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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45 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 ...
2
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1answer
59 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 ...
2
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1answer
52 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
139 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
44 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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0answers
18 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 ...
2
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2answers
102 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 $0$. ...
4
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1answer
62 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 ...
6
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1answer
90 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
67 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
64 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
51 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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0answers
19 views

Showing Brownian motion is measuable

How can I prove Brownian motion is measurable with respect to the corresponding product sigma algebra? I am struggling to extend the measurability from holding for rational times to all times using ...
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0answers
24 views

Solve this problem involving Geometric Brownian Process

The price of a stock follows a geometric Brownian process with annual expected return rate of 20% and volatility 50%. The initial stock price is 10 euros. Compute the probability that the stock price ...
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0answers
33 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
58 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 ...
3
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1answer
98 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
89 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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2answers
104 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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35 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
57 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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0answers
30 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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0answers
50 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 ...
2
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0answers
49 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 ...
2
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0answers
26 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 ...
1
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1answer
67 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
32 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
63 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
80 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
35 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 ...
2
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1answer
35 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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0answers
18 views

Why is $l(t,x,\omega)=\lim_{\varepsilon\downarrow 0}\frac{1}{2\varepsilon}\int_{0}^t1_{[x-\varepsilon,x+\varepsilon]}(X_s(\omega))ds$

Currently I am reading the book "Brownian motion and stochastic flow systems" (Harrison) and in chapter 1 paragraph 3 he states the following deep theorem about Brownian motion: Theorem Let ...
2
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2answers
61 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
40 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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45 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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0answers
46 views

Mistake in using Monotone convergence theorem

I am getting a contradictory result and can't find my mistake. I hope you can help. We have the following result by Spitzer (see (1) or Port) $\lim_{t\to ...
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1answer
23 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] & ...
2
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0answers
34 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, ...
0
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1answer
51 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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0answers
40 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
54 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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0answers
41 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
48 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
82 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 ...
2
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1answer
33 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 ...
2
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1answer
72 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
36 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 ...
2
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1answer
54 views

Laplace transform of stopping times

I am nearly done with a question: Let $(B_t)$ be a Brownian motion on $\mathbb{R}$. For a fixed $x >0$, let $\tau$ be a stopping time defined by $$ \tau = \inf \{t \geq 0 : B_t \not \in (-x,x) ...
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1answer
30 views

Sign of differentiable function near critical point

Background : We know that Brownian path oscillates infinitely often changing signs in any neighbourhood of $0$. I was trying to understand if this property holds because that Brownian paths are not ...
3
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1answer
71 views

Hitting time process of Brownian motion [closed]

I am stuck with this problem: Let $(B_t)$ be a standard Brownian motion in $\mathbb{R}$. For $t \geq 0$, let $$ H_t = \inf \{ s \geq 0 : B_s = t \}, \quad S_t = \inf \{ s \geq 0 : B_s > t \}. $$ ...