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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Sum of Brownian Motions

I've got a little problem: if $X_{t}$ and $Y_{t}$ are two indipendent Brownian motions, is then $$Z_{t}:=X_{t}+Y_{t}$$ a Brownian motion too? I've got some troubles only with showing that $Z_t$ is ...
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2answers
38 views

$E[e_te_s\Delta B_t\Delta B_s]$ for $\Delta B_t$ Brownian motion increments and $e_t(\omega)$ a measurable function.

Let $\Delta B_j=B_{t_{j+1}}-B_{t_j}$ where $B_t$ is Brownian motion, and $e_i(\omega)$ measurable with respect to $\sigma(B_{t_i})$. In Oksendal's 'Stochastic Differential Equations' he states: $$ ...
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1answer
68 views

Expected Return, Expected Value, and an Ito Process

I am reading John Hull's "Options, Futures, and Other Derivatives". I am currently in Ch. 31 on the HJM Model. Hull makes a statement which a need an explanation for. First, some notation. Let ...
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1answer
106 views

Transition density of Brownian bridge using generators

Let $X_{t}:=(1-t)\int_{0}^{t}\frac{1}{1-s}dB_{s}$. This satisfies SDE: $$dX_{t}=-\frac{X_{t}}{(1-t)}+dB_{t}$$ So the generator will be $A(f)=\frac{-x}{1-t}f'+\frac{1}{2}f''$ and so I think the pde ...
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1answer
51 views

A Question on the Scaling Invariance of Brownian Motion

I read the following paragraph. Let $B_t, \ t \in [0, \infty)$ be a standard linear Brownian motion. For each $q > 4$, define the following sequence of sets. $$ \Omega_k := \left\{\omega \in ...
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0answers
107 views

Cameron Martin Theorem

I am struggling with two versions of the Cameron Martin Theorem. 1) We define the measure spaces $(\Omega,\mathcal{F},P)$ and $(C[0,1],\mathcal{C},\mathbb{L}_0)$, where $\mathcal{C}:=\sigma(f\mapsto ...
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1answer
46 views

Zero hitting probability for positive measure sets in $\mathbb{R}^{d}$

In $d\geq 3$, we have that BM is transient a.s. i.e. $lim_{t\to \infty}|B_{t}|=\infty$. But does this imply $P_{x}(T_{A}<\infty)=0$ for some type of Borel sets $A\subset \mathbb{R}^{d}$ with ...
3
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1answer
534 views

Hermite Polynomials and Brownian motion

I am asked to prove the following : Let $B_t$ be a standard brownian motion. The $n$th Hermite polynomial is $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
0
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1answer
42 views

Equivalent Stopping Times for Brownian Motions

For standard Brownian motion $B$, define stopping time $T_1:=\inf\{t>0: B_t = 3\}$ and $T_2:=\inf\{t>0: B_t = -3\}$ and $T_3 := \min\{T_1, T_2\}$. Can I say that $T_3 = \inf\{t>0, B_t \in ...
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1answer
48 views

Testing numerical solvers with analytic solution to Ornstein-Uhlenbeck SDE?

I have an SDE I want to solve numerically that is fairly close to the Ornstein-Uhlenbeck process: $$ dx_t=θ(μ−x_t)dt+σdW_t $$ which has analytic solution $$ ...
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0answers
36 views

Probability that Brownian motion hits both hemispheres

The problem is to find: $P_{x}(\{T_{B_{1}}<\infty\}\cap \{T_{B_{2}}<\infty\})$ where $B_{1},B_{2}$ are the two hemispheres of sphere S shown below. There are two possible paths for first ...
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1answer
125 views

Brownian Motion inequality (related to Dvoretzky-Erdoes test)

i have the following question: Let $B(t)$ be a d-dimeansional Brownian motion $d\ge 3$, and $f$ be a monoton increasing function from the positive reals to the positive reals. Let $A_n=(\exists t\in ...
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2answers
989 views

Is this Stochastic integral a martingale ?

Let $(B_t)$ be a Brownian motion and set $X_t = \int_0^t B_t^2 dB_s$. Is $X_t$ martingale? My idea is to rewrite $X_t$ in terms of Ito's Formula $(f(x) = \frac{1}{3}x^3)$ $X_t = \int_0^t B_t^2 dB_s ...
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1answer
28 views

Brownian Motion and Progressive Process

Let $B_t$ be a Brownian motion. Define sign function as follows. $sign(0) = 0$ and $sign(x) = \frac{x}{|x|}, \forall x \neq 0$. I do not know how to show the following two questions, especially on the ...
2
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1answer
104 views

How to show the following definition gives Wiener measure

On the first page of Ustunel's lecture notes, he defines the Wiener measure in the following way: Let $W = C_0([0,1]), \omega \in W, t\in [0,1]$, define $W_t(\omega) = \omega(t)$. If we denote by ...
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1answer
124 views

Variance process of stochastic integral and brownian motion

Let $(W_t)$ be a Brownian motion with respect to a filtration $(\mathcal{F}_t)$. For all $t \geq 0 $ set $$X_t = \int_0^t W_s^2 \mathrm{d} W_s,\qquad Y_t = W_t^7.$$ Find the covariance process ...
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2answers
54 views

Brownian and Brackets

A continuous martingale with deterministic bracket must be a Brownian motion. Is this statement ture or not, please? If true, how to show it? If not, what is a counter example?
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1answer
32 views

Martingale property of negative Brownian motion

Let $B_t$ be Brownian motion, with $B_0=0$. Next define $M_t=-B_t$. Have I understood it correctly that $M_t$ is not a Martingale? $E[M_t]=0$ $E[M_{t+1}|M_t]=-M_t$ and therefore not a Martingale? ...
4
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1answer
194 views

Why is the canonical filtration of a Brownian motion left-continuous?

Let $\{W_t, t\geq 0\}$ be a Brownian motion, and has a.s. continuous sample paths. Let $\{\mathcal{F}^W_t, t\geq 0\}$ be the canonical filtration, i.e. $\mathcal{F}^W_t=\sigma(W_s, 0\leq s\leq t)$. ...
0
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2answers
43 views

simple stochastic differentiate

someone can help me to differentiate $$a(t-1)+bt+(1-t)\int_{0}^{t}\dfrac{dB_s}{1-s}?$$ I've tried but I really don't know how to do with the last part.. Thank you somuch for your help
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2answers
126 views

Stratonovich integral

I'm having some troubles to calculate the Stratonovich integral $I(sin)(t)=\int_{0}^{t}\sin{B_{s}}dB_{s}$. I've tried with the limit of ...
2
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1answer
74 views

Conditional Integral of Square of Brownian Motion?

I am struggling to compute the expectation and variance of the following, where $W(s)$ is a standard Brownian motion: $$ X := \int_{0}^{A}W(s)^2ds$$ $$ Y:= \int_0^AW(s)ds $$ $$E[X\mid Y] = \space ?$$ ...
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2answers
261 views

Proving Brownian Motion has Stationary Increments

In Oksendal's 'Stochastic Differential Equations', we define Brownian Motion as follows: Fix $x\in\mathbb{R}^n$ and define for $y\in\mathbb{R}^n$: $$p(t,x,y)=(2\pi ...
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1answer
419 views

The Brownian motion process in Sheldon M. Ross

Today I study Brownian Motion and Geometric Brownian Motion using textbook: An Elementary Introduction to Mathematical Finance, Third Edition by Sheldon M. Ross but I missed the class because I was ...
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1answer
132 views

How to solve a linear stochastic differential equation?

I don't know how to find a solution of this stochastic differential equation: $dX_{t}=(1+\delta \mu X_{t})dt+\delta X_{t}dB_{t}$ Where $B_{t}$ is a standard Brownian motion and $\mu$ and $\delta$ ...
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2answers
174 views

Prove $A_t := W_t^3-3t W_t$ a martingale

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let $(A_t)_{t \geq 0}$ where $A_t = W_t^3 - 3tW_t$. ...
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1answer
75 views

Cauchy distribution for Brownian motion

This is homework so no answers please Problem: Find distribution of $(B_{1}(T_{a}),B_{2}(T_{a}))$, where $T_{a}=inf_{t\geq 0}\{B_{2}(t)=a\}$ Any mistakes: $T_{a}=inf_{t\geq 0}\{B_{2}(t)=a\}$ has ...
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1answer
86 views

Random walk and Occupation measure

This is homework so no answers please I want to find for some $A\subset \mathbb{R}$ the limit $$\lim_{n\to \infty}\mu_{n}(A)=\lim_{n\to ...
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1answer
53 views

Help understanding a proof of non-differentiablity of Brownian motion

The following statement and proof are taken from the book Brownian Motion by Peter Morters and Yuval Peres. Since I initially didn't fully understand the proof I added some clarifications and I was ...
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1answer
88 views

Covariance of m-fold integrated Wiener process

The problem I'm trying to perform a Bayesian approach to the Maximum Likelihood Estimation procedure of Wecker and Ansley (1983). To this end, I need to compute the full likelihood of the data given ...
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0answers
538 views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
0
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2answers
120 views

Conditional expectation of Brownian motion given its absolute value.

Assume that $W_t$ is Brownian motion (1-D) and that $t<T$. How can I compute $$E(W_t||W_T|),$$ the conditional expectation of $W_t$ given $|W_T|$, i.e. with respect to the $\sigma$-algebra $F$ ...
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2answers
327 views

Itô process and covariance of two Brownian motion

I'm a novice in studying the stochastic different equation, and didn't know whether I have describe the question correctly. Here is the question: Suppose $$\begin{array}{rcl} ...
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1answer
44 views

mean hitting time of a level and growth rate of maximum process

Let $X_t$ be the absolute value of Brownian motion starting at $0$, let $\tau_x$ be it's first hitting time of the level $x>0$, and let $M_t$ be it's running maximum up to time $t$. Suppose we knew ...
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0answers
353 views

Expectation of absolute value of Brownian motion

I'm working on this problem that I can't seem to figure out. The problem involves a 1-dimensional Brownian motion, $B_t$, where the subscript denotes the time, and it asks me to show that the ...
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0answers
47 views

Ito's integral from the definition [duplicate]

I am doing Oksendal's book exercises one by one. I got stuck in 3.2. I need to prove, from the definition that $$\int_{0}^{t}B_s^2\text{d}B_s=\frac{B_s^3}{3}-\int_{0}^{t}B_s\text{d}s,$$ where ...
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1answer
42 views

Probability Brownian Motion doesn't hit a point in the limit.

This is a question from Revuz and Yor (exercise 3.18) for which I seem to get a different answer. Show that $\lim_{t \to \infty}\,t^{1/2}\,\mathbb{P}\{B_s\leq1\,\forall\, s\in[0,t]\}=\sqrt{2/\pi}$. ...
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1answer
118 views

Reference for the Construction of Brownian Motion

A common method for constructing Brownian motion is referred to as the Levy construction, the Levy-Ciesielski construction, the Ciesielski construction and sometimes seems to be attributed to Wiener ...
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0answers
12 views

Holder continuity, brwonian motion [duplicate]

Let $B$ stand for a brownian motion on a finite interval $[0,1]$. If i am not wrong, i think that there exists a positive constant $c$, such that almost surely, for h small enough , for all $0< t ...
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1answer
78 views

Distribution of hitting position of line by brownian motion.

What is known about the distribution of the hitting position of a line by a 2d brownian motion? I've tried to make some simulations of a 2d brownian motion where every computational step has a ...
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1answer
90 views

Brownian motion - Hölder continuity

Let $B$ stand for a Brownian motion on a finite interval $[0,1]$. If I am not wrong, I think that there exists a positive constant $c$, such that almost surely, for $h$ small enough , for all $0< t ...
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1answer
39 views

$\forall p>0$ we get $\displaystyle \lim_{n\to \infty} n^{a_{p}}\sum _{i=0}^{n}|B_{\frac{i+1}{n}}-B_{\frac{i}{n}}|^{p}=c_{p}$

This is a Homework question, so please do not answer it. Find real constants $a_{p},c_{p}$ s.t. $\forall p>0$ we get $\displaystyle \lim_{n\to \infty} n^{a_{p}}\sum ...
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0answers
38 views

Separation of variables and Fourier transformation

I know there's another question very similar to this argument. In the book Probabilità e Modelli Aleatori of Enzo Orsingher, at page 134, it shows that the transition function of an absorbing Brownian ...
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0answers
38 views

Hitting time and its distribution

÷I'm reading an italian book about casual process (Probabilità e modelli aleatori of Enzo Orsingher). At pag 105 there's the probability of the stopping time $T_\beta$. $$P\{T_\beta \leq ...
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1answer
20 views

Exclusion-Inclusion principle for Hitting times of two disjoint sets

Consider disjoint sets A, B and Brownian motion $B_{t}$ with $B_{0}\notin A\cup B$. Let $T_{A}:=inf_{t>0}\{B_{t}\in A\}$. Then, do we get $P(T_{A\cup B}<\infty)=P(T_{A}<\infty)+P(T_{ ...
2
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1answer
92 views

Simple question about the definition of Brownian motion

I have a question concerning the definiton of Brownian motion. Usually (e.g. on Wikipdia) one demands a brownian motion $\lbrace B_t\rbrace_{t\in[0,\infty)}$ to satisfy the following condition: ...
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1answer
61 views

Intuition underlying stopped martingales

Let $X$ be a martingale and $T$ a stopping time. Define the stopped martingale $X_{\min\{T,n\}}$. What is the intuition underlying this process? It is quite confusing here. $X$ is random and $T$ is ...
5
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1answer
193 views

Expectation of Square of Stopping Time

Let $B_t$ be standard Brownian motion and $a < 0 < b$. Define stopping time $T$ as follows. $$T = \min \{t \geq 0: B_t \in \{a, b\} \}.$$ The expectation of $T$ is $\mathbb ET = |a|b$ and can be ...
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1answer
32 views

Solve parameter from stochastic integral

how can I solve $\rho$ from the following: $\int_0^T dV_t = \int_0^T \kappa (\theta - V_t) dt + \int_0^T \sigma \rho \sqrt{V_t} dW_t + \int_0^T \sigma \sqrt{1-\rho^2} \sqrt{V_t} dZ_t$, where $W_t$ ...
1
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0answers
71 views

$E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t )ds$

I was trying to compute $E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t) ds$, $\mathcal{F}$ is associated to $W$. I tried the following. 1) Splitting the integral ...