# Questions tagged [stochastic-processes]

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

1,981 questions
Filter by
Sorted by
Tagged with
75 views

### 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 ...
41 views

67 views

58 views

### 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 ...
1 vote
55 views

### 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 ...
66 views

### 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-...
70 views

### 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 ...
1 vote
99 views

### 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 ...
28 views

### 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 $\mathbb R$...
91 views

### 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 ...
1 vote
42 views

### 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 $E$-...
190 views

### 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 $$dX_t = \sigma X_t \, dW_t$$...
132 views

### 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. ...
1 vote
40 views

### 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 ...
88 views

### 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 $E$...
79 views

### 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 $\tau$ ...
1 vote
30 views

58 views

231 views

### 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 ...
1 vote
247 views

### 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 ...
1 vote
188 views

### 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 ...
1 vote
62 views

### 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 ...
1 vote
102 views

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