Questions tagged [brownian-motion]

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28 views

L2-closure of absolutely continuous stochastic processes?

Assume we have a possibly multidimensional Brownian motion on a probability space $(\Omega,\mathcal F,\mathbb P)$ where $(\mathcal F_t)_{t\in[0;T]}$ is the Brownian standard filtration. Let $\Vert X\...
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2answers
118 views
+50

Bounding Brownian motion and an Ito process simultaneously

Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm :...
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0answers
64 views

Probability that a $d$-dimensional Brownian bridge is greater than a given parameter

Let $(W_t)_{t\in[0,T]}$ be a Brownian bridge such that $W_0=a$ and $W_T=b$, the probability that $\forall t\in[0,T],W_t\geqslant x$ given the parameter $x\leqslant\min(a,b)$ is well known : $$ \mathbb{...
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0answers
15 views

Question on the choice of boundary in the CUSUM test when we make some resampling [migrated]

Question on the choice of boundary in the CUSUM test when we make some resampling We are considering to make a CUSUM test for some economical time series $X = (x_1,..,x_n)$. Suppose $X$ contains many ...
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0answers
57 views

Decay rate of transition density of a SDE system

Consider the following SDE system $$dx_t = b(y_t)dt + dw^1_t, \quad dy_t = dw^2_t.$$ Here the drift $b(\cdot)$ is a smooth function that may decay slowly. For example, $|b(x)| \le C/|x|^\sigma$ for ...
1
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0answers
56 views

Extension of the Kelvin transform

Suppose $B=B(y,r)$ is ball in $\mathbb{R}^m$ ($m\geq2$), and $u$ a superharmonic function on a neighborhood of the closure $\overline{B}$ of $B$. We know that the Kelvin transform of $u$ with respect ...
2
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1answer
60 views

Generalized Fokker-Planck equation

Consider the diffusion process $$ d X = \mu(X, t) dt + \sigma(X, t) dY. $$ When $Y$ is a Brownian motion, we know that the density follows the Fokker-Planck equation. Here I'm considering the general ...
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0answers
57 views

Laplace Equation for Brownian Motion

So, I know that there is this theorem (taken from here): For Laplace's equation $\Delta u = 0$ on a domain $D$ and $u=f$ on $\partial D$ (and some regularity conditions on $D$), we have $$ u(x) = \...
1
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1answer
46 views

Bound moments wrt. known initial and final moments

Let $X$ be an $L^p$ random variable, where $p\in (0,1)$ and $W_t$ usual Brownian motion (with $W_t$ independent from $X$). I'd like to bound $$\mathbb E|X+W_t|^p$$ purely in terms of $\mathbb E|X|^p$ ...
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0answers
40 views

Superharmonicity of the distance function

Suppose $V$ is a convex open proper subset of $\mathbb{R}^m$ ($m\geq2$). It is known that the function $u(x)=$dist$(x,\partial V)$ is superharmonic on $V$. Is there a similar result without $V$ being ...
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1answer
67 views

Probability to cross an envelopp for 1D random walk?

Imagine we have an evolving sequence composed of 1 and -1 (ex: -1-11-111...) where the probability to get -1 or 1 is 1/2. n is the lengh of my sequence. I can make an analogy with random walk: let ...
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2answers
117 views

A question on minimum principle

Suppose $D$ be an unbounded domain of $\mathbb{R}^m$ for $m\geq3$, and $u$ is superharmonic on $D$. We know that if $\liminf_{x\to y}u(x)\geq0$ for all $y$ in $\partial^\infty D$ (the boundary of $D$ ...
1
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0answers
48 views

An open set whose complement is non-thin at infinity

Let $x^*$ designate the inverse of a point $x\in\mathbb{R}^m$ under the Kelvin transformation with respect to the circle of center 0 and radius 1. Recall that $$x^*=|x|^{-2}x.$$ For a set $E$, we set $...
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0answers
35 views

Differentiable approximation of Brownian diffusion with unbounded volatility

Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be ...
0
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1answer
41 views

Differentiable approximation of Brownian diffusion with bounded volatility

Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be ...
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0answers
41 views

Occupation time of SDE

Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation $$ X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...
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0answers
66 views

Is my quadratic variation derivative bounded?

Let $\{W_t\}_{t\in[0;T]}$ be a Brownian motion, let $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ be continuous, bounded and Lipschitz continuous in the second argument, let $X$ be the unique ...
5
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1answer
120 views

Superharmonicity at infinity

Some authors define superharmonicity at infinity in the following way. A function $u$ is superharmonic on an open set $V\subset\mathbb{R}^m\cup\{\infty\}$ (one point compactification), containing ...
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1answer
136 views

Associativity rule for integration against fractional Brownian motion

In Itô calculus, it is easy to construct an associativity rule. Namely, if $B_t$ is a Brownian motion and $M_t = \int_0^t X_s dB_s$ for suitable $X_t$, then we have the following associativity rule: $...
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32 views

Superharmonic extension 3

This question is related to the MO post Superharmonic extension 2. Let $u$ be a superharmonic function on $\mathbb{R}^m$ ($m>2$) such that for some $\alpha\in\mathbb{R}$ and $\beta$, $R>0$, $$u(...
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1answer
100 views

Probability to cross dynamic boundary for 1D-random walk?

context: Imagine we have an evolving bit sequence (ex: 001011...) where the probability to get 0 or 1 is 1/2. n is the lengh of my sequence (the number of bits) I can make an analogy with random walk: ...
1
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1answer
51 views

Is a stopped Ito-integral integrable if the Ito integrand is only square-integrable on an open interval?

Assume a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\in[0;T)}, \mathbb P)$ with an $\mathbb R^n$-valued Brownian motion $\{W_t\}_{t\in[0;T)}$ and the filtration $\{\mathcal F_t\}_{t\in[0;T)...
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2answers
91 views

Use stochastic process to express solution to Laplace equation in the whole space

Consider the Laplace equation in $\mathcal{R}^3$ \begin{equation} \Delta u = f, ~~~\lim_{x\to \infty} u(x) = 0. \end{equation} Here we assume $f$ is a smooth, compactly supported function. Of course, $...
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0answers
32 views

Quadratic variation of generalized stochastic integrals

My question is based on this paper: https://pdfs.semanticscholar.org/0b5a/e41096a3b16d0756a1d36da55143d861ed7c.pdf. In summary, this talks about the generalization of stochastic integrals to a two ...
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0answers
141 views

Local martingale but not martingale

For a 3-dimensional Brownian motion $B = (B_t, t ≥ 0)$ and $x ∈ \mathbb{R}^3 \backslash \{0\}$ define the process $Y = (Y_t, t ≥ 0)$ via $Y_t =\frac{1}{|B_t+x|}$ how come this is a continuous local ...
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0answers
34 views

2d interpolation minimizing the integral of the norm of the Hessian

It is well known that cubic interpolation is the solution of the interpolation problem that minimizes the integral of the square of the second derivative: $$ min_{f \text{ s.t. } f(x_i)=y_i} \int (f''(...
3
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1answer
265 views

proof that the covariance function for a fractional Brownian motion / fractional Gaussian free field is well defined

Given $0 < t_1 < \dots < t_n$, we can show that the matrix $\Omega$ whose entries are defined by $M_{i,j} = min(t_i,t_j)$ is symmetric definite positive. The proof is immediate once one ...
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0answers
72 views

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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1answer
85 views

Does convergence of a sequence of subharmonic functions imply the vague convergence of their Riesz measures?

Suppose $D$ is a bounded domain of $\mathbb{R}^m$ for $m>1$ and $\{u_n\}_{n\geq1}$ is a sequence of subharmonic functions on $D$. Assume $u_n\to u_0$ pointwise on $D$ and $u_0$ is subharmonic on $D$...
2
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0answers
65 views

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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54 views

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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1answer
89 views

A simple clarification on Riesz decomposition theorem

Let $D$ be a domain of $\mathbb{R}^{m}$ and let $K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. According to Riesz decomposition theorem (Hayman and Kennedy, "subharmonic functions&...
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0answers
107 views

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 + ...
1
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2answers
216 views

Expectation of Brownian motion increment and exponent of it

While reading a proof of a theorem I stumbled upon the following derivation which I failed to replicate myself. Let $\mu$ be a constant and $B(t)$ be a standard Brownian motion with $t > s$. Show ...
1
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1answer
96 views

Martingale derivation by direct calculation

I'm reading the proof of a theorem and stumbled across the following derivation which I cannot replicate myself. Let $W(t)$ be a $Q$-martingale and be given by $W(t) = B(t) + \mu t$ with $B(t)$ a ...
2
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0answers
70 views

Is the $\sqrt{{\rm time}}$ spread of a stochastic process about the global minima the ubiquitous phenomenon?

Given a function $f$ with a global minima at $x^*$, consider a stochastic process given as, $x_{t+1} = x_t - \nabla f(x_t) + \xi$ where $\xi$ is a random variable. Now we want to understand the ...
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1answer
81 views

Large deviation for Brownian occupation time

I am asking for reference about the large deviation principle (LDP) for the occupation time of a Brownian motion/bridge. Let $f:\mathbb{R} \to \mathbb{R}$ be smooth and compactly supported. My ...
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0answers
33 views

Backwards Regulated Branching Process with Browning Motion; duality

I am working on a problem which I have not well understood completely, so I can only give the intuition of it. Imagine that we have a population on the (unit) torus $\Bbb T\subseteq\Bbb R$ distributed ...
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0answers
73 views

“Return map” for Brownian motion

Consider a Brownian motion $W$ reflected at the boundary of a domain $D$ in Euclidean space. I want to look at the process obtained by "restricting" it to the boundary. I was thinking of ...
2
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2answers
198 views

Maximum eigenvalue of a covariance matrix of Brownian motion

$$ A := \begin{pmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\ \frac{1}{2} & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\ \frac{1}{3} & \frac{...
2
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0answers
61 views

Exit time for Brownian motion with stochastic barriers

I am interested in the expected exit time of a one-dimensional Brownian particle from a stochastically evolving interval as follows. Context: If $L_t$ and $R_t$ denote the distance to the left and ...
1
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0answers
35 views

Heat Equation Boundary Value Problem - alternative expressions for solution

Let $B_t$ be a Brownian motion, with with density function $f(t,x)dx = P(B_t \in dx)$. Then $f$ solves the heat equation $\partial_t = \frac{1}{2} \partial_{xx}f(t,x)$. Let for a fixed $u > 0$, $\...
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0answers
34 views

Weak convergence to fractional Brownian motion after transformation

Let $x_k$, $k = 1, 2, \cdots$, be a sequence of random vectors in $\mathbb{R}^l$ ($l > 1$), defined on the same probability space, and $f: \mathbb{R}^l \rightarrow \mathbb{R}$. I would like to ask ...
2
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1answer
273 views

Stochastic integral with respect to a random field

I came across a generalized Black-Scholes equation formulation in this paper. Let me highlight the basic idea below. Consider a random field $W(t,T)$ where for a fixed $T$, $W$ is a Brownian motion ...
2
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1answer
100 views

Natural way to thicken Brownian motion to 2D?

If we have a smooth plane curve (Hausdorff dimension 1), we can thicken it by a small amount to get a 2D set (all points within distance $\epsilon$ to the curve). What if we start with the graph of a ...
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3answers
228 views

How to prove that a Brownian bridge $\mathbb{P}(M[0, 1/2]\geq s)\leq 2\mathbb{P}(B(1/2)\geq s/2)?$

Consider a Brownian bridge $B: [0,1]\to \mathbb{R}$ with $B(0)=B(1)=0$. Let $M[0, 1/2]=\max_{x\in[0,1/2]}B(x)$. How to prove that $$\mathbb{P}(M[0, 1/2]\geq s)\leq 2\mathbb{P}(B(1/2)\geq s/2)?$$ ...
1
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1answer
56 views

Limit of an integral / Boundary behaviour of a Gaussian convolution / single layer potential

Let $k(t,x)$ be the transition density of Brownian motion $$ k(t,x) := \frac{1}{\sqrt{2 \pi t}} \exp \left\{ \frac{-x^2}{2t} \right\} , \quad t \geq 0, x \in {\mathbb R.}$$ Question Let $0 < x &...
3
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0answers
113 views

Correlation of stopping times for integral of Brownian motion increment

Let $\mu(x):=\int_{\epsilon}^{x}\exp\{B_{s+\epsilon}-B_{s-\epsilon}\}ds$, where $(B_{s})_{s\geq 0}$ is a Brownian motion (starting at $B_{0}=0$) and epsilon is small $0<\epsilon\ll 1 $. Consider ...
1
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0answers
103 views

2-d geometric Brownian motion hitting time distribution

I am trying to solve following problem: Given two independent geometric Brownian motions $\frac{d x_t}{x_t}=\mu_x dt + \sigma_x dw_t^x$ and $\frac{d y_t}{y_t}=\mu_y dt + \sigma_y dW_t^y$ and ...
3
votes
1answer
239 views

Orthonormal frame bundles on a manifold

Let $(\mathcal{M},g)$ be a torsion free compact Riemannian manifold of dimension $n$. Hence from the metric we know there is an associated horizontal sub-bundle $H_u F \mathcal{M}$ of the orthonormal ...

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