Questions tagged [stochastic-processes]

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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26
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446 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 ...
19
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5k 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; i\...
18
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437 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 $n$: $$R_n=|\{...
18
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0answers
294 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 ...
13
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1answer
1k 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 $\...
12
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0answers
560 views

Advanced stochastic process book (a bit flavor from real analysis)

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 (...
12
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3k 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 ...
11
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1answer
344 views

Is a vector of independent Brownian motions a multivariate Brownian motion?

Given a filtered probability space $(\Omega, \mathcal{F}, \mathcal{F}_{t\geq 0}, P)$: If $B_1, B_2, \dots, B_m $ are all real $\mathcal{F}_t$ Brownian motions, jointly independent. Is the resulting ...
11
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0answers
245 views

Convergence of a Stochastic Process - Am I missing something obvious?

In the paper On the Convergence of Stochastic Iterative Dynamic Programming Algorithms (Jaakkola et al. 1994) the authors claim that a statement is "easy to show". Now I am starting to suspect that it ...
11
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0answers
708 views

Schilling's proof of the Feynman-Kac Formula for Brownian motion

This is part of a proof to the Feynman-Kac formula from Schilling's Brownian motion. I need some help understanding the proof to this theorem. Theorem (Kac 1949). Let $(B_t)_{t\ge 0}$ be a $d$-...
11
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0answers
301 views

Limit distributions for Markov chains $X\to\sqrt{U+X}$

This question spawned from a recent, very interesting problem. Let $\varphi=\frac{1+\sqrt{5}}{2}$ and $T$ denote the map on the space of continuous probability density functions supported over $\...
10
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545 views

Intuition behind the renewal equation

So. I've acquired the unenviable task of having to learn renewal theory on my own. I'm finding most of it to be pretty intuitive, except for one thing. The intuition behind the renewal equation has ...
10
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571 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). ...
9
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227 views

$|𝐸[𝑋]|+𝜌(𝑋)≥1$?

Suppose $X_1, X_2, ... \sim X$ are i.i.d. random variables on $\mathbb{Z}$. Then the sequence $\{P(\sum_{i=1}^{d(X)n} X_i = 0)^{\frac{1}{d(X)n}}\}_{n=1}^\infty$ converges to some constant $\rho(X) \in ...
9
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189 views

Probability that a $d$-dimensional Brownian bridge is greater than a given parameter

Let $(W_t)_{t\in[0,T]}$ be a Brownian bridge such that $W_0=a$ and $W_T=b$, the probability that $\forall t\in[0,T],W_t\geqslant x$ given the parameter $x\leqslant\min(a,b)$ is well known : $$ \mathbb{...
9
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826 views

Meaning of right-continuity of a filtration

Given a set $\Omega$ and a filtration $(\mathcal{F_t}, t\in T)$ on $\Omega$, where $T\subseteq\mathbb{R}$, we say that such a filtration is right-continuous if for every $t\in T$ it holds that $\...
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1k 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 ...
9
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296 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 $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
9
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1answer
401 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 $b,\...
8
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218 views

A "reverse" maximal inequality

The following is an exercise from Pollard's "A user's guide to measure theoretic probability". I have to clarify that Pollard uses $\mathbb{P}$ to denote expectation and he omits the ...
8
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154 views

Probability that $\int_0^tX_s\,dW_s$ lies within $1/t$ of $X_t$

Consider the inequality $$f(x)-\frac1x\le f’(x)\le f(x)+\frac1x$$ on the positive axis. This tells us that $f(x)\sim e^x$ with infinitesimal deviation, and we can use identities such as Grönwall's ...
8
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106 views

Linear reward-inaction algorithm for two-armed bandit

Suppose there are two slot-machines. When playing one of them, you win with probability $p$ and while playing the other you win with probability $q$, where $0 < q < p <1$. A gambler ...
8
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258 views

First moments of Geometric Brownian Motion-like process with non-normal shocks

First consider a standard GBM process of the form, $$\frac{dS_t}{S_t} = \mu \, dt+ \sigma \, dW_t$$ but instead of the normal $W_t \sim N(0,1)$ , instead we have that, $$W_t \sim \operatorname{EMG}^...
8
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463 views

How to Prove the Stochastic Fubini Theorem? (Exercise 2.19 in Chapter IV of Revuz and Yor)

Here is the theorem statement: Let $B$ and $C$ be two independent standard Brownian motions. If $\phi$ is square integrable on the unit square ($\phi \in L^2([0,1]^2)$ ), by suitable filtrations, ...
8
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180 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 $(W_t)_{t\ge0}...
8
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203 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). ...
8
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2k 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 \...
8
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0answers
258 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 \...
7
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67 views

What is the probability that a random regular expression defines the language of all binary strings $\{0, 1\}^*$?

Suppose we generate a random regular expression $R$ in the following way: We start with a single meta-symbol $S$. Then each turn we independently replace all $S$ in our word with $\{0\}$, $\{1\}$, $(S ...
7
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339 views

Has Math of Finance education become unnecessarily inaccessible?

EDIT: Post shortened, following the suggestions of @Noah Schweber (Thank you!) EDIT#2: I have spent several hours trying to properly form these questions, so instead of voting this down, please ...
7
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1answer
187 views

Joint distribution of Brownian motion and its running maximum when time is different

Let $B_t$ be a standard Brownian motion and $B_t^*=\max_{s\leq t}B_s$. The joint distribution of $(B_t,B_t^*)$ is well known and its density function is given by $$ f(x,y)=\dfrac{2(2y-x)}{t}\cdot\...
7
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0answers
207 views

Solving root stochastic differential equation

I'm concerned with the following SDE: $$d Y_t= v \,dt + \sqrt {|Y_t|} \,d W_t$$ with $Y_0=-a$, $v>0$ being a constant, $a>0$ and $W_t$ as standard Brownian Motion. Do you have hints how to solve ...
7
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0answers
58 views

Constructing a nearly 0 stochastic process or stopping times

I'm trying to define a stochastic process $X_t$ with values on $\mathbb{R}$ which has the following properties, for the time interval $t \in [0,T]$: $X_t$ takes values in $[0,1]$ $m(\{t \in [0,T] : ...
7
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0answers
239 views

Brownian Motion in Confined space, any results?

I am searching for work regarding Brownian motion in a confined space, like a sphere or a cylinder, where the wall will serve as reflection boundary. I am wondering if it is possible to derive results ...
7
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0answers
293 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 ...
7
votes
1answer
325 views

Malliavin derivative under change of measure

Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$. Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative ...
7
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0answers
970 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 ...
7
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0answers
283 views

Regarding proof of converse to Girsanovs theorem

This is regarding an argument from Arbitrage Theory by Thomas Björk - Theorem 11.6, but is attempted self contained. Consider a Wiener process W on probability space $(\Omega,\mathcal{F},\{\mathcal{F}...
7
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0answers
3k views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
7
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0answers
193 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 model. ...
7
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0answers
766 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 $\textit{infinitesimal generator}$ $\...
7
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0answers
359 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, $$\frac3{(2\pi)^3}\...
7
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0answers
2k views

Infinitesimal generator of time-dependent Markov diffusion

If one talks about homogeneous Markov diffusion $$ \mathrm dX_t = \mu(X_t)\mathrm dt+\sigma(X_t)\mathrm dw_t $$ with $\mu,\sigma$ sufficiently differentiable and of appropriate dimensions, there is ...
6
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0answers
55 views

Existence and uniqueness of the solution to $dX_{t}=\left(2\chi_{\left\{ X_{t}>0\right\} }-1\right)\cos X_{t}dt+\cos X_{t}dW_{t}$

Let $\chi$ denotes an indicator variable and $W$ a Wiener process. What can we say about the solution of the following SDE:$$\begin{cases} dX_{t}=\left(2\chi_{\left\{ X_{t}>0\right\} }-1\right)\cos ...
6
votes
1answer
97 views

Martingale on compact is uniformly integrable

I tried to prove directly that a martingale indexed by a compact intervall is U.I without optional stopping, but without luck. I have a theorem stating that the $L^1$ exitance of $\lim_{t\rightarrow \...
6
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0answers
166 views

Continuity of the Random Variable Defining the Occupation Measure of Gaussian Process

Suppose $Z:\Omega \times [0,1] \to \mathbb{R}$ is a continuous Gaussian process with mean $\mu(t)$ and covariance kernel $C(t,s)$. Consider the random variable $$ X_\alpha = \lambda( \{t \; : \; Z(t) &...
6
votes
0answers
245 views

Understanding the Poincaré cone condition and its intuition (in probabilistic solutions to PDEs)

So, context. Theorem. Let $D\subseteq\mathbb R^d$ be bounded and satisfy the Poincaré cone condition and let $\varphi:\partial D\to\mathbb R$ be $C^0$ boundary conditions. Let $(B_t)_{t\geq0}$ be a ...
6
votes
0answers
129 views

What is the Girsanov density as a functional on the canonical path space $C[0,1]$?

I'll formulate the question via an example. On $( C[0,1], \mathcal{C} )$, where $C[0,1]$ is the set of continuous functions on $[0,1]$ and $\mathcal{C}$ the Borel $\sigma$-algebra given by uniform ...
6
votes
1answer
186 views

Markov Chain upper bound on the probability of hitting time

I encountered the following problem. $\{x_t\}$: Markov chain in discrete time; $\Omega$: a finite state space s.t. $|\Omega|=n<\infty$; $\tau_w\equiv\min\{t\ge 0\,|\,x_t=w\}$, $w\in\Omega$ (first ...
6
votes
0answers
224 views

Martingale problems and SPDEs

It is a classical result that if $X$ is a process with values in $\mathbb{R}^d$ and \begin{align*} M_t^i = X_t^i - \int_0^t b_i(X_s) ds \\ M_t^i M_t^j - \int_0^t a_{ij} (X_s) ds \end{align*} are both (...

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