The brownian-motion tag has no usage guidance.

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### 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 ...

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65 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 ...

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78 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 ...

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**1**answer

335 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 ...

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**3**answers

93 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 $\alpha<1/2$.
...

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**1**answer

222 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 ...

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49 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 ...

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54 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) \...

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**1**answer

74 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 ...

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108 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 ...

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84 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" ...

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79 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^...

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**1**answer

101 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-...

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169 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 $[0,1]$ ...

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67 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 ...

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99 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 ...

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**1**answer

141 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 ...

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52 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 ...

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**1**answer

103 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 ...

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84 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. ...

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329 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 ...

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81 views

### reflected Wiener process and Doob's h-transform

Is there a way to see a real valued Wiener process $W^R$ starting from $x \geq 0$ and reflected in 0 as the Doob's transform of a standard real valued Wiener process $W$?
Recall that
$$
W_t^R = x + ...

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94 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 $...

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274 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 $T=\inf \{t>0,|W|(t)= 1\}$. ...

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299 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 ...

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118 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,...

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82 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, ...

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488 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 ...

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190 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 ...

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143 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 ...

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86 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 ...

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123 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}{\...

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279 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 ...

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150 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 (...

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**1**answer

100 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 ...

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**1**answer

169 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 ...

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**1**answer

283 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 $W$ ...

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83 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 ...

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182 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 ...

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128 views

### Azema's martingale and quadratic covariation

Given a filtered space $(\Omega,\mathcal F,\mathbb F,\mathbb P)$ supporting a Brownian Motion $B$, where the filtration $\mathcal F$ is the augmented Brownian filtration, the Azema's martingale is ...

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49 views

### Matching Numbers in Ito McKean

Matching numbers are the basics Ito and McKean use to build out a bunch of stuff, like singular points and shunts. The four maching numbers $e_1, e_2, e_3, e_4$ are defined as
$e_1 = \lim_{b \...

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### constructing brownian motion adapted to a given brownian motion

Suppose we are given a Brownian motion $\{B_t,\mathcal F_t^B\}_{t\in[0,1]}$. I would like to know if it is possible to construct another Brownian motion $\{W_t,\mathcal F_t^W\}_{t\in[0,1]}$ from $B$, ...

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### Neat little proof using local time from Ito McKean

This is a cool little result, the proof of which uses the machinery of local time. On p. 72, Prob 1 asks to show that $\int_0^1 dt/x(t)$ exists, where $x(t)$ is a continuous time brownian motion.
In ...

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**1**answer

210 views

### Any modern/recent version of Ito & McKean?

This's a wonderful book[1] but the latest edition I have is dated 1973. Is there recent book(s)/rewrite(s) that covers the same subjects and elucidate with more explicit arguments and details of their ...

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176 views

### Explicit martingale representation for a Brownian bridge

Let $W$ denote a Wiener process, $\displaystyle M_t = \max_{0 \le s \le t} W_s$ its running maximum. The martingale representation of $M$ is known explicitly:
$$M_T = \sqrt{\frac{2T} \pi} + \int_0^T ...

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136 views

### Deterministic function in the support of Brownian motion

Is there an explicit example of a function in the (topological) support of the law of Brownian motion (with respect to the topology of uniform convergence of continuous functions)?
(You can take "...

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281 views

### Reference for Feynman-Kac

I would like to have a reference with more in deep explanation of Feynman-Kac than in Evan's An Introduction to Stochastic Differential Equations and, if possible, example of solution for equations ...

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**1**answer

74 views

### Equivalence of tail sigma-fields between two triangular arrays of Brownian motions

Let
$$\{B^{(i)}_t : 0 \leq t \leq T, i \geq 1\}$$
be a sequence of i.i.d. standard Brownian motions. For each $n,$ let $V^{(n)}_t, \,0 \leq t \leq T$ be a continuous bounded process adapted to the ...

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167 views

### Basis for $L^2(\mathbb{R})$ that Solves the Heat Equation

This is a less-than-serious question that I asked on math.SE, but I suspect it is slightly more appropriate to ask it here. Consider the heat equation $$
u_t = \frac12 u_{xx}
$$ On $\mathbb{T}$ with ...

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107 views

### Feynman-Kac formula for *general* Sturm-Liouville operator

One way to state (omitting technical requirements) the Feynman-Kac formula that I am familiar with is as follows.
Let $u$ be a solution to the pde
$$u_t(x,t)=-\frac{\sigma^2(x,t)}2u_{xx}(x,t)-V(x,t)u(...