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# Questions tagged [brownian-motion]

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### 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 ...
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### 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,...
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### "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 ...
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### Density of $W_t$ assuming it stayed above a line $L$

Let $W_t$ be a Wiener process with $W_0=0$, and let $L=\{at+by=c\}$ be a line with $c/b<0$ (i.e. the line crosses the $Y$-axis below $0$). Assume that $W_t$ stayed above $L$ up to time $T$. What is ...
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### How does the conditional Wiener measure work?

In the theorem below $P_D$ means the heat kernel in the open $D \subset \mathbb{R}^m$ and $P_m$ is the heat kernel in whole $\mathbb{R}^m.$ I know absolutely nothing about what Brownian bridges are, ...
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Let $(W_t)_{t\ge 0}$ be a standard Brownian motion and $\tau$ be a stopping time lying in $[1,2]$. For $x, y>0$, can we show $$\mathbb E\big[{\bf 1}_{\{x+\inf_{0\le t\le 2}W_t>0\}}(W_{\tau}-y)^+\... 0 votes 0 answers 44 views ### A two-dimensional variant of Bessel stochastic differential equation Let Z be a complex Brownian motion starting at 0. The stochastic integral$$W = \int_0^t \frac{Z_s}{|Z_s|} \mathrm{d}Z_s.$$yields a complex Brownian motion (starting at 0). The natural ... 0 votes 1 answer 94 views ### Does the convergence of f_n imply the convergence of \mathbb P[\inf_{0\le s\le t}(W_s-f_n(s))\le 0]? Let (f_n)_{n\ge 1} be a sequence of non-decreasing and continuous functions defined on \mathbb R_+ and taking values in [0,1]. Further, for each t\ge 0, n\mapsto f_n(t) is non-decreasing. ... 0 votes 1 answer 118 views ### Is this set negligible? Let (W_t)_{t\ge 0} be a standard Brownian motion starting at zero. Let f: [0,1]\to\mathbb R be a function that is righ-continuous with left limits. Set$$A:=\left\{\omega\in\Omega: \inf_{0\le t\le ... 131 views

### Mean of log-normal variable when exponent is replaced by runnung maximum of Ito-integral

Let $W=\{W_t\}_{t\in[0;1]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;1]}$ the filtration generated by $W$, augmented with the nullsets. Let $\{\sigma_t\}_{t\in[0;1]}$ be a continuous and ...
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### Conditional probability distribution of a Brownian particle surviving forever

Consider the drift Brownian motion $X_t:=1+bt+W_t$, where $(W_t)_{t\ge 0}$ is a Brownian motion starting at zero. Set $\tau:=\inf\{t\ge 0: X_t=0\}$. Assume $b>0$, then $\mathbb P[\tau=\infty]>0$....
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### Book: Continuous martingale and Brownian motion

I am reading the book "continuous martingale and Brownian motion" 1995_Revuz. It reads the following proposition 3.2 in Chapter VII. That confused me a lot. Where $T_r, T_l$ is the hitting ...
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### Characterization of Brownian motion: processes with right-continuous paths

I am looking for a reference with a proof for the following fact: If a right-continuous martingale $(X_r)_{ r \geq 0}$ is such that $X_0=0,(X^2_r-r)_r,(X_r^3-3rX_r)_r,(X_r^4-6rX_r^2+3r^2)_r$ are ...
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### Estimates on the density of hitting time for planar Brownian motion

Consider a polygon $\Pi \subset \mathbb{R}^2$, and let $T_{\Pi,x}$ be the (random) time a Brownian motion started at a point $x$ in its interior first crosses $\Pi$. For any such $\Pi$, do there exist ...
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### Endpoint of Brownian motion conditional on high maxima

Note: This question is closely related to an earlier question: A large noise limit. Let $W$ be a standard one dimensional Brownian motion. For every $\varepsilon > 0$, let $A_\varepsilon$ denote ...
The first Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E X^2<\infty$ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau$ is ...