 views

A stochastic process $X$ with values in a separable Banach space $E$ is a martingale iff $f(X)$ is a martingale for all $f\in E^\ast$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space and ...
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29 views

How do theorems like the optional stopping theorem generalize to Bochner integrable processes with values in a separable Banach spaces?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space ...
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11 views

Examples of state-space models that show strong homeostasis but also substantial change after critical threshold?

The question is, can can anyone provide examples of systems or math models that exhibit patterns of homeostasis but which can be exhibit substantial transitions or bifurcations after some critical ...
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27 views

Is there any difference between Correlation and Correlation coefficient?

I learnt in probability theorem class that correlation coefficient is $$ \rho=\frac{\sigma_{XY}}{\sigma_X \sigma_Y}=\frac{E\left[(X-\mu_X)(Y-\mu_Y)\right]}{\sigma_X \sigma_Y} $$ However, my ...
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1answer
36 views

Calculate probability of two different outcomes where history is governed by markov chain

Let the state space, $s_t$, be $\{0,1\}$ and be governed by a Markov chain with probability $\pi(s_0=1) =1$ for the initial state and time-varying transition probabilities $\pi_1(s_1=1|s_0=1)=1$, ...
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34 views

AR(1) process with exponential noise.

For the AR(1) process defined by $Z_t = aZ_{t-1} + \epsilon_t$, $\epsilon_t \sim Exp(\lambda)$, $a \in (0,1),\lambda >0$, compute $P(Z_t|Z_{t-1})$. I was only able to compute $E(Z_t|Z_{t-1}) = ...
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12 views

Statistical metric for assessing optimality

In a stochastic computational model, I'm given a limited number of parameter sets and hope to identify the one set of input that is optimal, defined by the values of its numerical outputs, i.e., ...
2
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1answer
46 views

Rigorous meaning of conditional expectation in Feynman-Kac formula/in general

In Wikipedia https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula and plenty of other books/sources, Feynman-Kac formula is expressed in a form of the type $$f(t,x)=E(f(T,X_T)\mid X_t=x)$$ What ...
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1answer
55 views

How does filtration model information?

Lets say you have a probability space $(\Omega, \mathcal{F},P)$ And a stochastic process on this space $\{X_t, t \in T\}.$ Assume that our process takes vaslues in $\mathbb{R}$. T is a totally ordered ...
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16 views

Malliavin Calculus: directional derivatives of cylinder functions exist in what sense?

Denote by $P_0(\mathbb{R}^d)$ the sets of continuous paths over $[0,1]$ started at $x=0$ with values in $\mathbb{R}^d$, we equip this space with the sup-norm and make it into a probability space by ...
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1answer
51 views

How to solve a SDE defined via a Markov Process?

I have to solve the following SDE. $$ \mathrm dY_t= f(X_t) \mathrm dt, \tag{1} $$ where $X_t$ is a two-state Markov Process possesses states $a$ and $b$. Moreover, I would like to solve $$ \mathrm ...
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1answer
24 views

Simulation of the variance of a typical waiting time W(q) in a queue

Write a computer programme that by means of stochastic simulation finds an approximation of the variance of a typical waiting time W(q) (in the queue) before service for a typical customer arriving to ...
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38 views

Random process involving CDF and PDF of standard normal.

Im studying old exams and found this one: Let $ \Phi(x)=\int_{-\infty}^{x} \frac{1} { \sqrt{2\pi} } e^{-y^2 /2} dy $ and $ \phi(x)=\Phi^\prime(x)=\frac{1} { \sqrt{2\pi} } e^{-x^2 /2} $ be the ...
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16 views

Proving bounded expectation and switching order of stochastic integrals

I'm working with the following set of stochastic differential equations: $$dx_i = x_i\left(b_i-\sum_{j=1}^n a_{ij}x_j\right) \,dt + \sigma_i x_i \, d\eta(t)$$ where $\eta$ solves the ...
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51 views

Geometric Brownian Motion Properties

I was reading Oksendal's book "Stochastic Differential Equations", fifth Ed., pp. 62-63 and came across some counter-intuitive properties of the Geometric Brownian motion (GBM). Let $\alpha,r>0$ ...
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29 views

How does Variance become an Autocorrelation Function?

"For a Gaussian stochastic process $X=\{X(t)|-\infty<t<\infty\}$ with mean function $\mu(t)=0$ for all $t$, its autocorrelation function is $$ E(X(t)\cdot X(s))=R(h)=\max(0,1-|h|), h=t-s. $$ ...
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18 views

Distribution MSR

We have $Y_i = \beta_0 +\beta_1(X_i -\bar X )+\epsilon_i$ for i=1,...,n $$\epsilon_i \sim N(0,\sigma^2)$$ We know that $SSR= Y^T P_xY - n\bar Y^2=Y^T (P_x -n^{-1} J_nJ_n^T)Y$ ...
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1answer
48 views

exponential martingale inequality

I stumbled across a claim I couldn't verify. Let $M_t$ be a continuous local martingale, $M_0=0$ a.s. and $\lambda>0$. Then $$ \mathbb{E}\left( \exp \left( \lambda M_t \right) \right) \leq ...
2
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1answer
30 views

Why does $(W_t)^2$ have mean $t$?

This is in the context of Ito Calculus. Here, $W_t$ is a $P$-Brownian Motion. My book says that "... $(W_t)^2$ has mean $t$, because of the variance structure of Brownian motion.." I understood that ...
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1answer
47 views

Stopping time wiener process

Suppose that $T=\inf\{t\geq 0: W_{t} \notin [-a,b]\}$ where $a,b>0$ and $W_{t}$ denotes a Wiener process. Now I'm wondering if this is a stopping time but don't know how to work this out.
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31 views

Can we apply the Itō formula to find an expression for ${\rm d}\eta_t(X_t)$ where ${\rm d}X_t=v_t(X_t){\rm d}t+\xi_t(X_t){\rm d}B_t$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(B_t)_{t\ge 0}$ be a $d$-dimensional $\mathcal F$-Brownian motion ...
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1answer
31 views

Solving equation with Wiener process

I want to show that if $E(f(X_{t}))=E(f(W_{t})e^{\lambda W_{t}-0.5*\lambda^2*t})$, where $W_{t}$ is a Wiener Process, then $X_{t}\sim N(\lambda t,t)$. Does anyone have a clue how to solve this ...
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26 views

expected value of a doubly stochastic matrix with i.i.d entries

I am now thinking a problem: what is the expected value of doubly stochastic matrix with i.i.d entries Each entries is i.i.d in $[0,1]$. Will the answer be a matrix with all entries ...
3
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1answer
72 views

Showing martingale for a Brownian motion $(W_t)_{t \geq 0}$

I want to show that $\dfrac{e^{W_{t}^2/(1+2t)}}{\sqrt{1+2t}}$ is a martingale with respect to $F_{t}$. We can use that $$E(e^{\alpha ...
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1answer
36 views

Law of total expectation well-defined?

Wikipedia states that this is a special case of the law of total expectation click me. Given a partition $A_1,...,A_n$ of the outcome space, we have for a random variable $X$ that ...
2
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1answer
29 views

How to find Kolmogorov Forward Equations, given generator matrix Q?

I am having difficulty in forming Kolmogorov Forward Equations. I understand how the KFE is derived and that $$\frac {d}{ds} p_{ij} (s) = \sum_{k \neq j} p_{ik} (s) \lambda_{k} r_{kj} - p_{ij} ...
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31 views

Autocorrelation function of a Wiener process & Poisson process.

Can anyone possibly explain step 3 and 4 in this solution?
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26 views

Simulating “Nested” Stochastic Differential Equations

I haven't had much luck over in Stats SE, so I'm going to try over here. I doubt many people here have experience with R, so I would like to know what are some methods to simulating a set of "nested" ...
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55 views

Kolmogorov's continuity criterion Ornstein-Uhlenbeck process

Let $(X_t, t \in \mathbb{R})$ be an Ornstein-Uhlenbeck process, i.e. $X_t$ is defined by $$X_t = \sigma \int_{-\infty}^t e^{-\theta(t-s)} dW_s$$ for $t \in \mathbb{R}$ for parameters $\theta, \sigma ...
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22 views

Noise with heavy tails

The main type of noise I know other than white noise is a colored noise (Ornstein-Uhlenbeck) of the form: $$d\eta = \lambda \eta dt + \alpha dW_t$$ with exponential correlation. I'm interested in ...
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41 views

Show that the solution to a stochastic differential equation is satisfied by the following

I am confused on how to get from the first statement to the second. Getting from the second statement to the third would just a simple case of substituting s=0. The solution sheet says to use ...
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1answer
39 views

Characteristic function notation

Having a rather basic understanding of probabilities I would like to ask you what exactly means the following notation. I am looking at the Gardiner's handbook for stochastic methods and am interested ...
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1answer
45 views

What is a “continuous modification”? And can we always modify an almost surely continuous process, such that every path is continuous?

Let's motivate the question by a classical result: Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ which ...
3
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1answer
52 views

Is this process a martingale?

Given $X_t=\int_0^t s W_s dW_s$ and the process $M_t=X_t^3-\int_0^t X_sY_s ds$. Find $Y_t$ such that $M_t$ is a martingale. I started thinking that $X_t$ can be seen as: $dX_t=tW_tdW_t$ then by ...
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21 views

Mixing process in statistics vs. mixing in classical ergodic theory texts

In dynamical systems a transformation $T$ is strongly mixing if $\lim_{n\rightarrow \infty} P(A \cap T^{-n} B) = P(A)P(B)$ (e.g., Patrick Billingsley's Ergodic Theory and Information) For stochastic ...
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42 views

Methods of SDE calibration

There is somewhere summary of methods that can be used to estimate parameters of SDE? I currently using MLE and regression due to linear dependence between samples. I searching for something ...
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24 views

Quasistationary distribution for the Moran model.

The Moran model is a model for genetic drift. Basically, it is a finite Markov chain (more precise: a birth-death chain) with state space $S:=\{0,...,N\}$ and the following transition probabilites: ...
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37 views

Pathwise definition of stochastic integral consistent with the Ito isometry

My definition of the stochastic integral is that it it is the image of the Ito isometry. Now we also prove Ito's formula and then apply it pathwise and get a pathwise definition in some cases. But in ...