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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182 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 ...
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123 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 $n$: ...
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177 views

How well can the maximum of a Gaussian process be approximated by a finite-dimensional Gaussian variable?

Consider a compact set $K$ in $\mathbb{R}^p$, and let $W$ be a mean zero continuous Gaussian process on $K$, meaning that $W$ takes its values in the space of continuous functions from $K$ to ...
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194 views

Sufficient condition in terms of stopping times for a stochastic process to be a local supermartingale

(Question edited in response to Nate's comment) 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 ...
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277 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, ...
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80 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 $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
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118 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 ...
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384 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 ...
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473 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 ...
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287 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). ...
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316 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 ...
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41 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 ...
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49 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). ...
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158 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; ...
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241 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": $(dt)^2=0$ ...
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79 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 ...
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341 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 ...
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1k 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 ...
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117 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 ...
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783 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 ...
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561 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 $dX(t)=\mu(t)dt+\sigma(t)dB(t)$ ...
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178 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 ...
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38 views

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

Let $\{X_n(t) \in \mathbb{Z}^+\}$ 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 ...
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47 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 ...
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41 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 ...
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98 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 ...
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103 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 $L^1$ ...
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70 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 ...
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54 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 ...
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161 views

6-digit password - a special decoding method

Consider the situation of decoding a 6-digit password that consists of the symbols A to Z and 0 to 9, where all possible combinations are tried randomly and uniformly. Consider the ...
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147 views

Dose “optional stopping theorem” imply “optional sampling theorem”?

Suppose $X$ is a martingale,$\tau$ and $\sigma$ are two stopping times which satisfy (a)$\sigma\le\tau$ and (b)the "optional stopping theorem" holds,that is to say: $$\mathbb E[X_\sigma]=\mathbb ...
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140 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 ...
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63 views

Markov chains and natural filtration

I have the following problem Consider a homogeneous Markov chain $(X_n)$ with countable state set $E$. Suppose that $A$ is a proper subset of $E$ and consider the stopping times $\tau^0=0 $ and ...
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178 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
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153 views

Determine if this is a Martingale

I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$. We know that $S_t$ is a local martingale because if we stop it ...
4
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64 views

2-D exponential functional brownian motion

I'm looking for the distribution of $X = \int_0^T e^{-W_t} dt \int_0^T e^{W_t}dt$ and $Y = \frac{\int_0^T e^{-W_t} dt}{ \int_0^T e^{W_t}dt}$ (where $W_t$ is a standard brownian motion) On most ...
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600 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 $\sigma$? $X_t$ ...
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185 views

Time scaling of Brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion and $A_t$ be an increasing continuous process adapted to the filtration generated by the Brownian Motion and $A_0 = 0$. I am trying to prove ...
4
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175 views

Characterization of the law of a stochastic process by its finite dimensional distributions

Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space. Let $(X_t)_{t \in [0,T]}$, $(Y_t)_{t \in [0,T]}$ (real-valued) centered Gaussian processes such that the finite dimensional distributions ...
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41 views

Intuition on continuty in probability/mean square of a process

How to explain that a process is continuous in probability? I know the definition, but what does it mean? The same with continuity in mean square.
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516 views

Use of Martingale Representation Theorem

I am working on the following problem, and struggling with it. Can anyone help? Let $$H=e^{\int_0^T B_s\,ds}$$ where $T>0$. Show first $E[H^2]<\infty$. Then find an adapted process ...
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332 views

Spectral process for the Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck process $X(t)$ is a centered, Gaussian process with covariance function $$B(s,t) = e^{-\vert t-s \vert /2}$$ The spectral measure is abs. cont. w.r.t. the Lebesgue measure ...
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205 views

Brownian motion integral

Let $(B_t)$ be a standard Brownian motion, $f$ a continuous function and $X_t = \int_0^t f(s)B_s ds$. I was able to prove that $(X_t)$ is a Gaussian process with zero mean and trying to find the ...
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310 views

Can infinitesimal generator be defined by the time-inhomogeneous stochastic process?

The following is the definition of infinitesimal generator from Oksendal. Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ ...
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88 views

For $X_{t}=\exp\left\{\left(\mu-r-\frac{\sigma^{2}}{2}\right)t+\sigma W_{t}\right\}$, do we have $\mathbb{E}[\int_{0}^{\tau_{b}}X_{s}dW_{s}]=0$?

Let $X_{t}$ denote the solution to the SDE: $$dX_{t}=(\mu -r)X_t dt+\sigma X_t d W_{t}, \ X_{0}=1$$ i.e. $X_{t}$ is the process: $$X_{t}:=\exp\left\{\left(\mu-r-\frac{\sigma^{2}}{2}\right)t+\sigma ...
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151 views

Calculating $\mathbb{E}[\int_0^T N_{t-} dS_t]$ - an expectation of a simple stochastic integral.

I came across some nasty stochastic integral of which I'd like to calculate the expected value" $\mathbb{E}[\int_0^T N_{t-} dS_t]$ where $N_t$ is a Poisson process and $S_t$ is, say, a geometric ...
4
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131 views

Harmonic measure or harmonic kernel

In the theory of discrete-time stochastic processes on a measurable space $(\mathscr X,\mathscr B(\mathscr X))$ one usually starts with a Markov kernel $$ P:\mathscr X\times \mathscr B(\mathscr ...
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935 views

First order variation and total variation of a function/stochastic process

The notions of first-order variation and total variation of a function or a stochastic process are equated in this book. However, I found their definitions different from two other sources: In ...
4
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187 views

Observable and unobservable parameters of stochastic processes

Consider the following diffusion process $$ dX_t = \mu\,dt+\sigma(t,X_t)\,dW_t $$ where $X,W$ are 1-dimensional and. Is it true that given a history $(X_s,s\leq t)$ for each $s< t$ one can find ...
3
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55 views

A Markov Chain probability, conditioned on a random time.

My question: Upon reading theory about diffusion processes, i came across an argument which i believe simplifies to this: Say we have a Borel measurable set $A$ (if it matters you can set $A=\lbrace ...