Questions tagged [brownian-motion]
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292
questions
1
vote
0answers
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\...
5
votes
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 :...
0
votes
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 :
$$ \mathbb{...
0
votes
0answers
15 views
Question on the choice of boundary in the CUSUM test when we make some resampling [migrated]
Question on the choice of boundary in the CUSUM test when we make some resampling
We are considering to make a CUSUM test for some economical time series $X = (x_1,..,x_n)$. Suppose $X$ contains many ...
2
votes
0answers
57 views
Decay rate of transition density of a SDE system
Consider the following SDE system
$$dx_t = b(y_t)dt + dw^1_t, \quad dy_t = dw^2_t.$$
Here the drift $b(\cdot)$ is a smooth function that may decay slowly. For example, $|b(x)| \le C/|x|^\sigma$ for ...
1
vote
0answers
56 views
Extension of the Kelvin transform
Suppose $B=B(y,r)$ is ball in $\mathbb{R}^m$ ($m\geq2$), and $u$ a superharmonic function on a neighborhood of the closure $\overline{B}$ of $B$. We know that the Kelvin transform of $u$ with respect ...
2
votes
1answer
60 views
Generalized Fokker-Planck equation
Consider the diffusion process
$$
d X = \mu(X, t) dt + \sigma(X, t) dY.
$$
When $Y$ is a Brownian motion, we know that the density follows the Fokker-Planck equation. Here I'm considering the general ...
1
vote
0answers
57 views
Laplace Equation for Brownian Motion
So, I know that there is this theorem (taken from here):
For Laplace's equation $\Delta u = 0$ on a domain $D$ and $u=f$ on $\partial D$ (and some regularity conditions on $D$), we have
$$
u(x) = \...
1
vote
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 $\mathbb E|X|^p$ ...
1
vote
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 ...
0
votes
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 ...
0
votes
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
vote
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 $...
1
vote
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
votes
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 ...
1
vote
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
$$
X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...
1
vote
0answers
66 views
Is my quadratic variation derivative bounded?
Let $\{W_t\}_{t\in[0;T]}$ be a Brownian motion, let $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ be continuous, bounded and Lipschitz continuous in the second argument, let $X$ be the unique ...
5
votes
1answer
120 views
Superharmonicity at infinity
Some authors define superharmonicity at infinity in the following way. A function $u$ is superharmonic on an open set $V\subset\mathbb{R}^m\cup\{\infty\}$ (one point compactification), containing ...
0
votes
1answer
136 views
Associativity rule for integration against fractional Brownian motion
In Itô calculus, it is easy to construct an associativity rule. Namely, if $B_t$ is a Brownian motion and $M_t = \int_0^t X_s dB_s$ for suitable $X_t$, then we have the following associativity rule: $...
0
votes
0answers
32 views
Superharmonic extension 3
This question is related to the MO post
Superharmonic extension 2. Let $u$ be a superharmonic function on $\mathbb{R}^m$ ($m>2$) such that for some $\alpha\in\mathbb{R}$ and $\beta$, $R>0$,
$$u(...
0
votes
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
vote
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)...
2
votes
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, $...
0
votes
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 ...
0
votes
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 ...
0
votes
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
votes
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 ...
2
votes
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 ...
0
votes
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
votes
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 \...
0
votes
0answers
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 ...
0
votes
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&...
1
vote
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
vote
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
vote
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
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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
votes
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
votes
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
vote
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 $u > 0$, $\...
0
votes
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
votes
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
votes
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 ...
0
votes
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
vote
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
votes
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
vote
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 ...
