Questions tagged [stochastic-processes]
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
2,460 questions
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On Riemann integration of stochastic processes of order $p$
Let $x:[a,b]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process, where $\Omega$ is the sample space from an underlying probability space. Let $L^p$ be the Lebesgue space of random variables on $...
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What do we know about Poisson boundaries of arbitrary Riemannian manifolds?
For closed manifolds, we know that the Poisson boundary is trivial due to compactness and for radially symmetric manifolds for which diffusion is one dimensional, there are A Brief Introduction to ...
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Positive definiteness of a matrix-valued function
This question is a repost from math.se, where I didn't receive an answer.
Are there simple conditions on an $d \times d$ matrix B under which
$$
f(t, s)
=
\begin{cases}
\exp(-B |t - s|^\alpha), &...
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What (continuous) stochastic processes have path measures that are absolutely continuous w.r.t. Wiener measure?
Suppose I have a stochastic process $\{Z_t\}_{t \in T}$ for which I know the sample paths to be a.s. continuous (we can also assume some usual stuff, such as $T$ a compact metric space, $Z$ having ...
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Self-adjointness of generator and semigroup of an SDE
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\eps}{\...
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Intensity and compensator for a jump process
Set-up and assumptions. Let $(\mathscr{F}_t, t \geq 0)$ be a right-continuous complete filtration. Let $(X_t, t\geq 0 )$ be a pure jump $\mathbb{R}$-valued process with unit jumps, that is,
$$
X_t = \...
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Does point process ordering ever imply conditional intensity ordering?
Let $N$ and $N'$ be regular/non-explosive point processes on $[0,\infty)$. I will take the view that these are collections of random arrival times: $N=(t_n)_{n\in\mathbb N}$ and $N'=(t_n')_{n\in\...
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Does the convergence of drifted Brownian motion imply the convergence of expectation?
Let $(f_{\epsilon})_{\epsilon>0}$ be a family of non-increasing and continuous functions on $\mathbb R_+$ s.t. $f_{\epsilon}(0)=1$ and $f_{\epsilon}(\infty)=0$. Assume that $\epsilon\mapsto f_\...
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Asymptotics of number of running maxima of iid random variables
Let $\{X_i\}_{i \geq 1}$ be a sequence of iid non atomic random variables, that is, their CDF has no jump discontinuities.
Given a realisation $\omega$ of the random variables, we say that $X_i (\...
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Convergence of random functions
Suppose I have a sequence of random continuous functions, $f^{n} : [0, t] \to \mathbb{R}$. Suppose there also exists a random continuous function, $f: [0, t] \to \mathbb{R}$, defined on the same ...
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Looking for a citation for this simple generalization of the Markov bound to non-negative super-martingales
Does anybody know a reference for the following theorem?
Theorem 1. Let $(X_t)_{t=0}^\infty$ be a non-negative supermartingale.
Then, for any constant $c > 0$, the event $(\exists
> t)\, X_t \...
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Quantifying the effect of noise on the posterior variance in Gaussian processes / multivariate Gaussian vectors
Consider a real-valued Gaussian process $f$ on some compact domain $\mathcal{X}$ with mean zero and covariance function $k(x,x') \in [0,1]$ (also known as the kernel function). This question concerns ...
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Convergence in probability results with still open point-wise versions
In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the ...
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Maximizing expectation of gaussian process over covariance matrix with fixed trace
Let $\mathcal{A} = \{\Sigma \in PSD_{n\times n}(\mathbb{R}), \wedge \forall i,\Sigma_{ii}=1\}$. Then $\mathcal{A} \subset M_{n\times n}(\mathbb{R})$ is convex, closed, and bounded.
For each $\Sigma \...
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On the stationarity of Gaussian processes
I am trying to understand and prove the statement:
The normal (or Gaussian) process is stationary in the wide sense if and only if it is strictly stationary.
I know the following:
A strictly ...
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Limit (convergence) of stopping times
Let $B=(B_t)_{0\le t\le T}$ be a continuous semi-martingale and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Denote by $\mathcal C_b(\Omega\times \mathbb R_+)$ the space of ...
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$\alpha$ stable processes without jumps
Levy processes with jumps can be formulated following the Levy-kinchkine representation, which provide a decomposition of the characteristic function into three factors corresponding to the diffusion (...
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Joint law of the time integral of Brownian motion and its maximum
Suppose $W_t$ is a standard one dimensional Brownian motion. Let $M_t$ and $I_t$ be its running maximum and time integral, respectively:
$$M_t=\max_{0\leq s\leq t}\,W_s$$
$$I_t=\int\limits_0^tW_s\,\...
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Relationship between transition density function and local time
Assume the local time is $L(t,y)$ and we know $P_x(L(t,y) \in d\tau)$ where $P_x$ denotes the probability measure for a stochastic process starts at $x$. Can we then derive the transition density ...
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Is integral of adapted separable process adapted?
Assume $f(t,\omega)$ is (i)separable, (ii) measurable as function from $((0,T)×\Omega)$ into $R$ and (iii) is adapted to the filtration $F_t, 0<t<T$
Also $\int_0^Tf^2(s)ds<\infty$ almost sure....
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On stochastic integration
This questions has been asked on math.stackexchange
I have two questions on stochastic integration.
(1) Constructing the Ito integral, there is the following remark in Jacod/Shiryaev (page 46, 2nd ...
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Drift of reverse SDE with Lévy processes ($\alpha$ stable distributions)
Given an SDE with a Lévy process with a drift $b(x,t)$ the reverse SDE will have a drift, $\tilde{b}(x,t)$, given by the relation:
$$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
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Designing an SDE satisfied by $\frac{B(t)}{1+t}$
Let $B$ be the Brownian motion. I want to find a stochastic differential equation satisfied by the process $$X(t) = \frac{B(t)}{1+t}.$$ I am trying to use Itô's lemma for $f(x,t) = \frac{x}{1+t}$ but ...
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The Ornstein-Uhlenbeck process from modified integrand
Suppose that $\alpha > 0$ and $\sigma \in \mathbb{R}$ are fixed. Define $Y(t), t \geq 0$ to be an adapted modification of the Itô integral
$$
Y(t) = \sigma e^{-\alpha t} \int_0^t e^{\alpha s} dB(s)
...
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One flip coin game
Nate has $n \geq 2$ coins $\{C_i\}_{0 \leq i \leq n-1}$ that each turn up heads with probability $\frac{i}{n-1}$ each, but he is not sure which ones are which.
He has \$1 with which to bet with. On ...
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What happens to an SDE conditional on the underlying Brownian motion being close to $f \in C[0, T]$?
The so called forgery theorem for Brownian motion says that for any continuous $f: [0, T] \to \mathbb R^d$, with $f(0) = 0$, the $d$ dimensional Brownian motion $W$ has a nonzero chance of staying $\...
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Reconstruction of law of diffusion process from call option values
Let $X_{\cdot}$ be a $1$-dimensional diffusion process. If I know the value of the
$$\big\{\mathbb{E}[\max\{X_t,c\}\big| X_0 =x\big]:\, c\in \mathbb{R} \text{ and } \,\, t\in (0,1] \big\}.$$
Then, ...
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Weighted sum of standard Brownian bridges
Let $\{B_j\}_{j=1}^k$ be a sequence of Brownian bridges.
Let us consider $$X(t)=\sum_{j=1}^m w_j(t)B_j(t),$$ where $w_j$ are positive weight functions.
Then what can we say about (distribution or may ...
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Existence of an invariant measure on an infinite dimensional space via Lyapunov functional
Set-up.
Assume that we have a complete separable metric space $\mathcal{X}$ that is not locally compact. Let $V: \mathcal{x} \to [0; +\infty]$ be a functional such that $K_r :=\{x \in \mathcal {X} : V ...
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Can we define the divergence of a stochastic process?
Suppose I have a stochastic process $(X_t)_{t\in \mathbb{R}^d}$ with infinitesimal generator $\mathcal{A}$, for example $\mathcal{A}f(X) = -\mu f'(X) + \frac{1}{2}\sigma^2f''(X)+\lambda \int (f(X')-f(...
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Existence of linear stochastic differential equation given solution
Normally if you have a linear SDE given such as
$dx_t = (A(t)x_t + a(t))dt + \sigma(t) dW_t$, we want to find $x_t$, more precisely we want to find the mean and variance of $x_t$ at each timestep $t$. ...
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Why optional stopping theorems require continuity conditions of martingales?
If we want to prove some form of optional stopping theorem (with a stopping time $T$) for continuous time martingales $M_t$, a typical strategy is to assume that $\mathbb E[M_{T\wedge n}] = \mathbb E[...
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Bound on the radon-nikodym derivative between two stochastic processes at a time point
I have two stochastic differential equations on $\mathbb{R}^d$ adapted to the same filtration evolving for finite time $t\in [0, T]$ at the same start distribution:
\begin{align*}
dX_t &= b(t, X_t)...
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The unique weak solution to some SDE yields the unique strong solution?
For some filtered probability space $\big(\Omega,\mathcal F, (\mathcal F_t),\mathbb P\big)$, consider a stochastic differential equation (driven by a real-valued Brownian motion $W$) for $X=(X_t)$, ...
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Asymptotic mixing time and Euclidean probability distance for path graphs
We are given a simple path graph $P(V,E)$ with vertex set $V$ and edge set $E$, having $n=|V|$ nodes. Given an initial distribution $\mathbf{\mu}$ over $V$, let $d_t(\mathbf{\mu},\pi)$ be defined as $\...
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Approximate the adjoint generator of the discretization of an SDE
Let
$d\in\mathbb N$;
$\sigma\in\mathbb R^{d\times d}$;
$p\in C^1(\mathbb R^d)$ be positive with $$c:=\int p(x)\;{\rm d}x<\infty\tag1$$ and $$b:=\frac12\Sigma\nabla\ln p;$$
$(X_t)_{t\ge0}$ denote ...
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Quasi-invariance of $\Phi_3^4$ under translation by nonsmooth shifts
In https://hairer.org/Phi4.pdf Hairer shows that the $\Phi_3^4$ measure is mutually singular with respect to any nonzero smooth shift. Is it also mutually singular with respect to any nonzero ...
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Cramér–Rao type bound for absolute estimation error
Let $\{X_1, X_2, \dotsc, X_n\}$ be independent and identically distributed (i.i.d.) random variables sampled from a common distribution with density $f_{\theta}(x)$, where $\theta$ is an unknown ...
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Reference for PDEs from system of SDEs
I'm working with a system of SDEs
\begin{align*}
dX_t &= b(X_t, t) + \sigma dB_t\\
dY_t &= c(X_t, Y_t, t) + \sigma dB_t.
\end{align*}
Here, the Brownian motion is the same.
I know that ...
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Conditional Expectation in Diffusion Process
Consider a $d$-dimensional diffusion process $\mathbf{X}=(\mathbf{X}_t)_{t\in [0,T]}=([X^1_t,...,X^d_t])_{t\in [0,T]}$ that is the unique strong solution of the following SDE:
$$\left\{\begin{matrix}
...
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On the Markov property of a limit process
Let $E$ be a locally compact separable metric with countable base. We consider a sequence of Hunt processes $\{X^{(n)}\}_{n \in \mathbb{N}}$ on $E$. That is, each $X^{(n)}=(\{X_t^{(n)}\}_{t \in [0,\...
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Independent stationary increment process but with finite propagation speed
Intuitively, standard Brownian motion has infinite propagation speed, as it has a non-zero probability of reaching any point in any arbitrarily short time. This is due to the fact that the probability ...
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Is this card shuffling process weakly mixing?
Consider the following continuous analogue of a card shuffling process:
Let $Y_i, Z_i$ ($i \in \mathbb Z^+$) be sequences of jointly independent uniformly distributed random variables on $[0, 1]$. ...
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A mean field SDE with hitting time
Let $b\in \mathbb R$ and $\sigma>0$ be given. For a fixed probability distribution $\mu_0$ on $\mathbb R$ s.t.
$$\int_{(0,\infty)}\mu_0(dx)=1,$$
consider the mean field SDE :
$$dX_t = \mathbf{1}_{\...
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Why is $\mathbb R^{\mathbb N}$ not high-dimensional enough?
In this paper [1], the authors consider the limiting distribution of $$S_{n,p}:=\frac{1}{\sqrt n}\sum_{k=1}^nX_k$$ for $p\rightarrow\infty$ as $n\rightarrow\infty$, where $X_1, X_2,\dots, X_n$ are ...
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Lower bounding an alternating series with signs from a martingale difference sequence
Let $\epsilon_n \in \{-1, 1\}$ be a martingale difference sequence, in the sense that
$$M_n := \sum_{i = 0}^n \epsilon_i$$
is a martingale.
We assume $\epsilon_0 = \pm 1$ with probability $\frac{1}{2}$...
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Higher or lower? (#2)
$N \geq 2$ players play a game - at the start of the game, they are each given independently and uniformly a number from $[0, 1]$. On each round, they are to guess whether their number is higher or ...
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Gibbs measure as stationary distribution of SDEs
I have been trying to understand how one can mathematically explain some of the results from statistical mechanics, especially regarding certain distributions like the Gibbs distribution. It would be ...
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Absolute Continuity of the Karhunen-Loeve expansion coefficients
The Karhunen-Loeve theorem (see these notes or the wikipedia page, for example) states the following:
Theorem: For a continuous, square-integrable, centered stochastic process $(X_t)_{t \in T}$ (with ...
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Distribution of Brownian motion conditional on linear growth
Let $W$ be a standard $d$-dimensional Brownian motion with $W_0 = 0$ almost surely.
Fix a constant $\lambda > 0$ and timeframe $T > 0$, and consider the event
$$ E_T := \{|B_s| \geq \lambda s\ \...
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