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Comparison theorem for SDEs driven by a continuous martingale

Consider the well-known comparison theorem for SDEs, versions of which appear in several textbooks, e.g., Karatzas and Shreve, Proposition 5.2.18, or Revuz and Yor, Theorem IX.3.7. The result states ...
ColorfulLion's user avatar
5 votes
1 answer
331 views

Equilateral triangle in a Brownian path

I am curious about the following simple problem but I couldn't do any progress on it. I would like to know whether it is possible to prove (with probabilistic proof) that a brownian trajectory ...
NancyBoy's user avatar
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7 votes
2 answers
253 views

PDE for the probability of Brownian motion staying in an area (reference request)

I am looking for a (preferably some monograph) reference on the following fact: $$ u ( t, x ) = \mathbb{P} \{ x + B_s \in A \ \text{for all} \ s \leq t \} $$ satisfies the heat equation $$ \frac{\...
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1 vote
0 answers
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A question about one Malliavin derivative calculation

Recently, I've asked here a question. While trying to find an answer on my own, I found an idea which I now will briefly describe below. I am not familiar enough with the Malliavin calculus, so my ...
tsnao's user avatar
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2 votes
0 answers
82 views

Non-selfadjoint operators and physical systems

There are plenty of examples of non-selfadjoint operators modelizing physical phenomena: to name a few, let's quote the the heat equation (\ref{HEAT}, see below), the Navier-Stokes system for ...
Bazin's user avatar
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1 vote
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Expectation of $B_u \operatorname{argmax}_t B_t$

This question is a repost from math.stackexchange. The question turned out to be harder than I initially thought, so I decided to try my luck here. Yesterday I asked a question about the joint law of ...
tsnao's user avatar
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4 votes
1 answer
181 views

Solution of SDE at finite time, continuity of pdf

I'm looking at the Langevin dynamics described by the following SDE $$d X_t = - \nabla U(X_t) \, d t + \sqrt {2 \Sigma} \, d B_t,$$ where $X_t \in \mathbb R^d$, $\nabla U(\cdot)$ has some regularity ...
Simone256's user avatar
2 votes
0 answers
118 views

Brownian motion reflected at a trailing barrier

Let $X_t$ be a Brownian motion with positive drift starting at 0. The process with reflection at fixed barrier $b<0$ (sometimes called a "regulated Brownian motion") is: \begin{equation} \...
Dale123's user avatar
  • 21
7 votes
1 answer
417 views

What happens when the diffusion term in an SDE becomes zero?

Consider this time-homogeneous SDE, in the Ito sense: $$dX_t= -(X_t-a)\,dt+\sigma(X_t)\,dW_t,$$ where $W_t$ is standard Brownian motion, $a<b\in\mathbb{R}$, $X_0\leq b$ a.s., and $\sigma(b)=0$. ...
ColorfulLion's user avatar
1 vote
1 answer
91 views

Reference for the 'Brownian Representation Formula'

I am reading a paper ('Hydrodynamics of the N-BBM Process', by De Masi, Ferrari, Presutti, Soprano-Loto) which quotes the 'Brownian representation formula' to represent the solution of a free boundary ...
user1598's user avatar
  • 177
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2 answers
121 views

Find the distribution of maximum of $B_t-t$

Let $B_t$ be a standard Brownian motion. It is easy to show that $\sup B_t-t<\infty$ a.s. . The question is, can we determinate the distribution of $\sup_{t\in [0,\infty)}B_t-t$?
Tiancheng's user avatar
4 votes
0 answers
132 views

Algebraic area of Brownian half-plane excursion

Is anything known about the distribution of the algebraic area, à la Lévy's stochastic area, of a Brownian excursion in the half-plane? To be precise, letting $x>0$, we consider the path $(X_t,Y_t)...
Timothy Budd's user avatar
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3 votes
1 answer
243 views

Laplace transform of Brownian motion functional

Let $(B_r,r\geq 0)$ be a standard Brownian motion on $\mathbb{R}$ started at $0$. I am interested in the quantity $$g(s,t) = \mathbb{E}_0\left[ \exp \left(- \beta \int_s^t \left\vert \frac{B_r}{r}\...
David Geldbach's user avatar
3 votes
1 answer
156 views

Are the paths of the Brownian motion contained in a suitable RKHS?

Let $H_B$ be the reproducing kernel Hilbert space (RKHS) of the Brownian Motion $(B_t)$ on $[0,1]$. It is well known that with probability 1 the paths of $(B_t)$ are not contained in $H_B$. But is ...
Mueller's user avatar
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0 answers
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Conditioned random walk over a graph

I want to solve for a conditioned random walk over a graph. I have a directed graph $G$. The random walkers start at a fixed node, Source. They all need to end up at fixed node, Sink. So the random ...
highBandWidth's user avatar
5 votes
1 answer
418 views

On the convergence of a martingale

Let $W$ be a standard one dimensional Brownian motion and let $A$ be the process defined by : $$\forall \ t\geq 0: \quad A_t := \int_0^t\left(1 + e^{W_s}\right)\mathrm{d}s$$ and for $t\geq 0$, we ...
Greyearl's user avatar
0 votes
1 answer
203 views

Polar form of 2D Brownian motion

Consider two independent unidimensional Brownian motion $w_1$ and $w_2$. What is the polar form of $(w_1,w_2)$? If $r(t)$ and $\phi(t)$ satisfy $(w_1,w_2) = r(t)(\cos(\phi(t)),\sin(\phi(t)))$, how to ...
happy hello's user avatar
2 votes
1 answer
194 views

Macroscopic sets - a notion of largeness for Lebesgue null sets

Let $E$ be a measurable subset of $\mathbb R$. We say $E$ is $\alpha$-macroscopic, for $0 \leq \alpha \leq 1$, if there exists an $\alpha$-Holder continuous function $f: \mathbb R \to \mathbb R$ such ...
Nate River's user avatar
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3 votes
0 answers
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Stochastic braids

I am definitely not a probability guy, but I'd like to have a heuristic answer to the following question: do $n$ independently moving points in an open, connected, bounded region $R$ tend to "...
Andrea Marino's user avatar
4 votes
1 answer
119 views

Reflecting Brownian motion in disk

What is the transition density function of a reflecting Brownian motion in $\mathbb D \overset{\mathrm{def}}= \{z \in \mathbb C : \lvert z\rvert < 1\}$ and how to compute it? The transition density ...
Focus's user avatar
  • 75
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0 answers
42 views

From large deviations to finite time probability tails

Cross-Post from Math.SE Let $(B_t)$ be a standard $d$-dimensional Brownian motion. It is well-known that $$\mathbb P(\sup_{s\in[0,t]}|B_s|\ge \alpha) \le 4de^{-\alpha^2/2dt}.$$ One possibility to ...
Benjamin's user avatar
  • 235
2 votes
2 answers
210 views

Weak convergence of measures on continuous function spaces

Let $S$ be the unit sphere of $C[0,1], \|\cdot\|_{\infty})$, let $(B_{t})_{t}$ the brownian motion. I would like to show that the measure $\mu_r$ defined on $\mathbb{B}(S)$ by $\mu_r(A):=P\Big(\frac{...
Paul's user avatar
  • 21
2 votes
2 answers
374 views

SDE driven by fractional Brownian motion

Let $B^H$ be a fraction Brownian motion of Hurst parameter $H$. Consider the SDE driven by $B^H$ as below: $$dX_t = b(t,X_t)dt + a(t,X_t)dB^H_t,\quad \forall t\ge 0.$$ I am looking for references that ...
GJC20's user avatar
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1 vote
0 answers
123 views

Generating realizations from $n$-dimensional geometric Brownian motion where the variables are constrained to sum to 1

Is there a way to simulate an $N$-dimensional geometric Brownian motion i.e. variable $$x_i, i \in [1, N] $$ is diffusing in log-space such that $$\log (x_i)$$ follows a Brownian motion with a given ...
arrhhh's user avatar
  • 21
2 votes
1 answer
132 views

Measurability of two hitting times at the stopped $\sigma$-algebra

Let $\mathcal{F}=(\mathcal{F}_t)_{t\ge 0}$ be the complete filtration generated by the Brownian motion $B $, and let $a<0<b$. Define the stopping times $\tau_a=\inf\{t\ge 0\mid B_t=a\}$ and $\...
John's user avatar
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2 votes
0 answers
152 views

Fractional Brownian motion covariance with a twist

Let $H \in (0, 1)$, $D \in \mathbb{R}$ and assume that the following function $$ r ( t, s ) = \frac{1}{2} \, \Big[ t^{2H} + s^{2H} - | t - s |^{2H} \Big] + D \, t^H s^H, \quad t, \, s \geq 0 $$ is ...
tsnao's user avatar
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0 answers
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Prove that $\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$

I'm folowing the proof of corollary 1.8 page 5 of Mörters - Sample path properties of Brownian motion. I want to show that $$\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$$ where $B$ is ...
sara's user avatar
  • 11
2 votes
0 answers
62 views

Joint tail for Brownian motion $P[B_{t_1}>g_1,...,B_{t_n}>g_n]$

Maybe not surprisingly there seems to be a lack of in-depth study of sharp estimates for the joint tail of Brownian motion over different times $$P[B_{t_1}>g_1,...,B_{t_n}>g_n]$$ for strictly ...
Thomas Kojar's user avatar
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2 votes
1 answer
263 views

If $u$ is harmonic, $\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x) \leq \alpha |x|+\beta,$ then $u$ is affine

We consider a harmonic function $u:\mathbb{R}^d \to \mathbb{R}$ $(\Delta u=0).$ Suppose that $$\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x)\leq \alpha |x|+\beta.$$ Therefore $u-u(...
mathex's user avatar
  • 441
1 vote
1 answer
146 views

Resources to understand Lebesgue measure of Brownian motion's path [closed]

[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47] Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
sara's user avatar
  • 11
0 votes
0 answers
94 views

Application of Ito's formula to Liouville's theorem

Liouville's theorem for bounded harmonic functions could be proved using Ito's formula, martingale convergence and Blumenthal's 0-1 law. I tried checking the classical books on Brownian motion and ...
mathex's user avatar
  • 441
2 votes
1 answer
141 views

Is every simply connected domain regular?

Recall that a domain $D \subseteq \mathbb C$ is called regular if for each point $x \in \partial D$, we have $\mathbf P_x\lbrack \tau_D = 0\rbrack = 1$, where $\tau_D = \inf\{t > 0 : B_t \notin D\}$...
Focus's user avatar
  • 75
3 votes
1 answer
382 views

Each diffusion SDE is associated to a *unique* family of transition kernels

I consider an SDE of the form $dX_t=b(X_t) \, dt + \sigma(X_t) \, dW_t$, with $b$ and $\sigma$ globally Lipschitz on $\mathbb{R}^n$. How can I prove that there exists a unique family of transition ...
No-one's user avatar
  • 1,077
1 vote
0 answers
49 views

Locality and restriction properties for self-avoiding and loop-erasing random walks

This question has been cross-posted from math.stackexchange.com : https://math.stackexchange.com/questions/4742746/locality-and-restriction-properties-for-self-avoiding-and-loop-erasing-random-wa I ...
Testcase's user avatar
  • 541
1 vote
1 answer
91 views

Characteristic exponent after Girsanov transformation

Let $B$ be a standard Brownian motion. Its characteristic exponent (or Fourier transform) is easily calculated to be $$ \mathbb E [e^{ixB_t}] = e^{-\frac 12 x^2 t}. $$ Now I want to apply a Girsanov ...
Benjamin's user avatar
  • 235
3 votes
1 answer
174 views

Recurrence of Drifted Brownian Motion Conditioned to not hit Moving Barrier

Suppose we have a Brownian motion $X$ with $X_0>0$ and drift $\mu$ conditioned to be less than a barrier $R$ which has behaviour $R_0 = r$, $dR_s = \nu \, ds$, where $\mu > \nu > 0$. Can we ...
user1598's user avatar
  • 177
1 vote
1 answer
95 views

Volterra Processes (integration wrt Brownian motion): reference request

I need some references about Volterra processes $Y=(Y_t)_{t\geq0}$ defined as $$ Y_t:=\int_{0}^{t} g(t,s)dB_s, \ \ t\geq 0,$$ where $B=\left(B_t\right)_{t\geq0}$ is a brownian motion and $g$ satisfies ...
Joegin 's user avatar
4 votes
0 answers
207 views

Schrödinger Bridge for other costs

Stochastic control formulations of the Schrödinger bridge problem between $\mu,\nu$ are well known (e.g Chen et al Eq. 4.23) $$\inf \limits_{p_t, v_t} \int_0^T \int \frac{1}{2}\lvert v_t\rvert^2 p_t ...
nico's user avatar
  • 91
7 votes
2 answers
445 views

Fractional Brownian motion of Riemann-Liouville type is not a semimartingale

Given a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ satisfying the usual conditions, $B$ a standard one-dimensional Brownian motion and $H\in(0,1/2)$. Consider the process $...
El_mago's user avatar
  • 99
2 votes
1 answer
165 views

Local martingale with increasing process

Here is a problem in stochastic calculus: If $M_t$ is a continuous process and $A$ an increasing process, then $M$ is a local martingale with increasing process $A$ if and only if, for every $f\in C^2$...
Liu Wei's user avatar
  • 21
0 votes
0 answers
75 views

Expand White Noise and Brownian Motion in Haar basis: which version of Haar basis?

Start with the Haar basis of $L^2(\mathbb{R})$, namely, the functions $$ \chi(t-k) \text { and } 2^{j / 2} h\left(2^j t-k\right), j \geq 0, k \in \mathbb{Z}, \quad \quad \quad (1) $$ where $\chi(t)$ ...
Mark's user avatar
  • 297
2 votes
0 answers
273 views

Identify two continuous martingales in law as time-changed Brownian motions

Let $W$ be a Brownian motion and $\alpha$ be a progressively measurable process taking values in $\mathbb R_+$. Set $\beta_t:=\max(\alpha_t, 1)$ for all $t\ge 0$. Define respectively $X$, $Y$ by $$X_t:...
Fawen90's user avatar
  • 1,051
2 votes
1 answer
261 views

Joint distribution for sticky Brownian motion

$\newcommand{\R}{\mathbb R}$The one-dimensional Sticky Brownian Motion (SBM in short) is an $\R$-valued Markov process given by \begin{gather*} dX_t=1_{[X_t\neq 0]}dB_t\\ L_t(X)=\int_0^t 1_{[X_s=0]}ds,...
leo monsaingeon's user avatar
5 votes
3 answers
941 views

"Practical" use of time-continuous stochastic processes like Wiener process or Poisson (point) process?

If one uses the Wiener process as an ingredient to model something, then for practical purposes one could just as well take a simple discrete random walk (with sufficiently fine scale). If one uses a ...
Mr H's user avatar
  • 59
1 vote
0 answers
57 views

Probability that a Lévy process "closely" follows a predefined trajectory

For a Brownian motion $(B_t)_{t\geq 0}$ it is well-known [Thm 38, David Freedman, Brownian motion and diffusion], that if $f:[0,1] \to \Bbb R$ is a continuous function with $f(0)=0$ then for $\...
Falrach's user avatar
  • 131
1 vote
2 answers
209 views

Converse Cameron-Martin theorem for shifts by adapted processes

Let $W$ be a standard one dimensional Brownian motion, $\mathcal F_t$ its natural filtration, and $\mathbb P$ be the induced Wiener measure on $\Omega := C[0, 1]$. Given a $C[0, 1] $ valued random ...
Nate River's user avatar
  • 5,325
4 votes
1 answer
398 views

Derive the solution of the diffusion equation from the solution of a random walk

Summary The probability distribution (pdf) of a random walk in 1 dimension is represented by a Bessel function. On the other hand, the pdf of a Brownian motion in free space is represented by a ...
papad's user avatar
  • 272
1 vote
1 answer
89 views

Brownian motion hitting open set starting from its boundary

Let $\{W(t),\,t \in [0,1]\}$ be a standard Brownian motion in $\mathbb{R}^d$, starting from $0$. Let $U$ be a non-empty open set such that $0 \in \partial U$. Which conditions on $U$ are necessary and ...
ssss nnnn's user avatar
2 votes
0 answers
166 views

Wiener sausage of a Brownian motion with coordinates scaled differently

The Wiener sausage of a standard Brownian motion $\{W(t),0 \leq t \leq T\}$ in $\mathbb{R}^2$ is the set $S(T,R)=\bigcup_{0 \leq t \leq T} W(t)+B(0,R)$, where $B(x,r)$ denotes a ball in $\mathbb{R}^2$ ...
ssss nnnn's user avatar
8 votes
2 answers
392 views

Regularity of translations for Brownian motion

Let $B_t$ be the classic Brownian motion. I understand that, if $s>1/2$, almost surely $B_t$ is nowhere $s$-Hölder continuous i.e. almost surely for no point $x$ it happens that $B_t\in C^s(x)$. ...
pipenauss's user avatar
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