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 $t-s$.

learn more… | top users | synonyms

0
votes
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 ...
0
votes
0answers
31 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.
1
vote
1answer
50 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
votes
1answer
72 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 ...
1
vote
1answer
54 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.
-2
votes
2answers
68 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 $\mathbf{E}[T_{a,b}]$?
0
votes
2answers
29 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} | < ...
0
votes
1answer
45 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 ...
1
vote
0answers
24 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 ...
1
vote
0answers
35 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
votes
0answers
47 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
votes
0answers
20 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
vote
1answer
58 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 ...
0
votes
1answer
29 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 ...
-1
votes
1answer
39 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} - ...
1
vote
2answers
56 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 ...
0
votes
1answer
34 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
votes
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 $\max_{n\leq s\leq n+1}\left|\frac{B_{s}-B_{n+1}}{n}\right|=\frac{1}{n}\max_{n\leq s\leq n+1}\left|B_{s}-B_{n+1}\right|$ ...
0
votes
0answers
15 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 ...
1
vote
2answers
57 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\} ...
1
vote
1answer
25 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) $\theta_t$ ...
0
votes
0answers
26 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} ...
0
votes
0answers
44 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 ...
0
votes
1answer
18 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
votes
0answers
28 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
votes
1answer
49 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 ...
1
vote
0answers
37 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 ...
1
vote
1answer
51 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} ...
0
votes
0answers
40 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 ...
1
vote
1answer
40 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 ...
1
vote
2answers
76 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
votes
1answer
26 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
votes
1answer
64 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): ...
1
vote
1answer
30 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
votes
0answers
49 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) ...
1
vote
1answer
28 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
votes
1answer
52 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 $$ H_t = \inf \{ s \geq 0 : B_s = t \}, \quad S_t = \inf \{ s \geq 0 : B_s > t \}. $$ ...
4
votes
1answer
61 views

Stopped process of Brownian motion

I am baffled about the following problem: Let $(B_t)$ be a standard Brownian motion. Let $$ \tau:= \inf\{ t \geq 0 :B_t = x \} \wedge \inf\{ t \geq 0 :B_t = -y \}$$ be a stopping time, where $x,y ...
1
vote
1answer
48 views

Showing that the Brownian Bridge is Gaussian

Take $X_t = (1-t)B_{t/(1-t)}$ for $t\in[0, 1)$ where $B_t$ is a $1$-dimensional Brownian motion. I want to show that $X_t$ is Gaussian. I have actually never been able to find a precise definition ...
1
vote
0answers
45 views

Conditional expectation involving Brownian Bridge

I have no ideas on this problem: Let $(B_t, 0 \leq t \leq 1)$ be a standard Brownian motion in $1$ dimension. Let $Z^y_t = yt+ (B_t -tB_1)$. We call $\{Z^y_t\}_{0 \leq t \leq 1}$ a Brownian Bridge ...
1
vote
1answer
62 views

Position of Brownian motion at exit time from the upper half plane

I am currently reading some books on SLE and struggling on some problems regarding Brownian motion. For a Brownian motion in $\mathbb{R}^2$ starting from $(x,y)$, I don't know how to find the ...
3
votes
1answer
81 views

What is the explicit obstruction to almost sure convergence in stochastic integrals?

Let $B(\omega,t)$ be a Brownian motion defined on some appropriately filtered probability space $(\Omega,\mathcal{F}_{t},\mathbb{P})$, and let $f(\omega,t)$ be a stochastic process defined on $\Omega$ ...
3
votes
1answer
54 views

Determining if some random variable is a stopping time

I am stuck on this issue: Let $(B_t)$ be a Brownian motion. We know that since $\{0\}$ is a closed set in $\mathbb{R}$ and that $(B_t)$ is a continuous adapted process, $$ \tau:= \inf \{ t\geq 0 : ...
1
vote
1answer
63 views

Verifying that a certain process is not a Brownian motion

Let $B$ be a standard Brownian motion in $1$ dimension. Define \begin{equation} \tau = \inf \bigg\{ t \geq 0 : B_t = \max_{0 \leq s \leq 1} B_s \bigg\}. \end{equation} We want to show that $(B_{t+ ...
1
vote
2answers
43 views

probability of a brownian motion being equal to the running maximum

Let $B$ be a standard Brownian motion on $\mathbb{R}$. I would like to show that $$ \mathbb{P} \bigg\{ B_1 = \max_{t \in [0,1]} B_t \bigg\} =0 .$$ I argue that since $\max_{t \in [0,1]} B_t $ has the ...
1
vote
2answers
37 views

Prove that lim sup of a function belongs to a certain sigma algebra

I am so baffled with this problem: Let $B$ be a standard Brownian motion, $\{ \mathcal{F}_t \}$ be the filtration generated by the Brownian motion. I would like to show that for any $k>0$, ...
-2
votes
1answer
84 views

Solve the SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ [closed]

Solve the following stochastic differential equations $ dX_t = \frac{1}{2 X_t} dt + dB_t$ or equivalently with a transformation $Y_t = X_t^2$ $ dY_t = dt + 2 \sqrt{Y_t} dB_t$ with $Y_0 = y_0 > ...
7
votes
2answers
154 views

A simple characterization of the Brownian Motion

A well-known characterization of the Brownian Motion says that it is the only continuous process $X_t$ (defined on $[0,\infty)$) such that $P(X_0=0)=1$, $E[X_t^2]=t$, $E[X_t]=0$ for any $t\ge 0$ the ...
0
votes
0answers
52 views

continuous random walks, wiener process, ito process: “snowballing” for high enough volatility?

I'm finishing a project for my ODE class and ran into some strange behavior involving a SDE (not exactly sure how to say this, but...) generated by an Ito process, using the Wiener process. I guess ...
0
votes
1answer
101 views

Variance of integrated squared wiener process

So I'm trying to figure out the mean and variance of $X = \int_{0}^{1} W^2(t) dt $ where $W$ is the Wiener process. The mean I've worked out easily to be $\frac{\sigma^2}{2}$ but I'm having ...