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Questions tagged [brownian-motion]

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2
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1answer
77 views

How far away is $\max_{x: x \in \{0, \ldots, N\}} |W(x/N)|$ from $\max_{0 \leq t \leq 1} |W(t)|$ ($W(t)$ a Wiener process)?

How far away is $$\max_{x: x \in \{0, \ldots, N\}} \left|W\left(\frac{x}{N}\right)\right|$$ from $$\max_{0 \leq t \leq 1} |W(t)|$$ In other words, if you simulate a Wiener process over a finite ...
0
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0answers
41 views

$P(\max_{0 \leq t \leq 1} \|W(t)\| \leq x)$ has no closed-form expression… right?

$P(\max_{0 \leq t \leq 1} \|W(t)\| \leq x)$ shows up in a formula for computing $p$-values for a certain statistic, where $W(t)$ is a $d$-dimensional (standard) Wiener process. My advisor says the ...
1
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1answer
78 views

Is there Brownian motion on Alexandrov spaces?

It is well known that there is a notion of Brownian motion on smooth Riemannian manifolds. I am wondering if there is a more general notion of Brownian motion on finite dimensional Alexandrov ...
1
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1answer
67 views

Martingale representation theorem for symmetric random walk

Let $X(t)$ be a martingale w.r.t. filtration generated by Brownian motion $B(t)$. There is a well-known theorem that states that there is a unique adapted process $H(t)$ such that $$ X(t) = \int_0^t ...
0
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0answers
30 views

Doob's h transform

Consider $B$ a one dimensional brownian motion, a probability $\mathbb{P}$ and its filtration $\mathcal{F}_t$. Let $t>0$, and define $$ H_0^{(t)} \triangleq \inf \{ \tau > t, \: B_\tau = 0 \} $$ ...
0
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0answers
79 views

Can we have Levy area for N dimensional process?

Consider a two dimension Brownian motion $(X_t,Y_t)$ and we can consider Levy's area as $\int_0^t X_sdY_s-\int_0^t Y_sdX_s$. Is there a equivalent area for N dimensional Brownian motion, if so what ...
0
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0answers
29 views

Supremum of a general Gaussian Process

I have a stochastic integral of the form \begin{align*} X(t) = \int_0^t h(v) W(v) dv \end{align*} where $W(v)$ is the standard Brownian motion and $h(v)$ is a positive, integrable function. While $X(t)...
2
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0answers
51 views

Mutual dependencies of BSDE solutions with markovian drivers with different starting points

Let $(\Omega,\mathcal F, P)$ be a complete probability space with a Brownian motion $(W_t)_{0\le t\le T}$ and the Brownian standard filtration $(\mathcal F_t)_t$ with $\mathcal F_T = \mathcal F$. Let ...
1
vote
1answer
105 views

conditional expectation brownian motion

$B=(B_t,t\in[0,1])$ a standard brownian motion on $[0,1]$. For $t\in[0,1]$, we define $$\mathcal{F}_t=\sigma(B_s,s\in[0,t]),$$ $$\mathcal{G}_t=\mathcal{F}_t\,\vee\,\sigma(B_1).$$ How can we show $$\...
2
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1answer
126 views

Gaussian sum VS Brownian motion

Given independent Gaussian $d$ dimensional vectors $G_i$, Let $ \sigma^2_n=\mathbb{E}(\sum_{i \le n} G_i) \cdot (\sum_{i \le n} G_i)^T$. $||\sigma_n^2||$ is norm of $\sigma_n^2$. Is there a $d$-...
2
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0answers
37 views

Floquet stochastic process

Let $X_t$ be defined by the SDE $$ dX_t = A(t, X_t)dt + dW_t $$ where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ ...
0
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0answers
46 views

quadratic variation on n-sphere

Is it true, and if so, is there an easy way to see that the quadratic variation of standard Brownian motion on n-sphere is $\leq$ t? Note: I am a novice in stochastic analysis.
-1
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1answer
71 views

Expected value of $W_{t_i} W^2_{t_{i+1}}$

I stuck in determining the expected value of the following product $E[W_{t_i}W_{t_{i+1}}^2]$ where $W_{t_i}$ and $W_{t_{i+1}}$ are Brownian with normal distribution, i.e. $W_{t_i}\sim N(0,t_i)$. I ...
1
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0answers
73 views

Schilder's theorem for brownian bridges

I am really not a probabilist and I apologize if my question is too naive or not appropriate, please feel free to migrate to SE. A bit of context: usually, Schilder's theorem tells us that the ...
0
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0answers
84 views

Concerning some Tauberian-type asymptotics of Laplace transform involving $e^{-\sqrt{s}}$

There are some well-known Tauberian theorems concerning the asymptotics of the original function (say as $t$ tends to $0$) and that of its Laplace transform (as $s$ tends to infinity). I want to ask a ...
2
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1answer
409 views

Is the ito integral $\int_0^t \operatorname{sign}(W_s)\mathrm{d}W_s$ a Brownian motion?

Consider the ito integral of the sign of the Brownian motion $W_s$ from $0$ to $t$: $$\int_0^t \operatorname{sign}(W_s)\,dW_s$$ This appears for instance in the Tanaka formula. I think this is a ...
3
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3answers
95 views

Moments of the Hölder norm of Brownian process

It is well known that for a brownian process $B(t),t\geq 0$, it holds $$ \sup_{0\leq s<t\leq T}\frac{|B(t)-B(s)|}{|t-s|^\alpha}<\infty $$ almost surely, for any $T>0$ and $\alpha<1/2$. ...
7
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1answer
297 views

Covariance function of Brownian motion and the second derivative operator

I recently noticed something about the covariance function of a Brownian motion that I don't quite understand, and I was wondering if anyone could help me. Suppose $W$ is a Brownian motion, and we ...
3
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1answer
114 views

What is the Wiener measure of the curves with Hölder index $\frac 1 2$?

One may show that the Wiener measure (for curves in $\mathbb R^n$) is concentrated on the Hölder-continuous curves of Hölder index $< \frac 1 2$. What happens to the curves of Hölder index ...
0
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0answers
55 views

Convergence of a stochastic process in probability

I came across the following. For any fixed $n$, let $\{X_{n}(s) \}_{s\geq0}$ be a stochastic process and let $\{B_n(s) \}_{s\geq0}$ be a Brownian motion. We wish to study the behaviour of $\{X_{n}(s) \...
5
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1answer
85 views

Convolution of two Brownian motions

Suppose $B_1(t)$ and $B_2(t)$ are two independent, standard Brownian motions. What is the distribution of \begin{align*} G(t) = \int_0^t B_1(\tau)B_2(t-\tau)d\tau \qquad \end{align*} Or, at least an ...
1
vote
1answer
112 views

Expected properties for a PDE whose solution is supposed to be something that doesn't exist

My understanding of Lecture #33, 34: The Characteristic Function for a Diffusion: As an alternative to directly computing the characteristic function of a random variable $X_t$ in a stochastic ...
4
votes
0answers
88 views

Who proved the reflection principle in random walks and Brownian motion?

I've heard Henry McKean say that the reflection principle is due to Désiré André. But the wikipedia page seems to say that André did not use a reflection principle. Does anyone know where the "modern" ...
4
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0answers
82 views

Normalisation in fractional integration and Brownian motion

Fractional Brownian motion comes in two forms (following Marinucci and Robinson 1998) for fraction $\alpha$ and Brownian motion $W_s$: Type II (Levy, Volterra, Riemann) $$ \tilde W^\alpha_t = \int_0^...
2
votes
1answer
112 views

Generator of Wiener process and its running maximum

This was originally posted on Math StackExchange a long time ago, but got no answer (even after a bounty). See https://math.stackexchange.com/questions/1274775/generator-of-wiener-process-and-its-...
3
votes
2answers
174 views

Supremum of difference of Brownian bridges: strictly positive wp 1?

EDIT: the original $\ge$ is now $>$ (sorry for the typo!) Let $B_1(\cdot)$ and $B_2(\cdot)$ denote independent, standard Brownian bridges, i.e., they are mean-zero Gaussian processes on $[0,1]$ ...
0
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0answers
68 views

Movement of a random walk in the limit (a particle in diffusion)

I asked this question in Math Exchange and obtained no answer. Let $X(t)$ be a stochastic process in time such that $X(0)=0$ and, at each increment of time $\Delta t$, it can move $h$ units in space ...
0
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0answers
105 views

Quadratic variation and DDS theorem

DDS theorem tells us, that for every continuous martingale M such that $⟨M⟩_{\infty} = \infty ~~a.s.$ then there exists a Brownian motion $B$ such that $M_t = B_{⟨M⟩_t}$. I have a question whether I ...
2
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1answer
148 views

Brownian sausage surgery of Poisson point process

Fix some $r >0$ and let $\mathcal P$ be a unit intensity Poisson point process on $\mathbb R^d - \mathbb B(0,r)$. Let $W_t = \cup_{s \leq t} \mathbb B(B_t,r)$ be the Brownian sausage around a ...
1
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0answers
55 views

Dynamics for sets related to Brownian motion: zero set, fast points

For sets like the Cantor set, we have preserving maps (eg. the shift-maps and conjugates to it) that allows us to study dynamical quantities such as invariant measure and entropy. I am wondering if we ...
3
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1answer
164 views

Conditional stochastic integration

Let's say we have two functions $h(s)$ and $g(s)$. We can easily simulate a stochastic integral, e.g. $$t \mapsto \int_0^t h(s) dB(s) \sim \mathcal{N}\bigg(0, \int_0^t h(s)^2 ds \bigg). $$ What is the ...
1
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0answers
92 views

Unique EMM & completeness in the Black-Scholes model

Consider the Black-Scholes model $$ dS(t) = \mu(t) S(t) dt + \sigma(t) S(t) dW^{\mathbb{P}}(t) $$ $$ dB(t) = r(t) B(t) dt$$ Steele shows now in "Stochastic Calculus & Financial Applications" (Ch. ...
1
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2answers
459 views

2 dimensional brownian motion hitting time

If we have two independent brownian motion in $x$ and $y$ direction. At time zero we sit at $(a,b)$ with $a>0, b>0$. What is the probability that we will hit positive $x$ axis before hitting ...
1
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0answers
99 views

Brownian motion on $[0,1]$

Let $\{W'_t\}_{t\in [0,1]}$ be the Brownian motion on the real line obtained by taking the standard Brownian motion $\{W_t\}_{t\ge 0}$ and conditioning on the events $W_1 = 1$ and $0\le W_t\le 1$ for $...
3
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2answers
280 views

Does the hitting time of +1/-1 of a Brownian motion posess a density?

The law of the hitting time of a 1-dimensional Brownian motion $W$ is well known, but I can't find any information on the density of the hitting time of $|W|$. I define $T=\inf \{t>0,|W|(t)= 1\}$. ...
6
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2answers
312 views

Reference for LIL for fractional Brownian motion

(Cross-posted to https://math.stackexchange.com/questions/2377810/law-of-iterated-logarithm-for-fractional-brownian-motion.) It seems strange but, even after consulting several books, and hours ...
2
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2answers
134 views

Isometry for the stochastic integral wrt fractional Brownian motion for random processes

Let us fix $(\Omega,\mathscr A,\Bbb P)$ a probability space. Let then $\Bbb F:=(\mathscr F_t)_{t\ge0}$ be a complete and right continuous filtration. Now if $B$ is an $\Bbb F$-standard Brownian motion,...
3
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0answers
83 views

Subordination process and Bismut's proof of asymptotic formula for the heat kernel

J. Bismut proved the asymptotic formula for the heat kernel of the Laplace-Beltrami operator $\Delta$ on a manifold $M$ in one of his well-known books. Later, in his paper on the index theorem, ...
6
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1answer
564 views

Brownian motion and its maximum and its minimum

Let $W_u, 0\leq u \leq t$ be Brownian motion. Let $m_t= min_{0\leq u\leq t} W_u$ and $M_t = max_{0 \leq u \leq t} W_u$. The fact that $(M_t , W_t)$ is absolutely continuous with respect to Lebesgue ...
4
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1answer
193 views

Uniqueness of a SDE with positivity constraint

We start by fixing some notation. If $x\in\Bbb R^N$, we denote the usual euclidean norm in $\Bbb R^N$ with $\|x\|$: we omit the reference to the space $\Bbb R^N$ or to the dimension $N$ since it ...
2
votes
1answer
149 views

Brownian motion and random walk

Let $M_{\Gamma}$ a Riemannian covering of a closed compact manifold $(M,g)$ with deck transformation $\Gamma$ (its neutral element will be denoted by $e$). If we denote by $p_t^{\Gamma}(x,y)$ the heat ...
1
vote
1answer
89 views

Literature on the total variation of fractal graphs/fractal Brownian motion?

I know that for standard Brownian motion, the total variation sampled at intervals of length $\Delta$ converges to $V(\Delta) = C \Delta^{-1/2}$ for some constant $C$. I wish to use this fact to study ...
1
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1answer
138 views

interpretation of the transition probability of a brownian motion in terms of the Wiener measure

Let $W(t)$ be a standard brownian motion in $E \triangleq \mathbb{R}^d$. The transition probability from a state $x \in E$ at time $t$ to a state $y \in E$ at time $T$ is $$ p(x,t;y,T) = \frac{1}{\...
2
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1answer
312 views

Exercise on a hitting time for a Brownian Motion

I'm following Chapter 3 of "Brownian Motion", by Peres and Mörters, about The Dirichlet Problem(DP). As it is known, in order to obtain existence and uniqueness of a solution for DP it is necessary to ...
2
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0answers
173 views

Weighted sum of Standard Brownian Bridges

Suppose, $\{B_j\}_{j=1}^k$ be a sequence of Brownian Bridges. Let us consider, $$X(t)=\sum_{j=1}^m w_j(t)B_j(t),$$ where $w_j$ are positive weight functions. Then what can we say about (...
3
votes
1answer
108 views

Covariation of the stochastic integral and the Wiener process

Let$^1$ $T>0$ $U,H$ be separable $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint operator with finite trace $\operatorname{tr}Q$ $(e^n)_{n\in\mathbb N}$ be an ...
7
votes
1answer
173 views

A Converse of the Skorokhod Embedding Theorem

I am wondering whether the following "sort of converse" of Skorokhod's embedding theorem holds: Suppose that $\{D_t\}_{t \geq 0}$ is a stochastic process with continuous paths, $D_0 = 0$, and suppose ...
6
votes
1answer
324 views

Moment bounds on exponential martingale

Consider the exponential martingale used in the Girsanov transformation of measure: $$Z(t) = \exp\Big(\int_0^tXdW - \frac{1}{2}\int_0^t|X|^2ds\Big)$$ so that $Z$ solves the sde $dZ = ZXdW$ where $W$ ...
1
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0answers
89 views

forward Ito integral

Forward integral is introduced by Francesco RussoPierre Vallois as a generalization of Ito integral. For simplicity, let $B$ be a standard Brownian motion and let $\phi$ be a measurable process. The ...
2
votes
1answer
190 views

“Brownian motion” without assuming continuity of path at origin of state space

This question is inspired partly by this question Any reference on Brownian Motion continuity. In this post, the author asked if the following three axioms can define a Brownian motion without ...