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
37 views

Transformation Stratonovich to Itô SDE (for BM on a surface)

The question arises from a section to Stochastic Differential Geometry in Rogers L.C.G., Williams D. Diffusion, Markov processes and martingales. Vol.2. Itô calculus. (31.22) Brownian motion on a ...
1
vote
0answers
20 views

conditional distribution : integral of BM

I have got a question and I have some ideas, but I don't know if I have got the right answer. The question is that Define $W_t=\int^t_0 B_s ds$ ,I have to get the distribution of $W_t$ conditional ...
2
votes
0answers
45 views

Electrostatic capacity of two spheres with changing radii

Although I have read a lot of questions and answers here, this is my first time actually posting. Feel free to suggest needed edits. My question is the following (in a simplified setting). All this ...
6
votes
1answer
44 views

Basic question about the stochastic integral $\int \limits_{0}^{t} X(s) \,dM(s)$

Suppose $(X_{t})_{t \geq 0}$ and $(M_{t})_{t \geq 0 }$ are stochastic processes, where the index is continuous and the probability space is $(\Omega, \Sigma, P)$. We say for each fixed $\omega \in ...
2
votes
0answers
30 views

Brownian motion, find minimum of function

Let $(\Omega,\mathcal{F},P)$ be a probability space, $(W(t),t \geq 0)$ a Brownian motion and $(\mathcal{F}(t),t \geq 0)$ its natural filtration. Suppose $0 \leq s \leq t$ and let $f:\mathbb{R} ...
1
vote
1answer
16 views

Expectation of Integral of Brownian Motion

I'm working through some stochastic analysis problems at the moment and I've come across a problem that is a bit tricky (to me) - does anyone know how to calculate this expecation? I'm not sure what ...
3
votes
1answer
100 views

$\frac{dy_t}{dt} = a \frac{dx_t}{dt} + x_t +y_t$ with $x_t$ Ornstein Uhlenbeck process - what to do? [UNRESOLVED]

I consider the following equation: $$\frac{dy_t}{dt} = a \frac{dx_t}{dt} + x_t +y_t, \tag{1}$$ where $a=$ constant and where $x_t$ follows an Ornstein Uhlenbeck process (see here under Alternative ...
0
votes
0answers
43 views

Calculating the generator of a weighted transition function

Let $(P_t)_t$ be the transition function of a Feller-Dynkin process $X$. The usual Banach space of functions that the semigroup $(P_t)_t$ is working on is $C_0(E)$, i.e. continuous functions that ...
2
votes
0answers
42 views

Cross Variation of two stochastic processes

I am currently working on a stochastic calculus exercise at the moment and I am slightly confused when it comes to finding cross variation. We are given that the process $X_t = W_t^3$ ($W_t$ is ...
1
vote
0answers
16 views

Brownian motion: Why $p\{B_u\neq 0\text{ for }0\leq u\leq t\mid B_0=a, B_t=b\}=1-e^{-\frac{2ab}{t}}$?

Let $(B_t)$ a Brownian motion. For $a>0$ and $b>0$, show that $$p\{B_u\neq 0\text{ for }0\leq u\leq t\mid B_0=a, B_t=b\}=1-e^{-\frac{2ab}{t}}.$$ In the correction they said: Let ...
1
vote
0answers
33 views

Brownian motion: Problem with some definition.

Let $(B_t)$ a Brownian motion. Let $f:\mathbb R\to\mathbb R$ such that $f\in\mathcal C^2(\mathbb R)$ and such that $f''$ is bounded. Show that $$\lim_{h\to 0^+}\frac{\mathbb E[f(B_{t+h})\mid ...
3
votes
3answers
66 views

Showing time changed brownian motion is martingale.

Let $W$ be a one dimensional Brownian motion and define, $$ X_t=W_{(\text{exp}(\beta t)-1)}\\ \hat{W}_t=\frac{1}{\sqrt{\beta}}\int_0^te^{-\frac{\beta s}{2}}dX_s $$ Show that $\hat{W}_t$ is a local ...
0
votes
2answers
31 views

Find parameters so that random variables (connected to Brownian movement) are independent.

$W_t\sim\mathcal{N}(0,t)$ is Brownian movement, find values of parameters $a, b$ for which $aW_1-W_2$ and $W_3+bW_5$ are independent. I don't even know where to start, so any hint is highly ...
1
vote
1answer
17 views

Find the distribution of some random variable connected to Wiener Process. Please, check my solution.

I need to find a distribution of $ 5W_1-W_3+W_7 $, where $W_t$ stands for Wiener Process $W_t\sim\mathcal{N}(0,t)$. Is this solution right? $E(5W_1-W_3+W_7)=5E(W_1)-E(W_3)+E(W_7)=0$ and since ...
1
vote
1answer
30 views

Ito's Isometry using Brownian Motion

Let $B_t$ be standard Brownian Motion. Could someone please help me to show that $$E[(\int_{0}^{t}B_sdB_s)^2] = \int_{0}^{t}E[B_s^2]ds$$ I am sure that it has something to do with Ito's formula but ...
0
votes
1answer
43 views

Distribution of Black Scholes call option price at time 0<t <T

Does anyone know how to find the probability law (distribution) under P* of a Black Scholes Call Option price $C_t$ for $0 < t < T $? (Under P*, $ dC_t = \frac{\partial c}{\partial s}\sigma S_t ...
1
vote
1answer
42 views

Construction on Ito Integral with Brownian Motion

I have just started learning stochastic calculus and my professor posed the following as exercises to help understand how we construct the Ito Integral. Let $B$ be a standard Brownian motion. Fix ...
1
vote
1answer
47 views

Ito Integral Properties with Brownian Motion

I am working out some of the properties for the Ito integral with Brownian motion and I am trying to use the definition to verify that $$ \int _0 ^t s \, dB_s = tB_t - \int _0 ^t B_s\, ds $$ and $$ ...
2
votes
2answers
51 views

Why $\mathbb E[B_t^2]=t\implies B_t\sim\sqrt t$?

Let $B_t$ a standard Brownian motion. Why $$\mathbb E[B_t^2]=t\implies B_t\sim\sqrt t\ \ \ ?$$
0
votes
0answers
13 views

Brownian motion: Why $p_x\{B_{T_{a,b}}=b\}=p_x\{\tau_a<\tau_b\}$?

Let $(B_t)$ a Brownian motion. I denote $\tau_a=\inf\{t\geq 0\mid B_t=a\}$, $T_{a,b}=\tau a\wedge \tau b$ and $p_x\{A\}=p\{A\mid B_0=x\}$. Why $$p_x\{B_{T_{a,b}}=b\}=p_x\{\tau_a<\tau_b\}\ \ \ ?$$ ...
0
votes
2answers
34 views

What is a valid range of applicability of Ito Lemma?

If I have e.g. such process $$ Z_{t}=t^{5}B_{t}+10\int_{0}^{t}sB_{s}ds $$ can I take $$ f(t,x):=t^{5}x+10\int_{0}^{t}sB_{s}ds $$ as a function to which I apply Ito formula? I'm concerned about ...
5
votes
1answer
37 views

$x_t := a_t -b_t c_t $ , with $dx_t = \theta (\mu-x_t) dt+ \sigma dW_t$

I would like to solve the following equation explicitly using Ito's lemma: $$ x_t := a_t -b_t c_t , $$ where $x_t$ is an Ornstein-Uhlenbeck process (see here) $$ dx_t = \theta (\mu-x_t) dt+ \sigma ...
1
vote
1answer
16 views

Brownian motion: Why $p\{-x\leq B_t\leq x \mid B_{t_n}=\pm x_n,…,B_{t_1}=\pm x_1\}=p\{-x\leq B_t\leq x\mid B_{t_n}=x_n,…,B_{t_1}=x_1\}$ [closed]

Let $(B_t)$ be a Brownian motion. Why: $$p\{-x\leq B_t\leq x \mid B_{t_n}=\pm x_n,...,B_{t_1}=\pm x_1\}=p\{-x\leq B_t\leq x\mid B_{t_n}=x_n,...,B_{t_1}=x_1\}\ \ \ ?$$
1
vote
1answer
31 views

Compute a conditional expectation with brownian motion

Let $(B_t)_{t\in [0,1]}$ be the standard Brownian Motion. Define $\mathcal{G_t}$ as $(\mathcal{F}_t\vee \sigma (B_1))_+$. Prove that for all $0s\leq t\leq 1$, $$ ...
3
votes
0answers
34 views

Does Ito's Isometry hold if the integrand has a brownian motion in it?

I am wondering what is the distribution of: $$ \int_0^tW_sdW_s $$ Solution: (Thanks to @muaddib) Applying Ito's Formula to $W_t^2$ gives $d(W_t^2) = 2W_tdW_t +dt$, and so: $$ \int_0^tW_sdW_s= W_t^2 ...
0
votes
0answers
87 views

Calculating the generator and domain of a scaled Brownian motion

Suppose we have transition functions $\{P_t\}_t$ that form a strongly continuous semigroup on a Banach space $\mathcal{S}$ (boundedness and closedness of all $P_t$). Define the domain as $$ ...
2
votes
0answers
18 views

Law of a supremum of random variables

Let $(B_t)_{t\geq 0}$ the standard brownian motion (with $B_0=0$), $p$ be a real number greater than $1$ and $q$ its conjugate number. Prove that $X_p=\sup _{t\geq 0}(|B_t|-t^{p/2})$ is a.s. strictly ...
1
vote
0answers
26 views

Schauder Basis and Fourier series

I'm looking at the constuction of the Brownian Motion given by Lévy-Ciesielki. We want to use Haar functions as basis of $L^2([0,1],\mathcal{B},\lambda)$. So on the n-th partition of $(0,1]$ ...
2
votes
0answers
18 views

Can we integrate brownian motion with respect to a deterministic function

Let $B_t$ be brownian motion at time $t$, and $x$ be some random variable. For instance, I know that $$\int_0^T 1 dB_t = 1(B_T-B_0)$$ And that $$\int_0^T \cos(B_t) dB_t$$ cannot be directly ...
2
votes
0answers
28 views

Construction of Brownian motion - differentiability

I'm working on the the construction of BM given by Lévy-Ciesielski. The author begin to prove another result and for this reason he assume that BM exists and that it is also differentiable. For this ...
1
vote
1answer
27 views

Girsanov's theorem corollary

Trying to understand the proof of the corollary on the page http://en.wikipedia.org/wiki/Girsanov_theorem It remains for me the show the equality of the quadratic variations $[W, X]_t = 2[[W, X], ...
2
votes
2answers
48 views

$\mathbb{E}[e^{2B_t -t}] = e^{2t-t}$ - why?

Let $B_t$ be brownian motion at time t. I know we can use properties of MGF, but I get a different answer: in general, for $X \sim N(\mu, \sigma^2)$ $$\mathbb{E}[\exp(\alpha X)] = \exp(\mu \alpha + ...
2
votes
0answers
52 views

Brownian Motion Hitting Times

I am reading through Walsh's Knowing the Odds book and came across this problem. Let $B_t$ be Brownian motion. Find the probability that $B_t$ hits plus one and then minus one before time one. I am ...
1
vote
1answer
49 views

Deriving a closed form expression for stochastic integral (to show its a martingale)

I have $B_s = $ brownian motion at time $s$. $$ \int_0 ^t B_s \, dB_s$$ $$0 \leq t \leq T$$ And want to check if it is a martingale, first from its closed form expression, and then via conditions on ...
3
votes
1answer
31 views

Stochastic calculus rules $d(B_t^2) = 2B_t\,dB_t + dt$ - why?

Let $B_t$ = Brownian motion at time $t$ I know that $(dB_t)^2 = dt$ and $d(f(x)) = f'(x)\,dx$ for some differentiable function. Now, I have that $$M_t = B_t^2 - t$$ $$dM_t = d(B_t^2) - d(t)$$ ...
1
vote
2answers
43 views

Why is this wrong - conditional expectation of brownian motion: $\mathbb{E}[B_1 | B_2]$

Trying to find $$ \mathbb{E}[B_1 | B_2]$$ $$\mathbb{E}[(B_1 - B_2 + B_2) | B_1] = \mathbb{E}[(B_1-B_2)|B_2] + \mathbb{E}[B_2 | B_2] $$ $$\mathbb{E}[ -(B_2 - B_1)| B_2] + B_2$$ Since $$ -(B_2 - ...
1
vote
1answer
39 views

How to check if integral wrt Brownian motion is a martingale

As in title, I have a process $$X_{t}=\int_{0}^{t}s^{2}dB_{s}$$ I found here a sufficient condition for such integral to be a martingale on the interval. But I am asked if it is a martingale, not ...
0
votes
1answer
48 views

Prove this expectation of Brownian motion?

Prove $E[(\Delta B_j)^4]=3(\Delta t_j)^2$ where the Delta stands for the change of something i.e $B_j-B_{j-1}=\Delta B_j$ and the $B_j$ stand for the standard Brownian motion I won't show my step ...
0
votes
0answers
12 views

Sample variance matlab geometric brownion motion

I have a question about the geometric Brownian motion. I want to sample many paths and then showing that the sample variance equals the exact variance: $$\mathrm{Var}\left[S(t)\right]=S_{0}^2 e^{2 \mu ...
1
vote
0answers
48 views

Brownian motion under Girsanov change of measure

i am struggeling with the following exercise Let $r,\mu,\sigma,T>0$ and consider the market model with a money-market account $B$ and one risky asset $S$ such that \begin{eqnarray*} ...
1
vote
0answers
12 views

Brownian motion, why $p\{B_{T_{x+h,x-h}}=x\pm h\}=\frac{1}{2}$?

Let $(B_t)$ a Brownien motion. Let $\tau_a=\inf\{t\geq 0\mid B_t=a\}$ (with $a\neq 0$) and $T_{a,b}=\tau_a\wedge \tau_b$. Suppose $x\in[a,b]$ and $B_0=x$. Let $h>0$ very small. Why do me have ...
1
vote
1answer
24 views

Absolute continuity and sample paths of Brownian motion

An offhand remark in Morters and Peres' book on Brownian motion says that Brownian motion is a.s. absolutely continuous on compact intervals (see page 147, immediately preceding the statement of ...
0
votes
1answer
30 views

Brownian motion: Why $p\{\max_{0\leq u\leq t} B_u\geq a\}=2p\{B_t\geq a\}$?

Let $(B_t)$ a standard Brownian motion (i.e. $B_t\sim\mathcal N(0,t)$). Let $a\geq 0$. Prove that $$p\left\{\max_{0\leq u\leq t} B_u\geq a\right\}=2p\{B_t\geq a\}.$$ The proof goes like this : ...
1
vote
0answers
22 views

jump-diffusion hitting time

Suppose I have a stochastic process $dS_t= rS_t dt + \sigma S_t dW_t + dJ_t$ where $W_t$ is a brownian motion and $J_t$ a compound poisson process of parameter $\lambda$ with lognormal jump size, ...
2
votes
0answers
91 views

Brownian motion stopped at the hitting time of an independent Brownian Motion

While I was working on the exit time of planar BM out of a square I came across the following observation, which I cannot grasp. I define this exit time as $$\tau = \inf\{t \geq 0: \lvert B(t)\rvert ...
0
votes
1answer
26 views

Problem with particular proof regarding infinite total variation of Brownian motion [closed]

I have some problems with a proof from the last page of this pdf: Brownian motion has infinite total variation. Could we say that variance is exactly $\frac{c_{1}}{n}$ for some constant $c_{1}$? ...
2
votes
0answers
32 views

2D Brownian Motion — Does this argument work?

Consider a 2D Brownian Motion $(X_1(t),X_2(t))$ starting at $(x_1,x_2) \in \mathbb{R}^2$. For every $s\geq0$, let $$\tau_s = \inf \left\{t \geq 0 \mid X_1(t) - x_1 > s \right\}\qquad Y_s = ...
0
votes
1answer
41 views

Finding $E[W]$ and $E[W^2]$, where $W = \int_{t=0}^T B_s$ $ds$

I'm trying to find a)$E[W]$ and b) $E[W^2]$, where $W_t = \int_{t=0}^T B_s$ $ds$ ($B_s$ denotes a Brownian motion). In addition, I'd like to find $E[Z_sZ_t]$, where $Z_t = \int_0^t B_s^2$$ ds$ ...
0
votes
1answer
52 views

Expectation of the product of Brownian motions

I'm new to Stack Exchange. I'd like to find the expectation of the product of three Brownian motions: $E(B(t_1)B(t_2)B(t_3))$ I know from a previous post here that ...
-1
votes
1answer
153 views

Independence of the components of a multidimensional Brownian motion

Let $B = (B^1, \dots, B^n)$ be an $n$-dimensional ($n \in \{1, 2, \dots\}$) Brownian motion (i.e. $B = (B_t)_{t \geq 0} \in \Omega \rightarrow (\mathbb{R}^n)^{[0,\infty)}$ has continuous paths, $B_0 = ...