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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Convergence of exponential Brownian martingale to zero almost surely

Define the exponential Brownian martingale as $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ which is a martingale with respect to the natural filtration of $W$ which stands for a standard Brownian ...
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8 views

Integral of Constant Parameter Martingale

What is the $\int_{1}^{t}W_1W_sdW_s$. This is the question solved by Kuo in his paper an extension of the Ito's Integral (2008) but there limit runs from $0$ instead of $1$.
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20 views

Distribution of a transformed Brownian motion

Let $W$ be a standard Brownian motion. From an earlier proven result I know that $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ defines a martingale on the natural filtration of $W$ for all $a \in ...
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1answer
11 views

$n$ times integrated Brownian motion

I have an identity that expresses the $n$ times integrated Brownian motion and I would like to prove that. First, I define what I mean by $n$ times integrated Brownian motion. $$V_1(t) = \int_0^tB_s\, ...
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23 views

Measurability of the event that Brownian motion hits a given set

Let $W$ be a Brownian motion in $\mathbb{R}^{2}$ on a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ . Let us assume $\mathcal{F}$ is the sigma-algebra on the path space ...
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Argmax distribution of Brwonian motion plus linear drift

I want to know the the density function or the tail of the density funcion of the following random variables: $$\underset{{t\in [0,+\infty]}}{\arg \max} \quad {W_t-t}.$$ Thank you very much
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Defining a stochastic process indexed by angle

I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin ...
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6 views

Derive Laplace Equation through Random Walk

I am looking for the solution of this problem: Consider a bounded domain $\Omega\subset\mathbb{R}^2$ and let $u(x,y)$ be the probability of exiting $\Omega$ starting at $p=(x,y)$, assuming that the ...
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0answers
22 views

Convergence in distribution of BM started in (x,y) to BM started in (0,0)

Let $B$ be a Brownian motion in $\mathbb{R}^{2}$ . Let $\mathbb{P}_{(x,y)}$ denote a probability measure under which $B$ is started at $(x,y)$ . Is it true in general that, for measurable set ...
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1answer
43 views

Proof of martingale representation theorem monotone class argument

Martingale representation theorem for reference: Theorem: (Martingale Representation) Let $M$ be a square integrable Brownian martingale with $M_0 = 0$.Then there exists a process $X$ which is ...
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22 views

Prove independent increments

The question I have is below; I have found one version of an answer for this so far here. Is the following a Wiener process? My question that I am asking is, "How would you show that the increments ...
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26 views

Two-parameters Wiener process

Two-parameters Wiener process $W(r, u), r \in [0, 1], u \in [0,1]$ is a stochastic process with a covariate kernel $\mathbb{E}\left[W(r_1, u_1) W(r_2, u_2)\right] = \min(r_1, r_2) \min(u_1, u_2)$. ...
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19 views

Basic Stochastic Calculus

Let $B_t$ be brownian motion. Then if I need to calculate $\mathbb{E}[2(B_2-B_0)+(B_2+B_1)(B_3-B_2)]$ is this simply $0$ as independence results in: $\mathbb{E}[2(B_2-B_0)] + \mathbb{E}[B_2+B_1] ...
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0answers
14 views

Functions of Brownian Motion and Time

Sorry, this will be a little long. I'm currently working on a problem where I basically have an SDE logistic equation: $$dX_t = diag(x_1,\cdots, x_n)[b+Ax-\lambda \eta(t)] dt + diag(x_1,\cdots, ...
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1answer
42 views

Law of Iterated Logarithms

I know the Law of Iterated Logarithms states the following almost surely: $$\limsup_{t\to\infty} \frac{B(t)}{\sqrt{2t\log\log t}} = 1 $$ I was wondering if there are similar ones. For example, if I ...
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1answer
73 views

Wiener process - proof of independent increments

I have defined the Wiener process to be a stochastic process $X_t$ with values in $\mathbb{R}$ such that $X_0=0$, the paths $t \mapsto X_t$ are continuous, and for any times $0<t_1<\dots<t_n$ ...
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28 views

covariance of two correlated integrated brownian motions

Assume we have two integrated Brownian Motions $\int_0^tf(t)dW_t$ and $\int_0^tg(t)dY_t$ where the $d$-dimensional Brownian Motions $W$ and $Y$ are correlated according to the positive semi-definite ...
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1answer
55 views

Determine for which values of some parameters a stochastic integral is a Brownian motion

Let $W_t$ be a Brownian motion on $(\Omega, F, (F_t)_t, P)$. Find all values of $a$ and $b$ such that the stochastic integral $$X_t=\int_0^t a+\frac{bu}{t} \;dW_u$$ is a Brownian motion. 1)So I need ...
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29 views

Transition density of a Geometric Brownian-motion

The solution to SDE $$dS(t)=\sigma S(t)dW_t$$ is $$S(t)=S(0)\exp(-\frac{1}{2}\sigma^2t+\sigma W_t)$$ the transition density for this martingale is $$p(S(t),t;S(0),0)=\frac{1}{S(t)\sigma \sqrt{2\pi ...
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1answer
43 views

Distribution of Integral involving wiener process [closed]

Given $W(t)$ as a standard Wiener process, i.e. $W(t) \sim \mathcal{N}(0,t)$. Prove the following statement: $$\int_{0}^{1}tW(t)dt \sim \mathcal{N}(0,\frac{2}{15})$$
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28 views

Mgf of first passage time of brownian motion

Define the $\tau_x=inf\{t:W_t = x\}$, where $W_t$ is a brownian motion. I know the distribution of $\tau_x$ is $$f_{\tau_x}(t)=\frac{|x|}{\sqrt{2\pi}}t^{-1.5}e^{\frac{-x^2}{2t}}$$, which is an ...
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41 views

Brownian motion with drift (stopping time and supremum)

Suppose $(B(t))_{t \geq 0}$ is a Brownian motion and $(B_{\mu}(t))_{t \geq 0}$ is a Brownian motion with drift, which is defined by $$B_{\mu}(t) := B(t) + \mu t, \ \ \ \mu <0. $$ With $T_{a} := ...
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1answer
46 views

Help with Semimartingale decomposition.

I'm having trouble with the following question: Let $\{W_t\}_{t\geqslant0}$ be a one-dimensional standard Brownian motion defined on a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal ...
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37 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 ...
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1answer
52 views

Relation between Hermite polynomials and Brownian motion (on martingale property) [duplicate]

Let us define Hermite polynomials as $H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n} e^{-x^2/2}.$ One can prove that $e^{\theta x-\frac{1}{2} \theta^2}=\sum_{n=0}^{\infty} \frac{1}{n!}H_n(x)\theta^n \quad ...
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2answers
28 views

Let $T_{a}$ denote the first time the Brownian motion process hits $a$. When $a>0$, then $P\{X(t)\ge a|T_{a}\le t\}=\frac{1}{2}$

Let $T_{a}$ denote the first time the Brownian motion process hits $a$. When $a>0$, then $P\{X(t)\ge a|T_{a}\le t\}=\frac{1}{2}$ I cannot see how it can be true, anyone could help me? Thanks very ...
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0answers
28 views

Question about zero set of Brownian motion

I was reading the posted to solutions to one of the questions on a probability midterm and couldn't figure out how to justify one of the steps. Let $\{B_t\}_{t\geq 0}$ be a Brownian motion and ...
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33 views

Variance of absolute value of brownian motion

Im wondering if anyone has this calculated, I cant seem to find it anywhere online. I am trying to find the variance of absolute value of BM. Here is my attempt: First, $f_{\lvert W_t \rvert} ...
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1answer
41 views

Proving Zero Covariance of Brownian Motion

Let $u>t>s\ge0$ I Want to know whether the following statement: $Cov[(W_t - W_s) -\frac{t-s}{u-s} (W_u-W_s),W_k] =0 ~~~~~~~~~~~~~~\forall K \in \{u,s\}$ implies : $Cov[(W_t - W_s) ...
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28 views

Meaning of equicontinuity in probability theory

I am reading a paper that says the following: Let $(B_t)$ be a standard one-dimensional Brownian motion and $t_0 >0$ and $c \in \mathbb{R}$ are fixed. Then the law of the process $(B_t + ct, t ...
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1answer
46 views

Conditional Brownian Motion

What is wrong with the following logic: let $0\leqslant s \leqslant t \leqslant u$, find $E[W_t | W_s, W_u]$ \begin{align*} E[W_t | W_s, W_u] &= E\left.\left[W_t - \frac{t}{u} W_u + ...
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1answer
50 views

Prove increment of Brownian motion is Brownian motion

I am trying to solve the following exercise in Oksendal's book: Let $B_t$ be Brownian motion and fix $t_0\ge 0$. Prove that $$\bar{B_t}:=B_{t_0+t}-B_{t_0};\quad t\ge 0$$ is a Brownian motion. I ...
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24 views

How to calculate density of first passage time of Brownian motion? [duplicate]

Let $B_t, t\ge0$ be a standart Brownian motion ($B_0 = 0$, $B_t - B_s$ ~ $N(0, t-s)$). $\tau_a = \inf\{t\ge0: |B_t| = a\}$. Density $$p_{\tau_a}(t) = \frac{\pi}{2a^2} \sum_{n=1}^\infty (-1)^{n-1} *n ...
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2answers
47 views

Limit of time integral of brownian motion

Can someone help explain the following, $$ \lim \limits_{t \to 0} \frac{1}{t} \int_0^t W_u\, du=\lim \limits_{t \to 0} \frac{W_0t}{t}=W_0=0\,? $$ Thanks!
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Simulation of Brownian Motion on Borel Spaces

I am studying stochastic calculus on my own, and currently stuck to the following issue. Say my probability space is $(\Omega, \mathcal F, \mathbb P)$. Now when my $\Omega$ has sequences of finite ...
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1answer
36 views

How to interpret the covariance matrix of Brownian motion

I'm reading Bernt Oksendal's "Stochastic Differential Equations". It says, Brownian motion $B_t$ is Gaussian Process, i.e. for all $0 \leq t1 \leq \cdots \leq t_k$ the random variable $Z = ...
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1answer
48 views

Distribution of Brownian motion before stoping time.

Let $B_{t}$ be a standard Brownian motion. Stopping time $\tau_{a} = \inf \{t \ge 0: |B_{t}| = a\}$. How to find $E[B_{\frac{\tau_{a}}{2}}]$? Or where is it possible to read about it? Thanks in ...
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16 views

Binomial Approximation to Black-Scholes Model / Brownian Motion

Above is my question. To be honest, reading it, it looks like it should be pretty straightforward. My method was this: Let's (wlog) take $S_0 = 0$ for ease. ($\sigma = s$ for question notation.) ...
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35 views

Stcochastic Integral and Ito Isometry

I am right now studying stochastic integral, and facing the following dilemma! I wjust want to check whether my understanding is right! The stochastic integral is defined by following: $I(t) ...
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1answer
19 views

Integral of square of Brownian motion with respect to Brownian Motion

While trying to compute $\int_0^TB_t^2\ dB_t$, $B$ being the standard Brownian motion, I got stuck at showing the following. $$\sum_{i=0}^{n-1}B_{t_i}(B_{t_{i+1}}-B_{t_i})^2 \rightarrow \int_0^TB_t\ ...
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2answers
50 views

Brownian Motion $dW_t \, dt=0$ proof!

I am facing a bit weird issue here. I am going through Shreeve book on stochastic calculus and faced the following theorem, while proving $dWdt=0$. $\sum_{j=0}^{n-1}(W(t_{j+1})-W(t_j))(t_{j+1}-t_j)$ ...
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1answer
97 views

Existence of a Continuous Modification of Fractional Brownian Motion

For a course on stochastic processes, I've been working on an exercise on fractional Brownian Motion. Showing that this process has a continuous modification is one of the final steps of the exercise, ...
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1answer
33 views

Radon-Nikodym Derivatives between Ito Processes

I am curious about the following problem: Let $B_t$ be a standard Brownian motion on $(\Omega, \mathcal F, \mathcal F_t, \mathbb P_a)$, where the filtration is generated by $B_t$. On a finite ...
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1answer
35 views

How to prove this Brownian motion convergence?

Let $W_t$ be a Brownian motion. How do I show the following? $$ \alpha > \frac{1}{2} \Rightarrow \lim_{t\rightarrow\infty} \frac{W_t}{t^{\alpha}} = 0 \text{ a.s.} $$ Showing convergence of this ...
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1answer
30 views

Explosion time of $dX_t=X_t(adW_1+bdW_2)$

I found in Karatzas & Shreve (1991), $dX=\sigma(X_t)dW_t$ cannot explode. But what about $dX_t=X_t(adW_1+bdW_2)$? Here $W_1$ and $W_2$ are independent. Feller's test for explosion seems to work ...
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25 views

How to compute the conditional expected value of a geometric brownian motion?

I'm working on a project, and I have to use the cumulative and conditional expected value of the variations of a stock following a Geometric Brownian Motion. I know that the cumulative is as follows ...
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30 views

What is the distribution of $\sup\limits_{t\geq 0}( B_t-xt)$

I would like to find the distribution of $\sup\limits_{t\geq 0}( B_t-xt)$, where $(B_t)_{t \geq 0}$ is a Wiener process and $x > 0$. I don't know how to begin. Any help is appreciated.
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0answers
30 views

Why does Brownian motion have finite $L^2$ norm?

The title might be a bit misleading. Sorry for that but here is the question. For predictable processes $X$, the $L^2$ norm over the set $[0,T]\times\Omega$ under the Doleans measure $\mu_M$, $M$ ...
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1answer
45 views

Strong Markov property of Bessel processes

I am thinking about the following: If $(B_t)_{t \geq 0}$ is a Brownian motion in $\mathbb{R}^3$, how can we show that the Bessel process (of order $3$) $(|B_t|)_{t \geq 0}$ has the strong Markov ...
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1answer
35 views

Proving that $X_t = W_t~ I (0<t\le T) + (2W_T - W_t) ~I(t > T)$ is a brownian motoin

The steps to showing that a process is a BM are as follows: (1)$X_0 = 0$ (2) $ \forall t ~~~X_t$ is continuous (3)$X_t \sim N(0,t)$ (4)$X_{t+s}-X_{s} \sim N(0,t)$ (5)$X_{t+s}-X_{s} \bot \mathscr ...