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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230 views

Does this Condition on Exit Times imply $X_t$ is a Local Supermartingale?

Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $$\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin (a,b)\}.$$ We can interpret $\tau_{s,a,b}$ as the ...
11
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208 views

Removing deterministic discontinuities from semi-martingales

Let $X:=(X_t)_{0 \le t \le T}$ be a solution of the SDE $$ X_t = X_0 + \int_0^t \sigma(s,X_s) dW_s + \sum_{i=1}^n f_i(X_{t_i^-}) 1_{\{t > t_i\}}$$ where $t_1,\cdots,t_n \in [0,T]$ and $(f_i)_{1 \le ...
10
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161 views

Expected range of simple random walk in $\mathbb{Z^2}$

Let $(Y_k)_{k\geq0}$ be a simple random walk process. The range of an $n$-step random walk, $R_n$, is a random variable that characterizes the number of distinct points visited at time $n$: ...
9
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556 views

Generated sigma algebra from Brownian Motion

Suppose that we have a Brownian motion and we define the P-augmented filtration by $$\mathcal{F}^W_t:=\sigma(\mathcal{F}^0_t \cup \mathcal{N})$$ where $\mathcal{F}_t^0:=\sigma(W_s;s\le t)$ and ...
7
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276 views

Sigma-algebra generated by a set of random variables

I know from standard textbooks that "Given the measurable functions $X_i:(\Omega,\mathcal{F})\rightarrow(\Omega_i,\mathcal{A}_i)$, the $\sigma$-algebra generated by a set of random variables $(X_i; ...
7
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127 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
7
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523 views

Generating a stochastic matrix with a given second dominant eigenvalue

I need a procedure (iterative or otherwise) that, given a positive integer $N$ and a (possibly complex) number $\lambda$ such that $0 < \vert \lambda \vert < 1$, will be able to generate an $N ...
7
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132 views

Percolation and number of phases in the 2D Ising model.

Update. As my previous figure had conceptual mistakes I decided to change the picture to another, more instructive After a long time I came back to try to understand an article on the Ising ...
7
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645 views

How to prove Brownian motion is Gaussian Process?

I'm reading Bernt Oksendal's "Stochastic Differential Equations" and this is one of the proof that I'm totally lost. This is from Ch2.2, page 12-13 (sixth edition). First, Brownian motion is ...
7
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2k views

Different versions of functional central limit theorem (aka Donsker theorem)?

I have seen several versions of functional central limit theorem (see the end of this post). I am confused, and hope someone could help to clarify their relations and differences. For example, I am ...
7
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1k views

1D Random Walk, with different step sizes in each direction.

A walker starts at a defined position greater than $0$, say $A$, and then makes a "decision" to walk either "$b$ steps to the right" or walk "$c$ steps to the left." He will choose the first option ...
7
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284 views

Is there a connection between the 3D random walk constant and the partition function?

In thinking about this question, I took a look at Pólya's random walk constants and was struck by the fact that an expression for the constant for a three-dimensional random walk, ...
6
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77 views

Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.

Let$^1$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q\in\mathfrak L(U)$ be nonnegative and symmetric operator on $U$ with finite trace ...
6
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91 views

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation. I'm searching for such a ...
6
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69 views

Advanced stochastic process book

I am looking for the book about advanced stochastic process . It may cover the following content: Stochastic matrices. Ex: $A(k)$, where $k$ is the time index. Stochastic process in space ...
6
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328 views

Potential theory: discrete-time Markov processes

Recently I've found lecture notes on "Analysis on Graphs" where the potential theory methods were used to study discrete-time, time-reversible Markov chains (i.e. the state space is countable). ...
6
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354 views

An application of the Optional Sampling Theorem

let $S(k), k\geq 0$ a discrete random process. Suppose $S(N)$ is with probability one either 100 or 0 and that $S(0)=50$. Suppose further there is at least a sixty percent probability that the price ...
5
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34 views

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 ...
5
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74 views

A strange-looking Feynman-Kac formula

I have a PDE which involves the predictable finite-variation parts of some semi-martingales and a quadratic-covariation process and I tried to derive a Feynman-Kac style expectation from the PDE. ...
5
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136 views

Equivalent definitions of Poisson process

Define a Poisson process with parameter $\lambda$ is a counting process $(N(t))_{t\ge 0}$ such that: (i) $N(0)=0$; (ii) It has independent increment property; (iii) $N(t+h)-N(t)$ has Poisson ...
5
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99 views

Is there an analytic solution for this Fokker-Planck equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial ...
5
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61 views

Why row vectors in stochastic processes?

It seems reasonable to state that column vectors $\mathbf{x}$ are the most frequently seen standard notation, often using $\mathbf{x}^\intercal$ to denote a row vector (transposed column vector). ...
5
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120 views

Modified Doob's $L^1$ inequality

Let $X_n$ be a non-negative submartingale. Show that for all $\lambda >0$ $$ P(\sup_{k\leq n} X_n \geq 2\lambda) \leq \frac{1}{\lambda} \int_{X_n \geq \lambda} X_n dP$$ In Doob's weak $L^1$ ...
5
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397 views

In stochastic calculus, why do we have $(dt)^2=0$ and other results?

I'm doing actuarial problems of Exam MFE and it covers some of the stochastic calculus (like Ito's Lemma). One of the frequently used results are the so-called "multiplication rules": $(dt)^2=0$ ...
5
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212 views

Brownian Motion and stochastic integration on the complete real line

I'm struggling to understand stochastic integration over intervals containing zero, i.e. integrals of the form $\int_{a}^{b} X_s \, d B_s$ where $-\infty \leq a < b \leq \infty$, $(X_t)_{t \in ...
5
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84 views

Upper bounding a Poisson Process with indicators of exponentials

Define $E_1,E_2,\ldots, E_i,\ldots E_n$ as i.i.d. exponentials with parameter $\lambda$. These define processes on some interval $[0,\delta]$ (think of $\delta$ as very small, it will come into play ...
5
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0answers
111 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
5
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0answers
832 views

Running maximum for Geometric Brownian Motion

Can anyone provide the expression and source for the running maximum $M_t$ for geometric Brownian motion $X_t$ as a function of the initial value $X_0$, drift $\mu$ and diffusion $\sigma$? $X_t$ ...
5
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130 views

Representation theorem for continuous process of finite variation

There is a martingale representation theorem If $M$ is a continuous $L^2$-martingale, there is a Brownian motion $B$ and a cadlag adapted function $\sigma$ such that $$ M_t = M_0 + \int_0^t ...
5
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582 views

Ito's lemma and application

Can someone help me apply Ito's lemma to the function $f(t,x,k)$ where t is the time and x,k dimensions where x and k refer to dynamics $dX(t)=\mu(t)dt+\sigma(t)dB(t)$ ...
5
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0answers
205 views

A Stopping Theorem for Right-Continuous Submartingales

Reading through the book "Brownian Motion & Stochastic Processes" by Karatzas and Shreve, I found the following problem (problem 3.24, page 20): Suppose that $ \{ X_t, \mathcal{F}_t \ | \ 0 \leq ...
4
votes
0answers
117 views

Convergence of an implicitly defined sequence of random variables

Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...
4
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33 views

Stochastic domination

Suppose we have two probability measures on a space $X$, $\mu$ and $\nu$, such that $\nu$ stochastically dominates $\mu$, i.e.there exist a coupling of $\mu$ and $\nu$ on the product space $X \times ...
4
votes
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42 views

For a simple random walk $S_n$ and for a stopping time $\tau$, what is the intuitive interpretation of $P(\tau < \infty) = 1$?

Suppose we have a simple random walk $S_n$ and we define a stopping time to be $\tau = min\{n: S_n = A \ \text{or} \ S_n = -B\}$. That is, we stop the first time we hit $A$ or $-B$. With this, I have ...
4
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0answers
58 views

Characterization of point process, given the number of points

For a point process with independent and identically distributed (i.i.d) inter-renewals, with distribution $p(x)$, we observed $N$ points on $[0,T]$. What is the joint probability distribution ...
4
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48 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 ...
4
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0answers
30 views

Clarification on the definition of progressively measurable processes

I am finding difficulties in becoming familiar with the definition of progressively measurable processes. Definition A stochastic process $X$ is called progressively measurable with respect to the ...
4
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0answers
109 views

Proof of strong Markov Property of double sided Levy Process

Let $(X_t)_{t\in\mathbb{R}}$ be a Levy Process, i.e. $X_0 = 0$ a.s., $X$ has independent and stationary increments, and almost all paths $t\mapsto X_t(\omega)$ are right continuous with left hand ...
4
votes
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49 views

Does there exist a continuous-time i.i.d. process on a standard probability space?

Is there are standard probability space $(\Omega, \mathcal{A}, P)$ and a process $X_t : \Omega \to \{ -1, 1 \}$, $t \in [0,1]$ such that $X_t$ is uniformly distributed on $\{ -1, 1 \}$ and all the ...
4
votes
0answers
77 views

Is $X_t = tW\left(\frac{1}{t}\right)$ a Martingale?If not, how could it be a Brownian Motion?

As is proved, $X_t = tW\left(\frac{1}{t}\right)$ is a Brownian motion. For example see Theorem 4.2 in this paper http://math.uchicago.edu/~may/REU2012/REUPapers/Leiner.pdf I'm just confused because ...
4
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103 views

Confusion about localization

I am a bit confused about the following result I read: Let $(\Omega,\mathfrak{A},\mathfrak{F},\mathbb{P})$ be a filtered probability space. Let $\left\{X_t\right\}_{t\in[0,T]}$ be a continuous and ...
4
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62 views

Correlation function of an asymptotically stationary AR process

I have a great confusion with the autocorrelation function of an AR process. Its derivation usually follows in this way (Haykin, 2007): The difference equation for an AR(M) process, $u(n)$, is ...
4
votes
0answers
85 views

Weak convergence of a sequence of stationary distributions to another stationary distribution

Let $\{X_n(t) \in \mathbb{R}^+\}$ for each $t \in (0,1)$ denote a discrete time Markov chain (with time index $n$ and parameterized by $t$). For each $t$, the Markov chain $\{X_n(t)\}$ has a unique ...
4
votes
0answers
78 views

Potential measure of the product of (independent $\alpha$-stable) subordinators

For a nondecreasing Levy process $\mathbf{X}$ with values in $[0,\infty)$ (i.e. a subordinator) Jean Bertoin defines the potential measure of $\mathbf{X}$ in his book "Levy processes" as follows (p. ...
4
votes
0answers
67 views

Brownian motion and associated martingales

Under the Wiener measure $\Bbb{W}$ the process $x(t)$ is a brownian motion. This means that $\Bbb{E}[{x(t)-x(s)\mid \mathcal{F}_s}]=0$. Let $P$ be a measure in $C([0,\infty),\Bbb{R}^d)$ such that ...
4
votes
0answers
119 views

Why is a predictable stochastic process called *predictable*?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $I$ be an index set $\mathbb F=(\mathcal F)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $X=(X_t)_{t\in I}$ be a stochastic ...
4
votes
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59 views

Gaussian process with deterministic bracket

Let $(M_t)_{t\geq 0}$ a continuous Gaussian process that is a martingale with $M_0=0$. Show that $\langle M,M \rangle _t=f(t)$ a.s. where $f$ is a continuous increasing function. In the particular ...
4
votes
0answers
214 views

Using Girsanov theorem to prove density of stopping time

Let $B$ be a standard Brownian motion and for $a>0$ and $b>0$, and set $$\sigma_{a,b} = \inf\{t\,:\, B_t + bt = a\}.$$ There are at least two ways to solve the following problem (the other one ...
4
votes
0answers
79 views

Convergence in distribution of stochastic equation solutions

I'm studying from Kurtz's book "Markov Processes Characterization and convergence" and I have a question about the convergence of processes in $\mathbb{Z}^d$ that are solution of some equation. (see ...
4
votes
0answers
66 views

Interchangeability of the malliavin derivative with a lebesgue integral

I was curious to know the most general conditions under which a malliavin derivative $\mathscr{D}_t \int^T_t F_v d\mu(v) = \int^T_t \mathscr{D}_t F_v d\mu(v)$ commutes with a lebesgue integral? I was ...