Questions related to Brownian motion, a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W(t)-W(s)$ is normally distributed, with $0$ mean and variance $t-s$.

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1answer
47 views

Second order Cone and Brownian motion

Denote $D = \{ x\in \mathbb{R}^{d+1}: \sqrt{x_1^2+\dots+x_d^2 }\leq x_{d+1}\}$ which is the second order cone ( ice-cream cone) in $\mathbb{R}^{d+1}$. My question is that if a standard brownian motion ...
3
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1answer
73 views

Details about Brownian motion

Unfortunately I could not make it to the last probability theory lecture and now I am reading through the notes and have some troubles understanding what is going on. So we defined Brownian motion as ...
1
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0answers
15 views

Distribution of a Brownian bridge

I am self studying some stochastic calculus material and come across this question to show that the distribution of $P(W(s)\in dy|W(t)=x)$ with $W(0)=0$ is normal with mean $\frac{s}{t}W(t)$ and ...
2
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0answers
20 views

Verify argument about random time being measurable w.r.t. a reversed BM increment process.

Let $B$ be a brownian motion (assume for convenient notation that it is two-sided). Let $T:=\sup\{t<2 :\vert B_t-B_1\vert\geq 1\}.$ Let $Y$ be the process $s\mapsto (B_T-B_{T+s})$. I want to argue ...
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1answer
47 views

Prove that the first hitting time $\tau_x:=\inf\left\{t\ge 0:B_t=x\right\}$ of a Brownian motion is almost surely finite

Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion and $$\tau:=\inf\big\{t\ge 0:B_t\in\left\{a,b\right\}\big\}$$ for some $a<0<b$. I want to prove, that $\tau$ is almost surely finite. Let ...
8
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1answer
177 views

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

Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial derivatives are uniformly bounded (or, more generally, have at most polynomial growth as $|x| \to ...
2
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0answers
17 views

$2$ d Brownian motion move from one point to another

Suppose $W_t= (X_t,Y_t)$ is a $2$d standard Brownian motion starting at $(-1,0)$. How do I show that there is a positive probability that $W_t$ moves from $(-1,0)$, to a neighborhood of $(1,0)$, say ...
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0answers
39 views

Brownian motion determining probability of success for binomial experiment

Suppose you have some particle that moves according to brownian motion in one dimension (x), in discrete time steps. Suppose you have, for each position x, a probability P(x), which specifies the ...
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0answers
30 views

Lebesgue Thorn and Brownian motion

Let $S$ be the unit sphere in $\mathbb{R}^3$ and $\Theta=\{(x,y,z), x\geq 0, z^2+y^2 \leq \frac{1}{10}e^{-1/x^2}\}$. Try to show that $\exists \delta>0,\forall x\in (-1,0)$,$$ P^x(W_t \text{ hit ...
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0answers
29 views

Correspondence between multi-dimensional Brownian motion and harmonic functions

Let $U\subseteq\mathbb R^d$ be a bounded domain. A continuous function $u:\overline U\to\mathbb R$ is called harmonic $:\Leftrightarrow$ $$u(x)=\frac 1{|\partial B_r(x)|}\int_{\partial B_r(x)} ...
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1answer
54 views

Proof that the stopping time for a Brownian Motion is finite for given target levels

Given a standard brownian motion $W_t$ and defining $\tau$ as: $\tau :=\inf\{t\geq0:W_t=1$ or $W_t=-2\}$ The proof below shows that the stopping time is finite: $$\begin{align*} P(\tau < t) ...
5
votes
0answers
32 views

Almost surely, for all $s \ge 0$, there exist $t$, $u \ge s$ with $B_t < 0 < B_u$?

Let $(B_t)_{t \ge 0}$ be a Brownian motion starting from $0$. Then, do we have that, almost surely, for all $s \ge 0$, there exist $t$, $u \ge s$ with $B_t < 0 < B_u$?
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0answers
25 views

Covariance of two stochastic integrals

Consider the stochastic integral $\int_{0}^{1}J(r)M(r,\lambda) dr$ where $J(r)$ is a demeaned Ornstein-Uhlenbeck process and $M(r,\lambda)=W(r,\lambda)-\lambda W(r,1)$ a Brownian Sheet, independent of ...
0
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0answers
26 views

Brownian motion (conditional density)

Suppose $(B_t)_{t\geq0}$ is a Brownian motion. (1) For $0\leq s<t$, give the conditional law: $B_{(t+s)/2}|B_s=x, B_t=y$. (Hint: start looking for coefficients $a,b\in R$ ...
1
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0answers
35 views

brownian motion conditioned on past and future observation

For a brownian motion, $W_t$ with scale $\sigma^2$ and for $t<s<u$, I want to find: $$ E(W_s|W_t=a,W_u=b) $$ I have the solution, but I am confused about two of the following statements: First ...
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0answers
17 views

Arithmetic Brownian Motion

Let X(t), t ≥ 0 be a Brownian motion process with drift parameter µ = 5 and variance parameter σ2 = 16. If X(0) = 5, find P(X(3) > 15). This question was giving me fits So I set it up like P ...
2
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0answers
29 views

Brownian motion : disjoint intervals with the same maximum

In this article, at the end of page 11, there is a proof that for a Brownian motion, almost surely, there exist disjoint intervals with the same maximum. The proof uses the random variables ...
1
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1answer
16 views

AR(1) process: Finding the distribution of the prediction error

I have the AR(1) process given by $$X_t = \rho X_{t-1} + \varepsilon_t, \ \ \ t=1,\ldots,T+h,$$ where the random variables $\varepsilon_t$ are independent, $\varepsilon_t \sim N(0,\sigma^2)$, and ...
0
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0answers
52 views

Product of two Ornstein Uhlenbeck processes : conditional distribution

Let $X(t)$ and $Y(t)$ be two independent OU processes (each with some fixed correlation time-scale), and let $S(t) = X(t)Y(t)$. Then is there a expression for the conditional distribution of $X(t)$ ...
1
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1answer
54 views

Proof of a stochastic process being a martingale

I have two questions on a rather basic topic on martingales and brownian motion. Now I can't seem to find a proof of this on the internet nor textbook, since they seem to skip this part of the ...
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0answers
11 views

Arithmetic Brownian Motion, Z score

Drift Parameter = 5 Variance Parameter = 16 X(0) = 5 Find P(X(3) > 15) The answer I came up with here is .0744 The first step being - P(N(20,16.3) > 10) The last step being- 1 - P(N(0,1) < 10 ...
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0answers
14 views

SLE theory, showing that $T_u$ is a.s. finite

I am currently reading this article, page 7 (enumerated 889). Let $g_t(z)$ be the solution for $t\in\mathbb{R}$ for $$\partial_tg_t(z)=\frac{2}{ g_t(z)-\xi(t)},\quad g_0(z)=z.$$ Where $\xi$ is a ...
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1answer
63 views

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

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 ...
2
votes
1answer
164 views

The expected value of stop-time for Brownian motion $\tau=\min_t\{B_t^2\geq t+1\}$.

Let $B_t,\;t\geq0$ be a standard Brownian motion. Define the stopping time $$\tau = \min_t\{B_t^2\geq t+1\}$$ Is the expected value $E(\tau)$ finite? Actually, my raw problem as following: $$\gamma ...
0
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0answers
29 views

How to prove that the normal pdf function satisfies the forward Kolmogorov equation

How to prove that the normal pdf function $ P(x,t)=\frac{1}{\sqrt{2\pi\sigma^2t}}e^{-\frac{(x-\mu t)^2}{2\sigma^2t}}$ satisfies the forward Kolmogorov equation $ ...
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0answers
26 views

Geometric Ornstein-Uhlenbeck process

Let the Geometric Ornstein-Uhlenbeck process be defined as: $dV_t=\theta(\mu-V_t)V_tdt+σV_tdW_t$ Does anyone know of a solution or a reference for where a solution may be found? Thanks!
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0answers
10 views

$ \sup_{t \geq 0}\big( |B_t|-t^{r/2}\big) \text{ and } \sup_{t \geq 0}\big( \frac{|B_t|}{1+t^{r/2}}\big)^\rho$ have the same law?

Let $r>1$ and let $\rho$ be its Holder's conjugate and I need to show that the following two random variables have the same law $$ \sup_{t \geq 0}\big( |B_t|-t^{r/2}\big) \text{ and } \sup_{t ...
2
votes
1answer
27 views

How can I show that $\sup_{t \geq 0}(B^*_t-\mu t^{r/2})=_{law} \sup_{v \geq 0}(\lambda B^*_{v}-\mu \lambda^rv^{r/2})$

I am trying to show that $\sup_{t \geq 0}(B^*_t-\mu t^{r/2})=_{law} \sup_{v \geq 0}(\lambda B^*_{v}-\mu \lambda^rv^{r/2})$ i.e the above two random variables have the same distribution where ...
4
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1answer
22 views

Why Brownian motion is a stationary process

I found out that I simply can't rigorously prove that Wiener process is a stationary process, i.e. its finite-dimensional distributions don't change under shift in time. Let $W_t$ be a Wiener process: ...
0
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0answers
26 views

continuity of brownian motion

Let $s \in ]0,1[$,$\epsilon>0$ with $(1+\epsilon)^2(1-s)>1+s$; $h(t)=\sqrt {2t\log(1/t)}$. Define $$K_n=\{(i,j) \in N^2:0 \leq i<j<2^n\text{ and }j-i \leq 2^{ns} \}\\ ...
0
votes
1answer
35 views

For a discrete Markov process $X$, the probability that $X$ started in $x$ returns to $x$ is always positive. So, there are no absorbing states?!

Let $E$ be an at most countable set equipped with the discrete topology and $\mathcal E=2^E$ $X=(X_t)_{t\ge 0}$ be a discrete Markov process with values in $(E,\mathcal E)$ and distributions ...
4
votes
1answer
49 views

Brownian motion, quadratic variation, existence of partitions?

Let $A_t$ be a standard Brownian motion. Where can I find a reference to/can anybody supply a proof of the fact that with probability $1$ there exists a sequence of partitions $\{t_{k, n} : k = 1, ...
3
votes
1answer
53 views

Quadratic Variation of Brownian motion indexed by cadlag increasing function

Let $(B_t)_{t \geq 0}$ a Brownian motion on $\mathbb R^n$ and $\ell_t \in \mathbb S$, where $\mathbb S$ is the space of all increasing càdlàg function from $(0,\infty)$ to $(0,\infty)$ with ...
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0answers
20 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 $0$, ...
3
votes
1answer
71 views

Representation theorem for local martingales

I want to prove the following local martingale representation theorem. For the statement of the theorems to come we fix a filtered probability space $(\Omega,\mathcal{A},\mathcal{F},\mathbb{P})$ where ...
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0answers
45 views

Limit of the solution to a SDE

I am trying to learn about SDEs and I don't seem to understand this claim: Assume $X_t$ is the strong solution to the following SDE: \begin{equation} dX_t=c\tanh X_t \, dt +dB_t \end{equation} Where ...
1
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1answer
27 views

waht does $dW^1dW^2=\rho dt$ exactly mean

I have a very simple question about a notation: $$dW^1dW^2 = \rho dt\tag{1}$$ where $W^1, W^2$ are BM. Usually this is translated to "correlated Brownian motion". So my first question: 1. Is $(1)$ ...
3
votes
2answers
107 views

Add a compensator to submartingale to create a martingale.

Let $W_t$ be a Brownian motion. By Jensen's inequality, $W_t^2$ is a submartingale. I was wondering if it were possible to add another process to $W_t^2$, say $X_t$, such that $W_t^2 + X_t$ is a ...
1
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1answer
58 views

Wiener measure on continuous paths

Let $\Omega$ be the space of continuous paths from $\mathbb{R}$ to $\mathbb{R}^n$. By a famous result, it is known that $\Omega$ is a measure space if we equip it with the "Wiener measure" (see this ...
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1answer
48 views

Wiener Process definition - Continuous paths?

A stochastic process $\{W(t): t \in T\}$ is a Wiener process if it satisfies the following properties: 1) $W(0)=0$ with probability $1$ 2) It has stationary and independent increments. 3) For every ...
3
votes
1answer
43 views

Probability associated with Brownian motion [closed]

I would like a nod in the right direction with the following problem. Let $0<t_1<t_2<\cdots<t_n<\infty$, and $\{a_i\}_1^n$ be real numbers, find a function $f(x_1,\ldots,x_n)$ ...
1
vote
1answer
68 views

How do we solve the laplace transform of the Heat Kernel?

I am interested in the value of $$\int_0^\infty e^{-\alpha t}\frac{e^{-\frac{|x-y|^2}{2t}}}{\sqrt{2\pi t}}\, dt $$ this is the laplace transform of the Heat kernel (changing the time variable) This ...
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0answers
31 views

Linear combination of two Brownian bridges

(I am trying to understand the proof of Theorem $3.1$ in this paper.) Denote by $B^{(x)}(t)$ a Brownian bridge process which starts at $0$ at time $0$ and ends at $x$ at time $t$, i.e. ...
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0answers
45 views

A martingale associated with a solution of a stochastic differential equation

Assume $B_t$ is a standard real brownian motion defined on $\{\Omega,\mathcal{F},P\}$ and $X_t(y)$ satisfies the following SDE: \begin{equation} dX_t=-\cot X_tdt+dB_t \text{ with }X_0=y ...
3
votes
1answer
48 views

Exist $\alpha < \infty$, $\beta > 0$ such that $\mathbb{P}\{T_\lambda > t\} \le \alpha e^{-\beta t}?$

Let $B_t$ be a standard one-dimensional Brownian motion. Suppose $\lambda > 0$ and let$$T_\lambda = \min\{t : |B_t| = \lambda\}.$$Do there exist $\alpha < \infty$ and $\beta > 0$ (which may ...
2
votes
1answer
104 views

$\langle X\rangle_t = t?$

Suppose $B_t$ is a standard Brownian motion in $\mathbb{R}^d$ and $X_t = |B_t|$. Do we have that$$\langle X\rangle_t = t?$$
2
votes
1answer
40 views

$L_t$ Hölder continuous of order $\alpha = 1/2$?

Let $B_t$ be a standard Brownian motion, $L(x, t)$ denote the local time at $x$ at time $t$, and $L_t = L(0, t)$. I know that with probability one $L_t$, $0 \le t \le 1$ is Hölder continuous of order ...
5
votes
1answer
171 views

$\limsup_{t \to 0} {L_t}/\sqrt{t} = \infty$ with probability one?

Let $B_t$ be a standard Brownian motion, $L(x, t)$ be the local time $x$ at time $t$, and $L_t = L(0, t)$. Do we have$$\limsup_{t \to 0} {{L_t} \over{\sqrt{t}}} = \infty$$with probability one?
2
votes
1answer
53 views

$\mathbb{P}\{B_2 > 0 \text{ }|\text{ } B_1 > 0\}$ [closed]

What is$$\mathbb{P}\{B_2 > 0 \text{ }|\text{ } B_1 > 0\},$$where $B_t$ is a standard one-dimensional Brownian motion?
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0answers
9 views

Integrating a Nested Generic Brownian Motion within a Specific Brownian Motion

Given that $R_t$ is a Brownian motion and the geometric Brownian motion $S_t=e^{\mu t + \sigma R_t}$, compute $E[S_t], Var[S_t], E_s[S_t],$ and $Var_s[S_t]$ for some $s<t$. On the expected value, ...