The brownian-motion tag has no usage guidance.

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### How to calculate the PSD of a stochastic process

This question was asked on math.stackexchange about 2 months ago, but it hasn't been very successful in attracting answers yet, so I'm posting it here.
Say we have a stochastic process described by a ...

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33 views

### Normality criterion based on Brownian motion

Consider analytic family $\mathcal{F}$ btw domains $U,V\subset \mathbb{C}$. For any $f\in \mathcal{F}$ we have time-changed Brownian motion $f(B_{t})=\widetilde{B}_{\int_{0}^{t}|f(B_{s})|^{2}ds}$. So ...

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49 views

### BM hitting times with exponential killing process

Assume a BM in 3d domain (infinite) with a small absorbing subdomain (cube, sphere, ect), centered at point $p_s=(x_s,y_s,z_s)$ . BM starts at point $p_0=(x_0,y_0,z_0)$ and when it riches the ...

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134 views

### Question on Wiener processes not hitting 0

Let $W_t$ be a standard Wiener process, and $0\leq a < b$. Let $\hat{W}_t:=W_{a+t}-W_a$. Then $\hat{W}_t$ is also a standard Wiener process. I think that the following should be true:
$$\mathbb ...

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30 views

### Distribution of stopped Brownian motion in $\mathbb R^2$

Let $B=(B^1_t,B^2_t)_{t\ge 0}$ be a standard Brownian motion in $\mathbb R^2$. Let $U=(U^1,U^2)$ be an independent random variable taking values in a circle $C_1\subset\mathbb R^2$ with uniform ...

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117 views

### Proof of Feynman Kac formula

I am trying to write a complete proof of the Feynman Kac formula in the multi-dimensional case. My starting point was the proof of the univariate form on wikipedia, at ...

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63 views

### Brownian motion increments

We know that if $W_t$ is a Brownian motion, $W_{t+t_0}-W_{t_0}$ is one too.
Does the "converse" holds : Let $t_0$ be a positive number. I have a Brownian motion $W_t$
and I seek another Brownian ...

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101 views

### Weighted global Holder property for Brownian motion paths

It is well-known that the Brownian motion (Wiener process) is almost sure locally $\alpha$-Holder for any $\alpha<1/2$. That is, with probability 1
$$
...

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134 views

### Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...

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73 views

### Brownian motion - probability of hitting an open subset of the sphere

Consider a Brownian particle in $\mathbb{R}^n$, starting at the origin. Let $\mathbb{P}_t(A)$ be the probability of the particle striking $A \subset S^{n - 1}$ within time $t$, where $A = \{ (x_1, ...

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171 views

### Brownian motion in $\mathbb{R}^n$, probability of hitting a set

Consider a particle undergoing Brownian motion in $\mathbb{R}^n$, starting at the origin, and let $B(t)$ denote its position at time $t$. Let $X$ be an arbitrary subset of $\mathbb{R}^n$. I am trying ...

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135 views

### Total absolute variation of brownian motion, with different sampling rates

Let $(B_t)$ be a brownian motion on [0,1]. For the following, let $\omega$ be fixed.
Let's compute the total absolute variation when sampling period = $\delta$ is fixed:
$$V(\delta) = ...

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102 views

### Expected value of product of Ito integrals

Assume that we have a process $F(t,T)$ that fulfills the following SDE.
$$
dF(t,T) = \sigma(t,T)F(t,T)dW(t)
$$
where $t$ is the running time and $T>t$ is called the delivery-time. $\sigma(t,T)$ is ...

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82 views

### Expectation, exponential of an additive functional of Brownian motion

I have a question about an additive functional of Brownian motion.
Let $d \in \mathbb{N}$. Let $b:\mathbb{R}^{d}\to \mathbb{R}$ be a measurable function and $(X_{t})_{t \in [0,\infty[}$ be a ...

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336 views

### Blumenthal and Kolmogorov 0-1 law

Blumenthal's 0-1 law see theorem 5.8/5.9 tells us that an event in the germ $\sigma-$ algebra has either probability zero or one with respect to a measure induced by a Brownian motion starting in some ...

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228 views

### Brownian motion in $n$ dimensions

Consider a particle starting at the origin in $\mathbb{R}^n$ and undergoing Brownian motion. Is there an expression known for the probability of the particle hitting the sphere $S^{n - 1}_r = \{x \in ...

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281 views

### Arcsine law for Brownian motion with drift

Let
$$X_t = m \cdot t + W_t$$
where $W_t$ is a Brownian motion. Let
$$Z = \sup \{ t\in [0,1] : X_t = 0\}.$$
It is known that if $m = 0$ then the distribution of $z$ is given by
$$\mathbb{P}[Z \leq y ...

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225 views

### Slight variation on law of the iterated logarithm

Let$$M_t = \max\{B_s : 0 \le s \le t\},\text{ }m_t = \min\{B_s : 0 \le s \le t\},$$where $B_t$ is a standard Brownian motion. My question is, does there exist $r$ such that with probability ...

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296 views

### Extension of Dynkin's formula, conclude that process is a martingale

This question was asked here, but it did not get enough attention, so I'm crossposting it to MO.
Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial ...

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78 views

### Concluding that the Poisson kernel is indeed the Cauchy distribution?

See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, ...

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339 views

### Between arithmetic and geometric Brownian motions: when are negative values possible?

Please note edits after original post changing the specific form of the setup
Let's say we have a stochastic differential equation:
$$
\mathrm{d}S_t = |S^\beta| {(\mu \mathrm{d}t + ...

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75 views

### Poisson kernel, follow-up question, follows that process $\left\{e^{i\theta X_t - \theta Y_t}\right\}$ is a martingale? [closed]

See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. For any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, ...

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138 views

### Poisson kernel, expectation, an absolute value comes in

See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, ...

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119 views

### Poisson kernel, $E^{(x, y)}\text{exp}\{i\theta X_t - \theta Y_t\} = e^{i\theta x - \theta y}$

Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. How do I see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, ...

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277 views

### In the plane, does complement of Brownian path have infinitely many connected components?

Let $d = 2$. Do we have that with $P_x$—probability $1$, for every $T> 0$ the complement $W[0, T]^c$ of the Brownian path up to time $T$ has infinitely many connected components?
I had seen this ...

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### Is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane almost surely?

Let $d = 2$. With probability $1$, is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane?

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128 views

### Poisson kernel is the Cauchy distribution, reference?

Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?

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145 views

### Brownian motion, “increase interval”, exists constants, bound,

Let $B_t$ be a standard Brownian motion. Let $J(j, n) = [j/n, (j+1)/n]$. We will call $J(j, n)$ an increase interval if$$B_s \le B_t,\text{ }0 \le s \le {j\over{n}},\text{ }{{j+1}\over{n}} \le t \le ...

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### Brownian motion, crossing intervals, possible usage of second moment method?

This is a followup to my question here.
Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le ...

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424 views

### Number of intervals needed to cross, Brownian motion

Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$and let$$K_n = \sum_{j = 2^n + 1}^{2^{2n}} ...

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122 views

### For which $r > 0$ is it the case with probability one, for all $n$ sufficiently large $M_n \le r\sqrt{\log n}$?

Let $B_t$ be a standard Brownian motion. Let$$M_n = \max\{|B_t - B_{n-1}| : n - 1 \le t \le n\}.$$For which $r > 0$ is it the case with probability one, for all $n$ sufficiently large$$M_n \le ...

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301 views

### Brownian motion, quadratic variation, existence of partitions?

Let $B_t$ be a standard Brownian motion. Does there with probability one exist a sequence of partitions $\{t_{k, n} : k = 0, 1, \dots, k_n\}$ $$0 = t_{0, n} < t_{1, n} < \dots < t_{k_n, n} = ...

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141 views

### Standard Brownian motion, Hölder continuous with exponent $\gamma$ for any $\gamma < 1/2$, not for any $\gamma \ge 1/2$

In some results on Hölder continuity with regards to standard Brownian motion, the following is asserted without proof.
It is not hard to see that for every $k < \infty$, and every $\epsilon ...

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145 views

### Large deviation for Brownian path on $[0,\infty)$

It seems strange to me that all we can find about Schilder's theorem in the literature is on a finite interval of Brownian path.
If we equip the space of continuous function starting from $0$, ...

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47 views

### $X_t = B_t^q$, $X_t = (\sin B_t)^q$, $X_t = B_t^q (\sin B_t)^r$, $dM_t = R_t\,M_t\,dB_t$ [closed]

What are the SDE's satisfied by the following processes?
$X_t = B_t^q$
$X_t = (\sin B_t)^q$
$X_t = B_t^q (\sin B_t)^r$
Assume $B_t$ is a standard Brownian motion with $B_0 > 0$ and the ...

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### Distribution of Brownian local time at first hitting times of $1$ and of $\pm1$? [closed]

Here, $(B_t)$ is a standard Brownian motion, and $(L_t)$ its local time at $0$. Consider $$T=\inf\{t : B_t = 1\},\qquad\tau =\inf\{t : |B_t| = 1\}.$$
What is the distribution of $L_T$?
What is the ...

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527 views

### Probability that planar Brownian motion doesn't “encircle” 0

Suppose $B_t$ is a standard Brownian motion in $\mathbb{R}^2$ and $T = \text{inf}\{t : |B_t| = 1\}$. Let $E$ denote the event that $0$ is contained in the unbounded component of $\mathbb{R}^2 ...

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### $M_t = f(B_{t \wedge \tau}) + (t \wedge \tau)$ local martingale, $\textbf{E}^x[\tau] = f(x)?$

Suppose $D \subset \mathbb{R}^d$ is a domain and $f: \overline{D} \to \mathbb{R}$ is a continuous function, $C^2$ in $D$, satisfying$$f(x) = 0\text{ for }x\in \partial D,$$$${1\over2} \Delta f(x) = -1 ...

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306 views

### Standard Brownian motion, limit, square of expectation bound

Let $J_t$ be a standard Brownian motion, let $X = \{t : J_t = 0\}$ denote the zero set, and let $I(j, n)$ denote the indicator function of the event$$\left\{\text{there exists }s \in ...

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### Law of motion when initial condition is perturbed

We know how to find the law of motion (Ito process) of the value function:
$$V_t(x)=E\Big{[}\int^{T}_te^{-r (s-t)}f(s,X_s)ds+e^{-r (T-t)}g(T, X_{T})|\mathcal{F}_t\Big{]}$$
such that
...

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84 views

### Continuity of expected payoff from a diffusion

Fix a discount rate $r>0$, and let $m,v,f:\mathbb{R} \rightarrow \mathbb{R}$ be bounded measurable functions of locally bounded variation, with $v$ globally bounded below by some strictly positive ...

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396 views

### Time Change of a Brownian motion

We know that for if $X$ is a stochastic integral of the form below -
$X_t = \int_0^t v(s,\omega) db(s,\omega)$.
then we can use time change formula to claim that
$X_t = W_{\alpha(t)}$ where $W$ is ...

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161 views

### Brownian bridge on a Lie group as a stochastic differential equation

Brownian motion $g_t$ on a compact Lie group satisfies the stochastic differential equation
$$dg_t = dB_t \circ g_t$$
where $B_t$ is Brownian motion on the Lie algebra and $\circ$ denotes ...

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403 views

### $\langle X\rangle_t = t$

Suppose $B_t$ is a standard Brownian motion in $\mathbb{R}^d$ and $X_t = |B_t|$. What is the easiest way to see that$$\langle X\rangle_t = t?$$I need this result for a simulation I am running...

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181 views

### Brownian motion, exists $c < \infty$?

Suppose $B_t$ is a standard Brownian motion. Does there exist $c < \infty$ such that with probability one$$\limsup_{t \to \infty} {{B_t}\over{\sqrt{t \log t}}} \le c?$$I need to know whether or not ...

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230 views

### A question about brownian motions

I would like to ask a question about Brownian motion:
Let $B$ be a standard brownian motion. How to show that $\mathbb P( \max\limits_{0 \leq s \leq t} B(s) \in (a,b) )$ decreases exponentially in t ...

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131 views

### Invariant and periodic measures of the random dynamical system on the circle generated by $d\theta_t=dW_t$

Here, I am considering one of the simplest random dynamical systems that one can consider, and yet I realise that I do not know the answer to one of the most basic questions that one can ask about it!
...

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124 views

### Scaling of First-passage times for Random Walk on integer lattices

Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots,
S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin.
Let $\tau_{N}$ be the first time $S_{n}$ exits ...

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135 views

### Reflected Brownian Motion with random barrier?

I am looking for a way to say something about
$$P\left(\max_{t\in[0,n]} W_t+|W_m|> x\right),$$
for $n>m$, where $W$ is a brownian motion with drift.

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171 views

### Joint law of a standard Brownian motion and its local time at a nonzero level

Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is
$$
P\left(B_t\in d y, L_t^0\in d v\right) = ...