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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Can we apply an Itō formula to the solution of a SPDE?

Let $V\subset H\subset V^\ast$ be a Gelfand triple $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(W_t)_{t\ge 0}$ be a ...
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25 views

Long term behavior of Brownian Motion

Let $(B_t)_{t \geq 0}$ be a Brownian motion. The objective is to prove that \begin{align*} \limsup_{t \to \infty} \frac{B_t}{\sqrt{t}} = \infty. \end{align*} By the scaling property of Brownian ...
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1answer
33 views

Why does the Borel-Cantelli lemma finish the job? - Law of Large Numbers Brownian Motion

The objective is to prove that \begin{align*} \text{$\lim_{t \to \infty} \frac{B_t}{t} =0 \qquad$ a.s.} \end{align*} By the strong Law of Large Numbers, we have that: \begin{align*} \text{$\lim_{t ...
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17 views

Brownian motion: Why $\xi_n$ and $\xi_m$ are independent.

Let $\{B_t\}$ a Brownian motion, $n\neq m$ and $$\xi_n=\sup_{s\in [n,n+1]}|B_s-B_{\lfloor s\rfloor}|\quad \text{and}\quad \xi_m=\sup_{t\in [m,m+1]}|B_t-B_{\lfloor t\rfloor}|.$$ We suppose WLOG that ...
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1answer
101 views

Discrete and continuous Girsanov

I'm trying to write a proof of the Girsanov theorem based on a discrete version of it. Discrete version Suppose that I have a random vector $X$ and two equivalent probability measures $\mathbb{P}, ...
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18 views

If $B_t - B_s, \ 0\leq s < t,$ is normally distributed, there are constants $C_n, \ E|B_t - B_s|^{2n}=C_n|t-s|^n$

I am working on the following problem: Show that if $B_t - B_s, 0 \leq s < t,$ is normally distributed with mean zero and variance $t-s$, then for each positive integer $n$ there is a positive ...
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1answer
24 views

Quadratic Variation of Wiener's process

I know I'm wrong, but I still don't understand why can't this operation be performed: $$ \sum_{j=0}^{n-1}(W(t_{j+1})-W(t_j))^2\le \max[W(t_{j+1})-W(t_j)]*(W(T)-W(0) )$$ which would have a limit of ...
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24 views

Show that $W(t)$ is almost surely non-differentiable at $t=0$

Show that $W_t$ is almost surely non-differentiable at $t=0$. Of course, $W(t)$ denotes a standard Wiener process. It is enough to show that $$P(\{\omega : \exists \epsilon>0 \: \forall \delta ...
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24 views

Compare moments of $\int_0^t h(B_s) ds$ and $\int_0^t h(\sqrt{s}Z)ds$ for $(B_t)$ Brownian motion and $Z$ standard normal

If we let $B_t$ be a standard Brownian motion and $\sqrt{t}Z$, where $Z$ is our standard normal random variable, we know that they have the same distribution. However, how can I show that the process ...
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38 views

How to find the variance of $\int_0^t B_s^2 ds$ where $B_s$ is a standard Brownian motion random variable?

I am trying to find the variance of $\int_0^t B_s^2 ds$ where $B_s$ is a standard Brownian motion random variable. My approach is to represent the integral as a sum. However, I am not sure how this ...
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25 views

Proving if $W(t)$ is a weiner process, then $W^2(t)$ is also a Weiner process [duplicate]

I'm trying to solve this question: For a stochastic process to be a Weiner Process it must have these properties: $W(0) = 0$ so $W^2(0) = 0$ $E(W(t)) = 0$ but $E(W^2(t)) = t$ I think this is enough ...
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18 views

What is meant by local time of BM on the boundary $\partial D$?

I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
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25 views

Probability of hitting a barrier

We have a stochastic process $ Y_t= \alpha t+ W_t$ where W is a standard brownian motion. Is there a way to calculate the conditional probability with respect to $Y_1$ for this process to hit a ...
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1answer
38 views

Hitting times for Brownian Motion (2)

In this post there is shown that for a standard Brownian motion $\mathbb{E}[\tau^p]<\infty$ for all $p \geq 1$, where \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ ...
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47 views

How can we identify $\omega\in\Omega$ with a path of Brownian motion $t\rightarrow B_t(\omega)$?

In the Stochastic Differential Euqations written by Oksendal(see page 12), As we shall soon see, the paths of a Brownian motion are (or, more correctly, can be chosen to be) continuous, a.s. ...
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24 views

PDE for Brownian Bridge Expectation?

Let $\displaystyle Y(t)=\int_0^t v(s)ds+B(t)$, where $B(t)$ is the standard Brownian motion and $v(t)$ a deterministic function. Compute $m(t,y):=\mathbf E\Big[\max\limits_{s\in[0,t]} ...
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2answers
47 views

Hitting times for Brownian motions

Let $B$ be a standard Brownian motion and let $\alpha, \beta > 0$. Let \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\}. \end{align} It can be shown by defining ...
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34 views

Last exit time of Brownian motion

I am trying to show that the last exit time of Brownian motion is a random variable, i.e. for $\tau$ defined as $$\tau = \sup\{t > 0 : W_t = 0\}$$ it holds that $\{\tau < t\} \in \mathcal{F}$ ...
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52 views

What is the solution to this graduate-level statistics problem?

I'm baffled as to how to explicitly solve this problem... I would normally just plug in problem-specific values and use Monte Carlo simulation to solve something complicated like this, but my ...
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1answer
37 views

Brownian Motion with rescaled time as an Ito process

I have a seemingly simple question that has me stumped. Suppose $(B_t)_{t\geq0}$ is a Brownian motion, and consider its rescaled version $(B_{\alpha t})_{t\geq0}$ for some $\alpha>0$. It seems ...
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28 views

Laplace transform of survival probability for stochastic diffusion

Let $Y_t$ be a killed process defined by \begin{eqnarray} Y_t = X_t \quad \mbox{if } t<\xi,\\ Y_t = 0 \quad \mbox{if } t\geq\xi. \end{eqnarray} where $\xi$ is a random time such that $$ \xi=\inf ...
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25 views

Expected time when Brownian motion leaves an interval

Let $S_t$ be standard Brownian motion (or a Wiener process) in one dimension. How do I formally derive the expected time that $S_t$ will leave a given interval $[-x, y]$ for some $x, y > 0$, given ...
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1answer
34 views

Deriving heat equation from brownian motion

Today my prof gave me an equation of random walk: $$p(x_i,t+\Delta t)=\frac{1}{2}(p(x_i-\Delta t)+p(x_i+\Delta t))-p(x_i,t)$$ Using this he get$$P_t=P_{xx}$$ when $\Delta t<<1$ But how and ...
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1answer
43 views

Martingale property for two stochastic processes

Let $(\Omega,F,P)$ be a probability space with filtration $\left\{F_{t}\right\}_{t\geq 0}$ generated by one dimensional Brownian motion $(B_{t})_{t\geq 0}$ defined on $(\Omega,F,P)$, assume that ...
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1answer
22 views

Applying ito integral to a specific example

Using the ito integral: $$x_t=\int_0^t b(s,x(s)) \, ds+\int_0^t \sigma (s,x(s)) \, dB(s)+x_0$$ I want to solve this: I have $$g(s,n(s))=\log(n(s))$$ Using ito's lemma taking the derivative I get ...
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1answer
17 views

GBM for the stock prices

Given the following SDE: $$ dS = \mu S dt + \sigma S dW $$ Why when we write this in discrete form: $$\Delta S = S_{i+1} - S_i = \mu S_i \Delta t + \sigma S_i \phi \sqrt{\Delta t}$$ The indices on ...
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1answer
82 views

Explaining a simple observation on Terry Tao's blog about the Wiener process

Quoting Terry Tao's blog: A simple but fundamental observation is that $n$-dimensional Brownian motion is rotation-invariant: more precisely, if $(X_t)_{t \in [0,+\infty)}$ is an ...
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Geometrical Brownian motion Passage time

Recently I have been self-studying stochastic analysis. One of the exercises was to calculate the probabilty of Brownian motion reaching certain level before time T given that W(t)=x. This wasn't that ...
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1answer
46 views

Solve the stochastic differential equation $dX_t = \alpha X_t \, dW_t + \sigma X_t \, dt$

We want to solve: $dX_t = \alpha X_t dW_t + \sigma X_t dt$ where the initial condition $X_0$ is given and $\alpha$, $\sigma$ are constant. The solution goes as follows - We rewrite the ...
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1answer
63 views

Measurability of the zero-crossing time of Brownian motion

I have the following random time $\tau = \inf\{t > 0: W_t = 0\}$ where $(W_t)_{t\geq 0}$ is Brownian motion with almost surely continuous paths and $W_0 = 0$ a.s. I need to prove that $\tau$ is ...
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65 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 ...
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29 views

Brownian motion - independence

I have not so difficult task - For Brownian motion $W(s)-W(t)$ is independant of $\sigma$-algebra $F(t)$ $0\leq t<s$. My goal is to show that for $0\leq t<s<u$, $W(u)-W(s)$ is also ...
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1answer
20 views

What is a general Brownian Motion?

This might be a dumb question, but no textbook ever defines what a "Brownian Motion" is, just what a "Standard Brownian Motion." I always assumed that a Brownian Motion is any random variable that can ...
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25 views

Brownian motion checking

$W(t)$ is a Brownian motion and $c>0$. I need to verify that 1)$X(t)=W(c+t)-W(c)$ and $X(t)=cW(t/c^2)$ are Brownian motions. 2) $Z(t)=tW(1/t)$ can be showm as $lim_{t-\rightarrow\infty} ...
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2answers
36 views

If $B_t$ is a standard brownian motion process, is $B_t^2 - \frac{t}{2}$ a martingale w.r.t. brownian motion?

If I have that $B_t$ is a standard brownian motion process, is $B_t^2 - \frac{t}{2}$ a martingale w.r.t. brownian motion? I know that $B_t^2 - t$ is but can't see it for the latter.
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Showing time inversion of a Brownian Motion $X_t = tB_{1/t}$ is continuous at $t=0$ USING the fact $X_t$ is BM on $\mathbb{Q}$? [duplicate]

I am reading the following paper on a rigorous construction of Brownian Motion: Brownian Motion. In the paper, they give a peculiar proof of the fact that the time inverted Brownian Motion is ...
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1answer
38 views

If $B_t$ is standard Brownian Motion, how to show that $X_t = B_t^2-t$ is a martingale?

If $(B_t, \mathcal{F}_t)$ is standard Brownian Motion, I would like to show that $X_t = B_t^2-t$ is a martingale. My attempted proof works as follows: \begin{align} E(X_{t+1}|\mathcal{F}_t) & = ...
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1answer
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For Brownian Motion $B_t$, and stop time $\nu = \inf\{t: B_t = r\}$, how to show $E(e^{-\alpha\nu}) = e^{-|r|\sqrt{2\alpha}}$?

If we have that $B_t$ is a Brownian Motion process, and we define a hitting time as $\nu = \inf\{t: B_t = r\}$ where $r \in \mathbb{R}$, how can I show that: $$ E(e^{-\alpha\nu}) = ...
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1answer
35 views

Is the Hölder random constant of the Brownian Motion Integrable?

Let $\{B_t:t\in [0,1]\}$ be the standard one-dimensional Brownian motion on the closed unit interval. Fix $\gamma\in (0,1/2)$. It is well known that there is a positive random variable $K\equiv ...
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1answer
34 views

Change of variable in $\varphi(s) = t$, effect in $\mathbb{d} W_t$

I'm a little confused here. If I have the stochastic integral $$ \int_0^T f(t)\,\mathbb{d} W_t $$ and perform the change of variables $t = \varphi(s)$, how will $\mathbb{d} W_t$ transform (where the ...
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1answer
121 views

Solution for SDE: $dF_t= \beta_t\left(F_t - \alpha\right)dW_t$

I am trying to derive the solution for the following stochastic differential equation, but I must be doing something wrong in my calculations because I can't arrive to the correct solution. The SDE ...
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1answer
31 views

Modification of Donsker Theorem

assume some i.i.d. random variables $(X_i)_{i \geq 1}$ with mean 0 and variance 1. Due to Donsker it holds that $$\left(\frac{1}{\sqrt{n}}\sum\limits_{i=1}^{nt}{X_i}\right)_t \rightarrow (W_t)_t$$ ...
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Variance of Brownian Integral when the end point is specified

Consider the Brownian $W_u$. Suppose you are only considering realizations of this brownian that verify both $W_0=0$ and, for a specific (given) $t$, $W_t=a$. Under these specific conditions, what is ...
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21 views

harmonic measure circle

I'm trying to compare the probability of a particle (performing Brownian Motion), starting a large distance away from a circle, passing through a specific section of that circle is approximately the ...
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29 views

Transience of Brownian Motion

I am reading a book by Morters & Peres on Brownian motion, and I got really confused with the definition of transience. At first, they give a simple definition that BM is transient if it converges ...
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1answer
49 views

Calculate $E[\exp(iu\int_0^ts \, dB_s)]$ for a Brownian motion $(B_t)_{t \geq 0}$

Since $X_t:=\int_0^ts \, dB_s$ is a process with independent increments, its distribution is infinitely divisible and its variance is $c_t=\frac{1}{3}t^3$. I think, its characteristic function ...
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1answer
57 views

Itos formula on a transformation of bessel Processes

Let $W$ be a Brownian motion and $z,\kappa>0$. Let $X_t(z)$ be a solution to the SDE $$dX_t(z)=dW_t+2/(\kappa X_t(z))dt.\quad X_0(z)=z.$$ The solution is well-defined on $t<\tau(z)$ where ...
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2answers
138 views

The stochastic integral $\int W_t dW_t$

I'm reading an introduction to Stochastic Calculus. I'm at the point where Ito integrals are developed and constrasted with the Stratonovich integral. Below is a calculation of $\int_0^T W_t d W_t$. ...
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0answers
41 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 ...