Questions tagged [brownian-motion]
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403
questions
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Lower-bound on zero-crossing probability of the nonstationary gaussian process $X(t) = tU+(1-t^2)^{1/2}V$, with $(U,V) \sim N(0,I_2)$
Let $(X(t))_{t \in [-1,1]}$ be a centered non-stationary smooth gaussian process with covariation function $\rho(t,s) = \mathbb E[X(t)X(s)]$. For $t_0 \in (-1,1)$ and $\epsilon \in (-1-t_0,1-t_0)$, ...
2
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1
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196
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Comparison of probabilities that drifted Brownian motion never hits barriers
Let $k , h: \mathbb R_+\to [0,1]$ be non-decreasing and right continuous s.t. $k(t)\le h(t)$ for all $t\ge 0$. Define $\tau_{k}$ (resp. $\tau_h$) by
$$\tau_k : = \inf\{t\ge 0:2+\beta t+ W_t \le k(t)\}\...
2
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40
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Continuity of translation operator in fractional white noise analysis
Fix $H\in(\frac{1}{2},1)$, and let $\Omega:=C_0([0,T],\mathbb R^d)$ be the space of $\mathbb R^d$-valued continuous functions. There is a probability measure $P^H$ on $(\Omega,\mathcal B(\Omega))$, ...
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209
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How to prove excursion process is a Poisson point process?
This question comes from book Ju-Yi Yen and Marc Yor P59 and P60,
On page 59, "Define $\mathcal{Z}_\omega=\{t:B_t(\omega)=0\},$ and $\tau_l$ is the inverse local time. The complement of $\mathcal{...
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111
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Estimate of cumulative probability of geometric Brownian motion
Let $B_\tau$ be the standard BM, $t$ be the initial time, $s$ be the time variable, $r$ and $\theta$ are positive constants. We also assume that $x$ is the initial position of the below geometric ...
4
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1
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554
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Random time change from Oksendal's SDE textbook
I have two questions related to the random time change introduced in Oksendal's textbook on SDEs (page 155-156). Specifically, for Lemma 8.5.6., I have no clue as to why we should define $t_j$ in ...
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1
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177
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How to prove the coupling version of the Donsker's Invariance Principle?
Donsker's invariance principle:
Let $X_1,X_2,...$ be i.i.d. real-valued random variables with mean 0 and variance 1. We define $S_0=0$ and $S_n= X_1+ ... + X_n$ for $n \geq 1$. To get a process in ...
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132
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Harmonic measure of a punctured disc
Let $D$ be a disc in $\mathbb{C}\cong\mathbb{R}^2 $ and $z_0$ a fixed point of $D$. Is the harmonic measure for $V=D\setminus\{z_0\}$ known? Any reference would also be welcome.
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Reference request (Brownian local time): for fixed $t$, $a\mapsto L_a(t)$ is a.s. continuous and with compact support
So the title is quite self explanatory.
In the book "Continuous Martingales and Brownian Motion" by Rebuz and Yor, in the proof of Proposition $(2.1)$ of chapter XIII it's stated that:
For ...
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184
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The quadratic variation of $\int_0^t\int_T^Sg(s,x) \, dW_s^x \, dx$
Consider the process $W^x_t$ which is a Brownian motion for every $x\geq 0$ such that
$$d\langle W_t^x,W_t^y\rangle=Q(x,y)\,dt$$
where $Q$ is some non-negative definite function. Now consider the ...
2
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65
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A polar open set in a topological subspace?
Suppose $U$ is a bounded open set in $\mathbb{R}^m$ with ($m\geq2$). Is it possible to have a non-empty set $E$ in the boundary $ \partial U$ of $U$ that is open in $ \partial U$ and is polar?
A set $...
5
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502
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Largeness of the set of zeroes of a Brownian motion
Definitions:
A measurable subset $S$ of $\mathbb R$ is said to be mesoscopic if there exists a continuous function $f: \mathbb R \to \mathbb R$ such that $f(S)$ is Lebesgue measurable and has nonzero ...
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204
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The long run average amount of time the deviation of Brownian motion spends above its expected value
Let $B_t$ be a standard one dimensional Brownian motion. Is it true that
$$\lim_{s \to \infty} \frac{\int_{[0, s]} \mathbf 1_{ \{|B_t| \geq \sqrt{2t/\pi} \} } \ dt}{s}$$
exists almost surely?
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75
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Why the distribution of M(t) is the same as X(t)?
Let $ B(t)(t\geq 0) $ be the standard Brownian motion and $ M(t)=\max_{0\leq s\leq t}{B(s)} $. If we define $ X(t)=M(t)-B(t) $ as a new stochastic process, how can I show that $ X(t) $ has the same ...
1
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118
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L2-closure of absolutely continuous stochastic processes?
Assume we have a possibly multidimensional Brownian motion on a probability space $(\Omega,\mathcal F,\mathbb P)$ where $(\mathcal F_t)_{t\in[0;T]}$ is the Brownian standard filtration. Let $\Vert X\...
5
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2
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264
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Bounding Brownian motion and an Ito process simultaneously
Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm :...
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167
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Probability that a $d$-dimensional Brownian bridge is greater than a given parameter
Let $(W_t)_{t\in[0,T]}$ be a Brownian bridge such that $W_0=a$ and $W_T=b$, the probability that $\forall t\in[0,T],W_t\geqslant x$ given the parameter $x\leqslant\min(a,b)$ is well known :
$$ \mathbb{...
2
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0
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79
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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 ...
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64
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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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1
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413
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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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146
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Laplace Equation for Brownian Motion [closed]
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
$$
u(x) = \...
1
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1
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50
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Bound moments wrt. known initial and final moments
Let $X$ be an $L^p$ random variable, where $p\in (0,1)$ and $W_t$ usual Brownian motion (with $W_t$ independent from $X$). I'd like to bound
$$\mathbb E|X+W_t|^p$$
purely in terms of $\mathbb E|X|^p$ ...
1
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0
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78
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Superharmonicity of the distance function
Suppose $V$ is a convex open proper subset of $\mathbb{R}^m$ ($m\geq2$). It is known that the function $u(x)=$dist$(x,\partial V)$ is superharmonic on $V$. Is there a similar result without $V$ being ...
0
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155
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Probability to cross an envelopp for 1D random walk?
Imagine we have an evolving sequence composed of 1 and -1 (ex: -1-11-111...) where the probability to get -1 or 1 is 1/2. n is the lengh of my sequence.
I can make an analogy with random walk: let ...
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2
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166
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A question on minimum principle
Suppose $D$ be an unbounded domain of $\mathbb{R}^m$ for $m\geq3$, and $u$ is superharmonic on $D$. We know that if $\liminf_{x\to y}u(x)\geq0$ for all $y$ in $\partial^\infty D$ (the boundary of $D$ ...
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An open set whose complement is non-thin at infinity
Let $x^*$ designate the inverse of a point $x\in\mathbb{R}^m$ under the Kelvin transformation with respect to the circle of center 0 and radius 1. Recall that
$$x^*=|x|^{-2}x.$$
For a set $E$, we set $...
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0
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68
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Differentiable approximation of Brownian diffusion with unbounded volatility
Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be ...
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136
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Differentiable approximation of Brownian diffusion with bounded volatility
Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be ...
4
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155
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Occupation time of SDE
Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation
$$
X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...
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0
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217
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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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1
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150
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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
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1
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244
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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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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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1
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146
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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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1
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166
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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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2
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409
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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
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592
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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 ...
1
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0
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60
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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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2
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1k
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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 ...
5
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240
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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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187
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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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0
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99
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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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78
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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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1
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134
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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
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0
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196
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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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2
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2k
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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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1
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274
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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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0
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75
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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
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1
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207
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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 ...
1
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0
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76
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"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 ...