# Questions tagged [stochastic-processes]

A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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### Stopping times for martingale

The nonnegative integer set is denoted by $\mathbb{Z}_+$.
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space and $\{\mathcal{F}_{n}\}_{n\in{\mathbb{Z}_+}}$ be an increasing sequence ...

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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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### L^2 approximation error in Gaussian Process Regression (finite data setting)

I am learning about Gaussian Process Regression. I would like to have some references or results regarding the distribution of the error between a given function, and the posterior obtained in ...

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### A simple procedure to simulate multifractional Brownian motion paths

In a paper by Peltier and Vehel the multifractional Brownian motion (mBm) was defined for the first time, and they also give a procedure to simulate mBm sample paths. Briefly, mBm generalizes the ...

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### techniques in studying moments of shifted integral process $\mu(T_{a},T_{a}+t)$

We have a strictly increasing measure $\mu$ on $[0,\infty)$ given by $\mu(0,x):=\int_{0}^{x}e^{X(s)-\frac{1}{2}\ln1/\epsilon}ds$, where $X(s)$ is a mean zero Gaussian field with truncated log ...

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### Counting overlaps of random intervals

We have a strictly increasing measure $\mu$ on $[0,\infty)$ given by $\mu(0,x):=\int_{0}^{x}e^{X(s)-\frac{1}{2}\ln1/\epsilon}ds$, where $X(s)$ is a mean zero Gaussian field with truncated log ...

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### What is famous mistake made by Feller?

I heard "Feller made a famous mistake in 1954 and fixed by A.D. Wentell in 1959" from one lecture. There is no further explain what is that mistake? Is there someone know it? Is it ...

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### Population growth with good and evil children - probability good outnumbers evil

Consider the following discrete-time population model. We start with a single "good" individual who reproduces asexually into $k$ children and dies in the process. At generation $t=2$, those ...

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### When is the mode of a stochastic process a better statistic than the mean?

This is a soft question.
I've been interested in Onsager-Machlup theory recently. Essentially, the Onsager-Machlup function serves the role of a density but it can exist on non locally compact spaces.
...

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### How to get $\lim_{N\to \infty} \sum_{i=1}^N e^{\lambda_i}u_i^2=\int e^{\lambda}d\sigma(\lambda)$?

I am reading the one lecture note Dynamics for Spherical Models of Spin-Glass and Aging.
On page 126. In the Sherrington-Kirkpatrick (SK) model, we suppose that there are $N$ people labeled as $[N]:=\{...

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### Chung's law of the iterated logarithm for Brownian motion

I am looking for a reference that gives a detailed proof of Chung's law of the iterated logarithm for Brownian motion: $$\liminf_{u\to +\infty}\sqrt{\frac{\ln(\ln(u))}{u}}\sup_{r \in [0,u]}|X_r|=\frac{...

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### Continuation : Uniqueness of the solution to some SDE with discontinuous coefficient

Consider the SDE below
$$X_t=X_0+\int_0^t b(s)ds+\int_0^t\frac{dW_s}{1+m(s){\bf 1}_{\{b(s)>0\}}},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X_0>0$ is square integrable, $b:\mathbb R_+\...

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### Probability that a geometric Brownian motion with additional determinstic drift ever hits zero

Let $W$ be a standard Brownian motion, and let $X_t$ be the solution to the following SDE
$$dX_t = (\mu X_t - Cke^{-kt}) \, dt + \sigma X_t \, dW_t$$
where $\mu, \sigma, C, k > 0$ are constants, ...

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### For some $\alpha>0$, $ e^L=P\left(\exp(\alpha\sup_{|s-t|\le\delta}\frac{|B_s-B_t|^2}{|s-t|})<\infty\right) $?

I am reading one lecture note Dynamics for Spherical Models of Spin-Glass and Aging by Alice Guionnet. On page 124, it says that
for some $\alpha>0$,
$$
e^L=P\left(\exp(\alpha\sup_{|s-t|\le\delta}\...

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### Is a Riccati BSDE explicitly solvable?

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $C\in (0;\infty)$ and $\{a_t\}_{t\in[0;T]}$ be a ...

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### Cover time of a box by SRW in $\mathbb{Z}^2$?

I am wondering can we say something about the cover time $T$ for a box, eg. $[-n,n]^2\cap \mathbb{Z}^d$, by the simple symmetric random walk on $\mathbb{Z}^2$ starting from zero?
For example, the ...

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### Kramers' escape problem: statistical physics vs. Large deviations

I'm almost not at all knowledgable in either Freidlin-Wentzel theory or Kramers' escape problem as it is known in the physics community, so please excuse some of my naivety.
One can use Freidlin-...

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### Running maximum/supremum of Brownian motion: add information to make it a Markov process?

Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it ...

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### Stochastic process on $\{0,1\}^{\mathbb N}$ domination of product measures, necessary and sufficient conditions

Let $X=(X_n)_{n\in\mathbb N}$ be a stochastic process in $\{0,1\}^{\mathbb N}$ with distribution $\mu$. I do not at first make any assumptions about $X$ being stationary or having any kind of ...

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### Rademacher complexity of function class $\{(x,y) \mapsto 1[|yf(x)-\alpha| \ge \beta]$ in terms of $\alpha$, $\beta$, and Rademacher complexity of $F$

Let $X$ be a measurable space and let $P$ be a probability distribution on $X \times \{\pm 1\}$. Let $F$ be a function class on $X$, i.e., a collection of (measurable) functions from $X$ to $\mathbb R$...

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### Infinitesimal generator of a Markov process acting on a measure

Short version: The transition operator of a Markov process can act on measures (on the left) or functions (on the right). The infinitesimal generator acts on functions. Is there a way to understand ...

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### How do we need to argue in this step of the Itō-Lévy-Khintchine decomposition?

Let
$E$ be a $\mathbb R$-Banach space;
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$;
$(X_t)_{t\ge0}$ be an $E$-...

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### Another large noise limit

Note: Here all processes take values in $[0, 1]$.
Let $W$ be a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.
Let $X$ be the solution to the SDE
$$dX_t = \sigma X_t \, dW_t$$...

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### Placing pins on a Galton board to approximate an arbitrary distribution

Inspired by this reddit post: https://old.reddit.com/r/math/comments/tv3cbg/how_do_you_unbell_curve_a_galtonplinko_board/
The Nth Galton Board, G(N), is a triangular lattice of pegs of height N-1.
...

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### Continuation : Does the density of a stopped drifted Brownian motion vanish at zero?

Let
$$Y_t:=1+\int_0^t b_sds + W_t,\quad\forall t\ge 0,$$
where $(b_t)_{t\ge 0}$ is a bounded adapted process and $(W_t)_{t\ge 0}$ is a standard Brownian motion. Denote $\tau:=\{t\ge 0: Y_t\le 0\}$ and ...

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### Proof of the Lévy–Itō decomposition in this paper

Let
$E$ be a normed $\mathbb R$-vector space;
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$;
$(X_t)_{t\ge0}$ be an $E$...

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### What is the difference $\{\tau\leq t\}\in (\mathcal{F})_t $ and $\{\tau<t\}\in (\mathcal{F})_t $ in the definition of stopping time?

Let $\tau$ be a random variable, which is defined on the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F})_{t\in T}, P)$ with values in $T$. In most cases, $T=[0,\infty]$. Then $\tau$ ...

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### Let $(X, W)$ be a weak solution to a SDE. Is $W$ a Brownian motion w.r.t. $\sigma(X_s : s \le t)$?

Let $(X, W)$, $(\Omega, \mathcal{F}, \mathbb{P})$, $\{\mathcal{F}_t\}$ be a weak solution to an SDE.
Per definition $W$ is an $\mathcal{F}_t$-Brownian motion and both $X$, $W$ are adapted to $\mathcal{...

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### Can integrals with respect to time-changed Brownian motion be seen as integrals with respect to Brownian motion?

Let $X_t:=W_{t\wedge \tau}$ for $t\ge 0$, where $(W_t)_{t\ge 0}$ is a standard Brownian motion and $\tau:=\inf\{t\ge 0: |W_t|=1\}$. It holds
$$X_t=\int_0^t {\bf 1}_{\{|X_s|<1\}}dW_s,\quad \forall t\...

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### Does a sequence of coin-tosses a.s. have a subsequence on which the remainder of the sequence can be identified with the position in the sequence?

Let $(X_n)_{n \geq 0}$ be an i.i.d. sequence of $\{0,1\}$-valued random variables $X_n \sim \mathrm{Bernoulli}(\frac{1}{2})$, i.e. a sequence of independent tosses of a fair coin.
Does there exist a (...

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### Where does the extra term in the density of a diffusion with respect to $c B(t)$ come from?

It is well known that for the diffusions
\begin{align*}
dX&=f(X)dt+&cdB\\
dY&=&cdB
\end{align*}
the density of the law of $X$ with respect to the law of $Y$ is
\begin{align*}
\frac{d\...

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### Investigating the properties of a stochastic process with infinitely divisible finite distributions

We know that if $X = [X_t , t \geq 0 ]$ is a Levy process, then any marginal distribution has the charasteristic function - ch. f. of $X_t$ - given by
$$\varphi_t(r)=e^{t \phi(r)}$$
whith:
$$\phi(r) =...

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### How does the probability of staying positive depend on the diffusion coefficient?

Let $X$ and $Y$ be two continuous martingales given as
$$X_t=z + \int_0^t a(s,X_s)\, dW_s,\quad \quad Y_t=z + \int_0^t b(s,Y_s) \, dW_s,$$
where $z>0$, $a,b$ are Lipschitz and bounded functions s....

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### Uniform bound for the occupation time of a diffusion

Note: We denote by $\mathcal L(U)$ the Lebesgue measure of a set $U$.
Let $\mu: \mathbb R^d \to \mathbb R^d$ and $\sigma: \mathbb R^{d} \to \mathbb R^{d \times d}$ be Borel functions.
Suppose the ...

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### On the growth of sample paths of Gaussian random fields

Consider a centered Gaussian random field on $\mathbb{R}^n$ with continuous covariance and a.s. continuous sample paths. What is known about the growth of the sample paths at infinity of such a random ...

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### Alternate proof of Levy’s characterisation of Brownian motion

Levy’s characterisation theorem for Brownian motion states that for a local martingale $X$ with $X_0 = 0$, $X$ is a Brownian motion if and only if it has quadratic variation $\langle X, X \rangle_t = ...

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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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### Autocorrelation function of Itô process

I'm working with a time independent (vector) Itô SDE such as:
$$
dX = a(X) dt + b(X) dW.
$$
I've looked (numerically) at several examples and it seems that the autocovariance function $r_{xx}(\Delta t)...

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### Math videos featuring interesting data animations

I am looking for interesting videos featuring pure data animations (not someone talking about math, but a video featuring some math phenomenon). I am interested in videos that tell a story, rather ...

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### What is the Cameron-Martin norm associated to $X(t)=\int_0^t B(s) ds+B(t)$?

The process $X(t)=\int_0^t B(s) ds+B(t)$ is a centered continuous Gaussian process. Therefore it defines a Gaussian measure on $C[0,T]$. Therefore there is a Cameron-Martin space with Cameron-Martin ...

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### A large noise limit

Let $f: [0, 1] \to \mathbb R$ be a bounded, continuous function, and $W$ a standard Brownian motion.
Denote $Y := \int_0^1 f(t) \, dW_t$.
For each $\varepsilon > 0$, consider the conditioned random ...

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### If $(N,g)$ is a stochastically complete Riemannian manifold and $f : M \to N$ is a submersion, is $f^{\ast} g$ stochastically complete?

Recall that a Riemannian manifold $(N,g)$ is stochastically complete if the weak Omori-Yau maximum principle holds, i.e., for every $u \in C^2(N)$ with $\sup_N u < \infty$, there is a sequence of ...

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### The input and output processes in a single-server queue

Consider an $M/M/1$ queue with the arrival rate $\lambda>0$ and the service rate $\mu>\lambda$ (so that it is stable), in the stationary regime. Let $A_t$ be the number of arrivals in the time ...

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### Uniqueness of the solution to some SDE of state-dependent coefficient

This is a continuation of my question posted in Uniqueness of the solution to some SDE
Consider
$$X_t=X_0 + t + \int_0^t \frac{\sigma(s,X_s)}{1+m(s)}dW_s,\quad \forall t\ge 0,\quad\quad\quad (\ast)$$
...

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### Has this "stochastic differential equation" been studied?

Update: Thanks to GJC20's answer on the existence and uniqueness. Let me reformulate my questions 3/4 as follows: There exists a unique non-increasing and continuously differentiable function $f:\...

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### Is identifying the best in randomly chosen $n$ elements, equivalent to identifying one from the best half of randomly chosen $2n$ elements?

Suppose we are given a set $U$, and a black-box objective function $f: U \mapsto [0, 1]$. The job is to maximise $f(\cdot)$. Now, for a given $\delta \in (0,1)$, consider the following randomised ...

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### The universal constant in Ossiander's bracketing theorem

Ossiander's bracketing theorem reads, for a general function class $\mathcal{F}$ with bracketing number $N_{[]}(\epsilon ,\mathcal{F}, L_2(\mathrm{P}))$,
$$
\mathbb{E} \sup_{f \in \mathcal{F}} \left \...

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### Harmonic function and Markov chain

Let $X=(X_k)_{k \in \mathbb{N}}$ be a Markov chain with countable countable state space $S$ and transition matrix $P.$
Let $\mathcal{T}$ be the tail $\sigma$-field of $X:\mathcal{T}=\bigcap_{k \in \...

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### Solution to a fully nonlinear SDE

Let $W$ be a standard one dimensional Brownian motion. Does the following (fully nonlinear) SDE admit a strong/weak solution?
$$dX_t = X_{t + W_t} \, dt \, ,\, X_0 = 1 \text{ a.s.}$$
Explictly, we ...

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### Compute the autocovariance function of a stationary process

Say we have a stationary process, but we observe samples at random times $\{t_n\}$ which itself is a stochastic point process (e.g. Poisson process). The resulting sample is also a stationary process (...