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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16
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457 views

Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
10
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0answers
393 views

Does this random variable have a density?

I have a persistent problem, which I'm almost certain can be answered using elementary probabilistic arguments, but for some reason I've been stuck for some time. Here is the problem. Let $(B_s, s \...
7
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121 views

How to calculate the PSD of a stochastic process

Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô ...
7
votes
0answers
259 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
7
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0answers
655 views

How to prove Brownian motion is Gaussian Process?

I'm reading Bernt Oksendal's "Stochastic Differential Equations" and this is one of the proof that I'm totally lost. This is from Ch2.2, page 12-13 (sixth edition). First, Brownian motion is ...
5
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0answers
58 views

Brownian Motion Third Power Martingale using Ito Integral

Let $(B_t)_{t \geq 0}$ be a standard Brownian motion and $M_t = B_t^2 - t$. According to this and this posts we know that \begin{align} [M] = [B^2] = 2 \int_0^t B_s^2\ ds. \end{align} Now, without ...
5
votes
0answers
88 views

Brownian motion on sphere proof?

proving the brownian motion on the sphere equation the stratonovich form differential equation $$\partial X=n(X)\times \partial B$$ the equation in ito's form becomes $$dX=n(X)\times dB+H(X)n(X)...
5
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0answers
37 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 $B_t < 0 < B_u$?
5
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56 views

Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$ E[B_t B_s] = \min\{s,t \}, $$ which can ...
5
votes
0answers
222 views

Brownian Motion and stochastic integration on the complete real line

I'm struggling to understand stochastic integration over intervals containing zero, i.e. integrals of the form $\int_{a}^{b} X_s \, d B_s$ where $-\infty \leq a < b \leq \infty$, $(X_t)_{t \in \...
5
votes
0answers
119 views

The probability that a linear Brownian motion will hit a curve

Summary I am trying to estimate the probability that a standard linear Brownian motion will hit some curve. To make things a bit simple, I can assume that the curve is a graph of a function, that is ...
5
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0answers
869 views

Running maximum for Geometric Brownian Motion

Can anyone provide the expression and source for the running maximum $M_t$ for geometric Brownian motion $X_t$ as a function of the initial value $X_0$, drift $\mu$ and diffusion $\sigma$? $X_t$ ...
5
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0answers
180 views

Intuition for the optimality of bold play

There is a standard result (I think originally by Dubins and Savage) that if one wants to maximise the probability of winning a certain amount in an unfair game of chance then an optimal strategy is “...
4
votes
0answers
87 views

Is $X_t = tW\left(\frac{1}{t}\right)$ a Martingale?If not, how could it be a Brownian Motion?

As is proved, $X_t = tW\left(\frac{1}{t}\right)$ is a Brownian motion. For example see Theorem 4.2 in this paper http://math.uchicago.edu/~may/REU2012/REUPapers/Leiner.pdf I'm just confused because ...
4
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0answers
79 views

Relationship of SDE and Feynman-Kac PDE

I am struggling with this problem: Given a stochastic differential equation $$ dX_t = b(X_t) dt + \sigma (X_t) \,dW_t $$ where $W$ is a Brownian motion and the functions $b$ and $\sigma$ are ...
4
votes
0answers
228 views

Using Girsanov theorem to prove density of stopping time

Let $B$ be a standard Brownian motion and for $a>0$ and $b>0$, and set $$\sigma_{a,b} = \inf\{t\,:\, B_t + bt = a\}.$$ There are at least two ways to solve the following problem (the other one ...
4
votes
0answers
718 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 ...
4
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0answers
204 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
4
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0answers
68 views

2-D exponential functional brownian motion

I'm looking for the distribution of $X = \int_0^T e^{-W_t} dt \int_0^T e^{W_t}dt$ and $Y = \frac{\int_0^T e^{-W_t} dt}{ \int_0^T e^{W_t}dt}$ (where $W_t$ is a standard brownian motion) On most ...
4
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0answers
250 views

Time scaling of Brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion and $A_t$ be an increasing continuous process adapted to the filtration generated by the Brownian Motion and $A_0 = 0$. I am trying to prove that $(...
4
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0answers
213 views

Integral representation of fractional Brownian motion

Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $${1\over C(H)}\int_\mathbb{R}\left((t-s)_+^{H-{1\over2}}-(-s)_+^{H-{1\over2}}\right)dB(s)$$ ...
4
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0answers
162 views

Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion

Let $ 0 < r < 1$, fix $x > 1$ and consider the integral $$ I_{r}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$ In the investigation of ...
3
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0answers
41 views

Hitting times in two-dimensional case: expectation of Brownian motion at a hitting time

Consider two Brownian motions $$X_{1t}=\mu t+\sigma_1B_{1t}$$ and $$X_{2t}=\mu t+\sigma_2B_{2t}.$$ Here $B_{1t}$ and $B_{2t}$ are uncorrelated. Let $\tau_1$ and $\tau_2$ be the stopping times: \begin{...
3
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0answers
49 views

Find $P(B_3>0,B_6>0)$ where $(B_t)$ is a Brownian motion

Suppose that $B_{t}$ is a standard Brownian Motion. What is the probability that both $B_{3}$ and $B_{6}$ take positive values? This is what I've tried but then I get stuck and I'm not sure how to ...
3
votes
0answers
24 views

covariance and expectional in proccess

Show that the process $X=(W_{\sqrt{t}}I_{(1,2)}(t))_{t \ge 0} \in \mathcal{L}_3^2$. ($W$- Wiener) Additionally calculate, for $t,s \in [1,2]$, $EX_t$ and $Cov(X_t,X_s)$ I have no idea how to start ...
3
votes
0answers
31 views

Stochastic process is brownian motion by Levy's characterization

I would like to know if $B_t=W_t-\int_0^t \frac{W_u}{u}du$ is a brownian motion. I know that $W_t$ is a brownian motion. For that i would like to use Levy's characterization, so I have to show that $[...
3
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0answers
44 views

Hitting Times for Brownian Motion - Levy Process?

Let $X$ be a Brownian motion and let $$H_a = \inf\{ s \ge 0 \mid X_s = a \} \;\ \text{and} \;\ S_a = \inf\{ s \ge 0 \mid X_s > a \}.$$ Now, I've shown that $H_a$ and $S_a$ are equal almost surely (...
3
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0answers
20 views

Expectation related to Wiener process using strong Markov property

Can you help me to understand a result I found in Krylov's book "Introduction to stochastic calculus". First, I will introduce some notations: $w_t,t\ge 0$ denotes a Wiener process. $\mathcal{B}(...
3
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0answers
40 views

Simple application of Donsker's theorem

I am trying to do exercise 5.15 in Moerter's book "Brownian Motion". It seems quite easy, but I can't solve it somehow: Suppose $S(j)_j$ is a SRW on the integers, started at zero. Show that: $$ \frac{...
3
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0answers
68 views

Proving how to reduce a Brownian walk on a plane to a line (2D to 1D)

I have a Brownian motion on a plane and would like to find the time of when it is expected to hit a set of parallel lines, i.e the hitting time. In order to do so, I understand that I can reduce the ...
3
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0answers
42 views

Holder Continuity of a Continuous Stochastic Process

I have recently read the proof that the Brownian Motion and Fractional Brownian motion are almost surely Holder Continuous. I was wondering if this can be extended to a higher class of continuous ...
3
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0answers
40 views

Clarification on the Augmented Filtration

Consider the following definition. Definition. Let $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ be a probability space and $W$ a Brownian motion. Let $\mathcal{F}^W_t=\sigma\left(\left\{W_s\mid s\...
3
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0answers
23 views

What does $W_{1}$ and $(W_{1},W_{2})$ mean under the context of Brownian motion $W_{t}$?

As part of some practice questions for a course I'm taking, I was given the definition of a Brownian motion $W_{t}$ as a unique continous-time stochastic process satisfying: $W_{0}=0$ The function $...
3
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0answers
50 views

What is the Skewness of a Geometric Brownian Motion?

Consider a GBM : $$S(t) = S(0)\exp\left({(\mu-\frac{1}{2}\sigma^2) t + \sigma W_t}\right)$$ $$d\log S(t) = (\mu-\frac{1}{2}\sigma^2) t + \sigma dW_t$$ $$\frac{d S(t)}{S(t)} = \mu t + \sigma ...
3
votes
0answers
78 views

What can you tell me about backward Brownian motion?

I'm trying to understand "backward Brownian motion" and how it relates to standard Brownian motion. In this paper, they construct a solution to Burgers Equation (transformed via Cole-Hopf) with ...
3
votes
0answers
109 views

Moment Generating Function for Brownian motion's exit of interval.

Let $B(t)$ be a standard BM. Consider the stopping time $T = \inf\{ t > 0: |B(t)| = a\},$ the usual first exit time of the interval $(-a, a).$ We can see that $\mathbb{E} e^{tT} < \infty$ for $...
3
votes
0answers
57 views

Does Ito's Isometry hold if the integrand has a brownian motion in it?

I am wondering what is the distribution of: $$ \int_0^tW_sdW_s $$ Solution: (Thanks to @muaddib) Applying Ito's Formula to $W_t^2$ gives $d(W_t^2) = 2W_tdW_t +dt$, and so: $$ \int_0^tW_sdW_s= W_t^2 ...
3
votes
0answers
129 views

Integral of Brownian Motion with respect to an independent Brownian motion

I have this seemingly simple problem which I haven't been able to solve. I have two standard Brownian motions, $B$ and $W$, on the same probability space and under the same filtration (I am not so ...
3
votes
0answers
72 views

No drift brownian motion problem

Given two same brownian motion with no drift and different variances: $$dG_1= \sigma_1 G_1 dW $$ $$dG_2= \sigma_2 G_2 dW $$ and two barriers $P_1 > P_2$ assuming that $ \sigma_1 > \sigma_2 $ ...
3
votes
0answers
70 views

Why is the black-scholes model arbitrage free when $\sigma >0$?

I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
3
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0answers
26 views

Best predictor of Brownian motion

Let $B_t$ be brownian motion at time $ t \geq 0$. Then I want to find the best predictor of $B_8 + 4$ given that there are observations of brownian motion up to time $t = 1$. Approach: Essentially, ...
3
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0answers
70 views

the 2D fractional Gaussian noise as derived from the 2D fractional Brownian motion

Let $X_n$ be a 1D discrete fBm. Then, its 1st order difference, $W_n=X_n-X_{n-1}$ is fractional Gaussian noise (fGn). This case is simple. But what happens in 2D? Let $Y(m,n)$ be a 2D fBm, then we ...
3
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0answers
83 views

Stochastic Integral and Ito Isometry

I am right now studying stochastic integral, and facing the following dilemma! I just want to check whether my understanding is right! The stochastic integral is defined by following: $I(t) =\int_0^...
3
votes
0answers
204 views

Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution? $$ \int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau} $$ where $\tilde{...
3
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0answers
50 views

Strong markov property in two dimensional Brownian motion

I don't understand the following claim from my book: Let $(B_t)$ be a standard Brownian motion. Let $u:\Omega \rightarrow \mathbb{R}$ be a continuous function, where $\Omega$ is a domain and $B(x, \...
3
votes
0answers
482 views

Expectation of Exponential of Stochastic Integral

Let $z$ be the standard Brownian motion, $\omega$ an element of the sample space. Is it true that $$ \mathbf E\bigg[\exp\Big(\int_0^t f(\omega,s)\,\mathrm dz(s)\Big)\bigg] = \mathbf E\bigg[\exp\Big(\...
3
votes
0answers
110 views

What is the distribution of the area between a Brownian Bridge and the x-axis?

Lets say that we have a Standard Brownian Bridge ($\sigma=1$) with endpoints $(0,0),(1,0)$ Is there a way to derive the distribution of the area between a sample path of this bridge and the x-axis?? ...
3
votes
0answers
62 views

Brownian motion, modifications vs indistinguishablity

In Protters book Stochastic Integration and Differential Equations And in uncountable other sources, they mention the continuous sample paths of the brownian motion. That is: It holds that $$t\...
3
votes
0answers
107 views

Law of iterated logarithms for BM

The law of iterated logarithms for the standard Brownian motion asserts that $(\ast) \limsup\limits_{h \downarrow 0} \frac{B(h)}{\sqrt{2h\log\log(\frac{1}{h})}} = 1$ I'm trying to prove the ...
3
votes
0answers
51 views

Conditional expectation and coupled set of ODEs

How to find a coupled set of ODEs and initial conditions for the deterministic functions $a$ and $b$ such that $$\mathbb{E}\left[e^{-\int_{t}^{T} W^2(u)du} | \mathcal{F(t)}\right] = e^{-a(T-t) - \frac{...