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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Application of Girsanov theorem

Let $f(t)$, $t \geq 0$ be a smooth function with $f(0) = 0$ and let $B(t)$, $t \geq 0$ be a brownian motion. Let $P$ and $Q$ be two measures on $C[0,1]$ corresponding to respectively, $B(t)$, $t \geq ...
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1answer
103 views

Computing quadratic variation and criteria for Brownian motion

Let $f(t)$ be a nonrandom and continuously differentiable function and $B(s)$ be the brownian motion. a) Computer the quadratic variation of : $X(t) = f(t)B(t) - \int_0^t f'(s)B(s)ds$ b ) For ...
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1answer
55 views

What does $s$ and $t$ stand for in this definition of fractional brownian motion?

$$B_H(t_2,\omega)-B_H(t_1,\omega) = \frac{1}{\Gamma(H+1/2)}\Bigg\{\int_{-\infty}^{t_2}(t-s)^{H-1/2}dB(s,\omega)-\int_{-\infty}^{t_1}(t-s)^{H-1/2}dB(s,\omega)\Bigg\}$$ It's taken from Mandelbrot & ...
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1answer
152 views

Probability that Brownian Motion hits $t+1$ before $t-1$

Compute the probability that a brownian motion starting at $0$ hits the line $t+1$ before the line $t-1$. Here is what I did: I figured it has to do with optional stopping theorem. The ...
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204 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 ...
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1answer
119 views

Prove identity in law for stochastic process driven by Brownian Motion

Let $B = (B_t)_{t\geq 0}$ be a standard brownian motion started at $0$. Consider the two following stochastic equations: \begin{equation} \begin{split} dX_t &=& (13 + 2X_t)\,dt + (6 + ...
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1answer
69 views

Sufficient and Necessary condition for the sum of brownian motions to be a brownian motion

The question has two parts : Part a): Let $(B_1(t), B_2(t), B_3(t))$ be standard a brownian motion in $R^3$. Write down necassary and sufficient condition for $\sum\limits_{i=1}^3 a_iB_i(t)$ to be a ...
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1answer
87 views

Bound probability Brownian motion stays in $[-1,1]$.

Let $T:=\inf\{t>0: |B_t|=1\}$ be a hitting time for standard Brownian motion. I want to show that $$\lim_{t\to\infty} e^{\frac{\pi^2}{8}t}\mathbb{P}[T\geq t]=\frac{4}{\pi}$$ I had a look at A ...
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1answer
68 views

Brownian motion and posterior distribution

I am a bit stuck on this question: Suppose that $X_t = W_t + \alpha t$, where $W$ is a standard Brownian motion, and let $\mathcal{F}_t = \sigma ( X_u: 0 \leq u \leq t)$. The drift is constant in ...
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1answer
144 views

A good book on Brownian motion

Can you suggest me a good book on Brownian motion, where it is introduced as a limit of measures on polish spaces like $C[0,1]$ and subsequently stochastic calculus is discussed?
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1answer
148 views

Considering stochastic processes as random variables (Brownian motion)

Define process $X$ by $X_{0}=0$ and $X_{t}= tB_{1/t}$ for $t>0$, where $B_t$ is a standard Brownian motion. I want to show that $X$ is continuous in zero. The suggested hint is: "think of $X$ and ...
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2answers
56 views

How can I solve this expected value?

Good evening, how can I solve this expected value? $$ E \Bigl[ B_1 \int_0^{x} B_u du\ \Bigr] $$ where $B_t$ is a standard Brownian Motion and x > 0.
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22 views

Sufficient condition of boundedness of diffusion process

I came across the following statement in Sebastian Bossu's book "Advanced Equity Derivatives", page 27. He says that the time-homogeneous diffusion process $dX_t=a(X_t)dt+b(X_t)dW_t$ (coefficients ...
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2answers
311 views

Product of stochastic integral and brownian motion

I am trying to compute the following expectation: $$ M_T = \mathbb E\left[W_T\int_0^T\,t\,d W_t \right] $$ where $0<t<T$ and $W = (W_t)_{t\geq 0}$ is a standard Brownian Motion started at $0$. ...
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1answer
80 views

Show that $E[X_t^2]<\infty$

Show that $E[X_t^2]<\infty$, where $$ X_t=e^{3W_t-\frac{3t}{2}}-3e^{W_t-\frac{t}{2}}\underbrace{\int_0^te^{2W_s-s}ds}_{A_t},\quad. t\geq0, $$ where $t$ is a fixed number and $W_t$ is Brownian ...
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1answer
182 views

Can anyone solve a stochastic differential equation - related to neuroscience research?

I'm a neuroscience grad student, and I'm hoping one of ya'll could help me solve this problem regarding particle diffusion. It relates to my research on molecular-level neural plasticity, but I've ...
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1answer
148 views

Doob-Kolmogorov Inequality

Denote by $(X(t),t\ge 0)$ a standard Brownian motion, i.e random variables with the following properties: $X(0)=0$. With probability 1, the function $t\mapsto X(t)$ is continuous on $[0,\infty)$. ...
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1answer
100 views

Why is Brownian Motion B_t distributed as N(0,t)?

Almost all textbooks define a Brownian Motion ($B_t$)using three / four points: $B_0 = 0$; it has stationary independent increments; for every $t>0$, $B_t$ has a normal $N(0,t)$ distribution; it ...
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1answer
112 views

Exponential of Brownian motion with negative drift

I am reading a text on Brownian motion and don't understand the following: Let $X_t = \exp \{ W_t - \frac{t}{2} \}$, where $W$ is a standard Brownian motion on $\mathbb{R}$. Let $T_n = \inf \{ t \geq ...
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37 views

How can I solve $E[B^4_t B^3_t]$?

How can I solve the following expected value: $$ E[B^4_t B^3_t] $$ where $ B_t $ is a standard Brownian Motion.
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1answer
79 views

Prove that the following process is a Geometric Brownian motion for every constant

Having some trouble understanding this problem: Given the dynamics of the geometric brownian motion $X_t$ where $(B_t)_{t\in\mathbf{R}_{+}}$ $$ dX_t = X_tdt+X_t dB_t,$$ $$X_0=1$$ for which value of ...
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1answer
404 views

How to compute $E[W_t^4]$, with $W_t$ being a standard Wiener process

I want to compute the fourth moment of a standard Wiener process: $E[W_t^4]$. My solution is not equal to the one in my textbook but I don't understand where I am wrong. I started by applying Ito's ...
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1answer
257 views

Expected Value of the exponential of a stochastic integral

What is the expected value of the following process: $$ e^{\int_0^t B_u\, du} $$ Thanks.
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322 views

Brownian motion: first-hitting-time with double barrier [closed]

Let $(B_t)_t$ be a standard ($B_0=0$) Brownian motion , and $$ T_{a,b} = \inf\{t>0 : B_t \not\in(a,b)\} $$ where $a<0<b$. What is the expected first-passage time $\mathbf{E}[T_{a,b}]$?
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42 views

Find the value of the real $\alpha$ for which exp($2B(t) - \alpha t$) is a martingale.

I tried to answer using the three conditions to be a martingale (measurability, integrability, and martingality), validating the integrability condition, which is $$ E |e^{2B(t) - \alpha t} | < ...
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1answer
230 views

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion? The answer seems like no to me. Using Ito's lemma I can write $$W^3(t)=\frac{3}{2}W^2(t)+\int_0^t3W(u)dW(u)$$ The second piece on the LHS is an ...
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44 views

Deterministic integrals involving a Brownian motion [duplicate]

I am trying to work out the following two integrals involving a standard Brownian motion started at $W_0 = 0$. The first expression is bewildering me a bit, since it seems like somehow the Itô ...
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78 views

Deriving mean and variance of a function of Gaussian process

Suppose $\mathbb{G}$ is a tight zero mean Gaussian process and $F$ is an absolutely continuous CDF $$Y=\int_a^b\frac{d\mathbb{G}}{1-F}+\int_a^b\frac{\mathbb{G} \, dF}{(1-F)^2}$$ I know that $Y$ is a ...
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59 views

Distribution of $(\sup_{0\leq s\leq t} W_s -W_t)$

I am interest in the law of the $(\sup_{0\leq s\leq t} W_s -W_t)$ where $W$ is a standard brownian motion. I know that $M_t:=\sup_{0\leq s\leq t} W_s \overset{\mathcal L}{=} |W_t |$ so its density ...
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34 views

Intersection of two independent 1-d Brownian motions.

I am interested in the first intersection of two independent 1-d Brownian motions. More precisely, what is the joint distribution of the intersection point and intersection time? Any help is ...
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1answer
94 views

Martingale property of Brownian motion with resprect to a different filtration

Let $W$ be a Brownian motion on $(\Omega,\mathcal F,\mathbb P)$ and let $N$ be a Poisson process on the same probability space. Denote by $\mathbb F$ the filtration that is generated by $(W,N)$. Now ...
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1answer
53 views

Malliavin Derivative

Motivation : We know that, if the randomness in the system is due to Brownian Motion then any contingent claim with mean zero can be written as Ito integral. (Of course, we need to have boundedness ...
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1answer
183 views

Is square of Wiener process an orthogonal process?

I'm trying to prove: Let $t_1 < t_2 \leq t_3 < t_4$ and $(X)_t$ is the square of Wiener process. Then $E(X_{t_2} - X_{t_1})(X_{t_4}- X_{t_3}) \neq 0.$ Progress Maybe the fact $E(X_{t_2} - ...
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2answers
425 views

Law of large numbers for Brownian Motion (Direct proof using L2-convergence)

In “Brownian Motion” by Schilling and Partzsc, they give a HINT to prove the Law of Large Numbers for Brownian Motion (not in their solutions, fyi) by (1) Noting that ...
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1answer
47 views

How to identify the future distribution of a stochastic variable from its SDE

I would like to know some common practice to identify the future distribution of a random variable modelled by an arbitrary SDE. Would you study it empirically (like generating Monte-Carlo ...
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1answer
42 views

$L^{2}$ -limit of expression involving Brownian Motion

Let $(B_{t})_{t\geq0}$ be a Brownian Motion. I would like to prove that $\max_{n\leq s\leq n+1}\left|\frac{B_{s}-B_{n+1}}{n}\right|=\frac{1}{n}\max_{n\leq s\leq n+1}\left|B_{s}-B_{n+1}\right|$ ...
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2answers
77 views

Brownian Motion and Continuity

Consider a Brownian Motion $(B_{t})_{t\geq0}$. In my lecure notes it says, without proof, that $\mathbb{P}\left(\sup_{t,s\leq N}\left\{ \left|B_{t}-B_{s}\right|:\left|t-s\right|<\delta\right\} ...
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1answer
82 views

A question on integration wr.t to a local martingale

In a lemma in my graduate level course on financial mathematics uses the fact that integral of a progressive portfolio process(which is almost surely lower bounded i.e it is admissible) $\theta_t$ ...
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108 views

Existence of local time of Brownian motion

Suppose we define the local time $L_0(t, \omega)$ of the standard Brownian motion $B(s, \omega): [0,t] \times \Omega \rightarrow \mathbb{R}$ by $$ L_0(t, \omega) = \lim_{\epsilon \rightarrow 0} ...
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1answer
36 views

Covariance of Wiener Processes on the same Brownian Motion

I am trying to solve $Cov(Tw_T,\int^{T}_{0}tdw_t)=\mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t]$, my attempt is as below: \begin{split} \mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t] & ...
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47 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, ...
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1answer
55 views

Continuous in probability of hitting times

Let $(B_t)$ be a standard Brownian motion. How can we show that the process $$ \tau_t := \inf \{ s \geq 0 : B_s >t \}$$ satisfies continuity in probability? $$\bigg( \text{i.e. } \quad \lim_{h ...
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46 views

Independence of increments of some processes

I am stuck on this question: Let $(B_t)$ be a standard Brownian motion. Define $$ (\tau_1)_t := \inf \{s \geq 0 : B_s = t \} ; \quad (\tau_2)_t := \inf \{s \geq 0 : B_s > t \}. $$ Any ideas how ...
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1answer
60 views

Equality relating $L^2$ convergence and martingales

I am baffled with this question: Let $(B_t)$ be a standard Brownian motion. For any $n \in \mathbb{N}$, let $(f_n)$ be a sequence of functions defined by $$ f_n(x) = \left\{ \begin{array}{lr} ...
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45 views

Is this stochastic differential equation wrong?

The following is an old exam question I think might be misstated. Consider the SDE $$dX(u)=(a(u)+b(u)X(u))\,du+(\gamma(u)+\sigma(u)X(u))\,dW(u)$$ where $W(u)$ is a brownian motion relative to ...
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1answer
88 views

Expectation of a function of Brownian motion at a stopping time

I do not understand the following claim from my book: Let $(B_t)$ be a Brownian motion on $\mathbb{R}^d$ starting at $x$. Let $\tau = \inf \{ t>0 : B_t \in \partial B( x, r) \}$. Also, let $u ...
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2answers
97 views

For every $\epsilon>0$, the probability of $W_t>(1+\epsilon)\sqrt{t\log(t)}$ tends to $0$ as $t\to\infty$

Can anybody give a hint to show for all $\epsilon>0$ $$\lim_{t \to \infty} P \left( \frac{W_t}{\sqrt{t\log(t)}}>1+\epsilon \right) = 0$$ with $W_t$ Brownian Motion? (Or W(t), a Brownian motion ...
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1answer
74 views

Hitting time of Brownian Motion on a line

Given a 3-dimensional Brownian motion $B_t$, we know that it is transient. But how can we show that if it starts outside a straight line, it will remain outside forever with probability $1$ ? Any ...
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1answer
90 views

Path properties of Brownian Motion: relation between its maximum and hitting time

Let $B(t)$ be a Brownian motion. $$T_a=\inf\{t>0,B(t)=a\}$$ $$M(t)=\max_{0\le s\le t} B(s)$$ There is a statement in Durrett's textbook (3rd last line in page 318, 4th edition): ...
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1answer
49 views

independence of stopping time and a sigma algebra

Let $(B_t)$ be a standard Brownian motion and $\{ \mathcal{F}_t \}$ be the filtration generated by the Brownian motion. For a stopping time $\tau$, we know that $\{B_{\tau + t} - B_{\tau}\}_{t \geq ...