Questions tagged [stochastic-processes]
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
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Randomized stopping times
On p. 75 of his Stochastic Calculus book, Michael Steele writes,
Here, $\tau$ is defined as the first time that a certain random walk leaves a certain random interval, and the subtlety is the ...
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2
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non-homogeneous counting process
Consider a counting process $\{N(t), t\geq 0\}$ where the time distribution between any two consecutive events, say $k$ and $k+1$ has a Poisson rate $\lambda(k)$, which is an explicit function of $k$....
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Malliavin derivative of stopped Brownian motion
Cross-posted from: "https://math.stackexchange.com/questions/3917971/malliavin-derivative-of-stopped-brownian-motion"
I have a small question concerning the Malliavin derivatives. It could ...
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Covering number after projection
In these lecture notes on Statistical Learning Theory we find the following definitions for covering numbers:
Definition. Let $(\mathcal{W}, d)$ be a metric space and $\mathcal{F} \subset \mathcal{W}$...
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How to get the mean, skewness of an Itō integral?
If $B_t$ denotes a standard Brownian motion, and let $X_t = \int f(s)dB_s$, $f(s)$ is a deterministic integrand. I know $B_t$ is a martingale. Is $X_t$ also a martingale? And how can I get the formula ...
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The integral of a Gaussian process on a unit sphere
Suppose there exist a zero-mean Gaussian process $\mathbb{G} f_u$ indexed by $u \in \mathcal{S}^{p - 1}$ with known covariance $\mathrm{E} \big[ \mathbb{G} f_u \mathbb{G} f_v \big]$ when both $u$ and $...
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About deriving the Fokker-Plank-Smoluchowski equation of a (homogeneous) S.D.E
We recall that given a $d-$dimensional stochastic process defined as a solution of a homogeneous S.D.E $dX_t = b(X_t)dt + \sigma(X_t)dB_t$ its corresponding infinitesimal generator ${\cal L}$ is s.t ...
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Invariant measures of Levy S.D.Es
Suppose we call a real valued stochastic process $\{Z_t\}$ to be distributed as ${\cal S}\alpha{\cal S}(\sigma)$ if each of the characteristic functions is $\phi_{Z_t}(u) = \exp\left\{-t\vert \sigma u ...
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The min of the mean of iid exponential variables
Let $X_1, \ldots, X_n, \ldots$ be iid exponential random variables with mean 1. It is well-known that $\min_{1\le j < \infty} \frac{X_1 + \cdots + X_j}{j}$ follows the uniform distribution U(0,1). ...
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Characterization of the generator of a Lévy process using martingale problems
Let $(X_t)_{t\ge0}$ be a real-valued Lévy process. Note that $$\mu_t:=\mathcal L(X_t)\;\;\;\text{for }t\ge0$$ is a continuous convolution semigroup$^1$. Let $$\tau_x:\mathbb R\to\mathbb R\;,\;\;\;y\...
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Existence of unique convolution semigroups of probability measures on more general spaces then $\mathbb R^d$
Let $E$ be a $\mathbb R$-Banach space, $\mathcal M_1(E)$ (resp. $\mathcal M_1^\infty(E)$) denote the set of probability measures (resp. infinitely divisible probability measures) on $E$, $\varphi_\mu$ ...
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Convolution in state space of a Markov process?
This question is regarding the convolution theorem. So far, I have only used it as a tool to solve some recurrence relations. A nice example that I am familiar with is the renewal formula for finding ...
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Hitting probability for mean-reverting stochastic process
I quote Delbaen and Shirakawa (2002).
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\right)dt+\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}dW_t\...
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Some doubts on proof of pathwise uniqueness of a stochastic differential equation
I quote a paper from Delbaen and Shirakawa (2002). I will write in italics my observations/questions.
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\...
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Define the convolution root of probability measures on a measurable group
Let $(G,\mathcal G)$ be a measurable group and $\nu^{\ast k}$ denote the $k$th convolution power of a probability measure $\nu$ on $(G,\mathcal G)$ for $k\in\mathbb N$.
Remember that a probability ...
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Generator of a Hilbert space valued Wiener process from the solution of a martingale problem
Let $H$ be a separable $\mathbb R$-Hilbert space, $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with $\operatorname{tr}Q<\infty$ and $(W_t)_{t\ge0}$ be a $H$-valued Wiener process on a ...
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If $W$ is a Markov chain and $N$ is a Poisson process, then $\left(W_{N_t}\right)_{t\ge0}$ is Markov
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $(W_n)_{n\in\mathbb N_0}$ be a time-homogeneosu Markov chain on $(\Omega,\mathcal A,\...
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Bounding the second moment of a stochastic process
Consider the stochastic process
$$
\theta_{t+1} = \arctan\left( \frac{\lambda_2 \sin \theta_t + \sigma w_t}{\lambda_1\cos \theta_t + \sigma v_t} \right),
$$
where $v_t, w_t \sim \mathcal{N}(0,1)$ are ...
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Random sequential adsorption of rectangles with random aspect ratio
Jian-Sheng Wang, in his 1994 article A fast algorithm for random sequential adsorption of discs (arXiv link), considered deposition only on (small) squares which are not fully covered by the excluded ...
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Convergence of stochastic process $X_n$
Consider the discrete time random process $X_n,n\in \mathbb N$, with
$$X_{n+1}=(1-K)\cdot X_n+K\cdot\frac{G_n}{c}\cdot X_n$$
where $G_n$ is a random variable with expectation $\mathbb E[G_n\mid X_n]=\...
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Estimating the hitting time for a SDE solution
Consider a the following OU process in one dimension,
$$dX = -\theta(X -x_0)dt + \sqrt{s}dW $$
Now one can define the time $t_x$ as the time it takes for the solution to reach the point $x$.
Then ...
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Existence of Gaussian random field with prescribed covariance
Suppose a function $G:\mathbb{R}^d\rightarrow\mathbb{R}$ is given.
What are some necessary or sufficient conditions on $G$ for there to exist a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and ...
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Convergence of the probability that hitting times being infinity
Let $X^n=(X^n_t)_{t\ge 0}$ and $X=(X_t)_{t\ge 0}$ be RCLL (right-continuous with left limits) processes such that
$$\lim_{n\to\infty}X^n=X,\quad \quad \mbox{almost surely},$$
where this convergence ...
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2
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If $\mu$ is an infinitely divisible probability measure on $[0,\infty)$, then the Lévy measure of $\mu$ is the vague limit of $n\mu^{*1/n}$
If $\nu$ is a finite measure on $(\mathbb R,\mathcal B(\mathbb R))$, let $\nu^{\ast k}$ denote the $k$-fold convolution¹ of $\nu$ with itself for $k\in\mathbb N_0$, $$\exp(\nu)\mathrel{:=}\sum_{k=0}^\...
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Growth rate of exponential sum of $S_j$
Let $X_j$, $1 \leq j \leq n$ be chosen i.i.d. uniform over $[0,2\pi)$. Denote $S_j \triangleq X_1 +X_2+\cdots+X_j$ and suppose that $c_j$, $1 \leq j \leq n$ are some constants such that $|c_j|=1$.
I'...
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Is the distribution of a Banach space valued Lévy process uniquely determined by its characteristic function?
Let $E$ be a $\mathbb R$-Banach space. Remember that if $\mu$ is a finite measure on $\mathcal B(E)$ then $$\Phi_\mu:E'\to\mathbb C\;,\;\;\;\varphi\mapsto\int\mu({\rm d}x)e^{{\rm i}\varphi(x)}$$ is ...
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Continuous time Markov chains and invariance principle
This question may be elementary for experts
Let $\{\xi_n\}_{n=1}^{\infty}$ be an i.i.d random variables on a probability space $(\Omega,\mathcal{F},P)$. We assume that the mean of $\xi_n$ is zero, and ...
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Path dependent Markov property
Let's consider a function $\Psi\in \mathcal{C}_B(\mathcal{C}[t,T])$ continuous and bounded
\begin{align*}
\Psi \colon \mathcal{C}[t,T] \longrightarrow [0,+\infty)
\end{align*}
Then my question is:...
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Poisson summation formula for infinite dimensional spaces
Let $M$ be an orientable, compact smooth manifold with a metric $g$ and $H^{-1}(M)$ be the dual space of $$H^{1}(M)=\{f:\int |f|^2+(\nabla f)^2 d\mu<\infty\}$$
I know it is well known that (see ...
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Cointegration of (multiple) time series
Some time ago I stumbled upon the notion of cointegration of time series (see the wiki for some basic fact).
Unfortunately, my knowledge of time series is a bit sketchy, and moreover I was able to ...
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Show an SDE's solution has positive probability to visit every set in the state space
Let $(\Omega, \mathcal{F},\mathbb{P})$ be a filtered probability space, let $b:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n$ be a continuous function and Lipschitz continuous in the space variable. For ...
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Probability of a particle surviving forever
Consider a particle whose position is driven by the following equation:
$$Y_t = y + t + W_t + C\min\big(1,(Y_t+1)^+\big)\Lambda_t,\quad \mbox{for all } 0\le t<\tau_*,$$
where $y>0$, $0<C<1$...
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Laplace transform of sum of random variables in first hitting time problem
Let me refer to the example here.
Suppose $X$ is a birth-death (BD) process (represents population size) that evolves by:
$X \to X+1$ if a birth occurs with rate $\mu$,
$X \to X-1$ if a death occurs ...
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The Itō isometry for Riemannian manifolds
If $\alpha$ is a real smooth $1$-form, and if $\mathcal C$ is the space of continuous functions $c : [0,1] \to \mathbb R^n$, endowed with the Wiener measure $w$, and if $I_\alpha : \mathcal C \to \...
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Independent increments for the Brownian motion on a Riemannian manifold
In am not a probabilist, but I must do some stochastic-flavoured work on a connected Riemannian manifold $M$. A nice thing about the Brownian motion on $\mathbb R^n$ is that we may talk about its ...
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Can one define a bounded noise process by conditioning standard Gaussian white noise on the assumption of boundedness?
Background of the question.
One of the problems that arises with Wiener-driven (and more general Lévy-driven) models of noisy systems is that, due to the extremely rapid decay of tails of infinitely ...
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Can derivatives of 2 stochastic processes be multiplied?
We understand SDEs like "$dX_t = b(t,X_t)dt + \sigma(t,X_t)dB_t$" for Brownian process $B$ to be formally the same as "$\frac{dX_t}{dt} = b(t,X_t) + \sigma(t,X_t)W_t$" where $W$ is ...
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Kac-Rice formula and Borell-TIS inequalities for gradient-flow of centered gaussian random field
Let $x\mapsto g(x)$ be a centered gaussian random field on $\mathbb R^m$. Let $x_0 \in \mathbb R^n$, and (assuming regularity conditions) consider the gradient-flow
$$
\dot{x}(t) = -\nabla g(x(t)), \;...
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Intersection of a Poisson bridge and a Brownian bridge
Take a Poisson process $N_t$, a Brownian motion $W_t$ and constants $T > 0$ and $a > 0$. Suppose $N$ and $W$ are independent. I'm interested in the probability that $W$ does not cross over $a + ...
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Central limit theorem for chi-squared random field on $\mathbb R^p$
Let $X:x \mapsto X(x)$ be a centered stationary Gaussian process on the $\Omega:=\mathbb R^p$, such that $X(x) \overset{d}{=}X(x')$ for all $x,x' \in \Omega$. Set $\sigma^2 := \mbox{Var}(X(0)) = \...
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Concentration inequality for the supremum of $L_2$ norm of a vector-valued Gaussian process with iid components
Let $\Omega$ be a compact subset of $\mathbb R^p$ and let $f_1,\ldots,f_k$ be zero mean identically distrubuted Gaussian processes on $\Omega$ such that $f_1(x),\ldots,f_k(x)$ are independent $x \in \...
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Characterizing 'very homogeneous' finitely valued stochastic processes
Fix a positive integer $n$. Let $X = \{X_i\}_{i \in \mathbb{N}}$ be a discrete time stochastic process such that each $X_i$ is a $\{0,\dots,n-1\}$-valued random variable. Suppose that the joint ...
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Recursive random number generator based on irrational numbers
Here $\{\cdot\}$ and $\lfloor \cdot\rfloor$ denote the fractional part and floor functions respectively. For a negative, non-integer number $x$, we use the following definition: $\{x\}=1-\{-x\}$. If $...
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Why are financial markets modeled by càdlàg processes?
When opening a book or reading an article on mathematical finance, financial markets (e.g. stock prices) are always modeled by càdlàg semimartingales. I was wondering why it is that these processes ...
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A scaled random walk on the number line
An agent $A$ is performing a random walk on the number line. Let $X_t$ be his position at time $t$. $X_{t+1}$ is calculated according to the following rules:-
$ X_{t+1} =$
\begin{cases}
...
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1
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Is the set of almost surely continuous points dense?
Denote by $D(0,T)$ the space of right continuous functions with left limits defined on $[0,T]$. Let $\mathbb P$ be a probability measure on $D(0,T)$. Define
$$cont(\mathbb P):=\Big\{t\in [0,T]:~ \...
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Poisson-like random walk expressed as Bernoulli-like random walks (splitting scheme)
In our problem we have the transition density for $x,y\in \mathbb{Z}$ and $t\in \mathbb{N}$
$$R_{t}(x,y):=e^{-t}\frac{t^{x-y}}{(x-y)!}1_{x\geq y},$$
which is the Poisson distribution pdf. (This is ...
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1
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662
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Question/References on the Skorokhod M1 topology
Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
2
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86
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Concentration inequalities for gradient flows induced by random fields
Let $G=(G(x))_{x \in \mathbb R^m}$ be a conservative random field with values in $\mathbb R^m$, for large positive integer $m$. That is, there exists a scalar random field $g=(g(x))_{x \in \mathbb R^m}...
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Kernel of the adjoint of the infinitesimal generator of Levy SDE
Consider S.D.Es driven by a combination of Brownian and non-Brownian Levy noise (like say Gamma). Then we know that the flow of the density of the S.D.E variable is given by the adjoint of the ...