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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Conditioning on Brownian motion

I was reading on conditional probability with respect to a partition of a sample space, and I came across the following example: Let $(N_t:t\geq0)$ be the Poisson process. Given fixed times $0\leq ...
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Diffusion Constant of a 1D Random Walk

Brownian motion(Wiener process) is a limit of Random walk. What is the diffusion constant for a Brownian motion that is a limit of a 1D Random Walk, with $\frac{1}{2}$ probability of moving to each ...
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Extension of Cameron-Martin formula via monotone class theorem

my question revolves around the Cameron-Martin theorem: Let $(\mathcal{C}_{(0)}[0,1],\mathcal{B}(\mathcal{C}_{(0)}),\mu)$ be the Wiener space (i.e. continuous functions starting in $0$, equipped ...
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1answer
39 views

Applying the Multivariate Ito Formula

I want to show that the stochastic process $$ S_t^i = S_0^i \exp\left( \int_0^t \left(\mu_s^i - \frac{1}{2} \sum_{j=1}^m (\sigma_s)^{ij} \right)^2 d s + \sum_{j=1}^m \sigma_t^{ij} S_t^i dW_t^j ...
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$\limsup$ of Brownian Motion Time Integral

The following are well-known: $\limsup_{t\rightarrow \infty} \frac{B(t)}{t} = 0$ $\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt t} = \infty$ $\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt ...
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Wiener process on 2D plane

You have a particle on a 2D plane. It starts at the origin (0,0). The particle moves according to the Wiener process (standard Brownian Motion) in both X and Y direction independently. What's the ...
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1answer
53 views

Donsker's theorem and multidimensional CLT

I want to prove that the linear interpolation $X_t^n(\omega):=\frac{1}{\sqrt{n}}\sum_{k=1}^{[nt]}{Y_k}(\omega)+\frac{1}{\sqrt{n}}Y_{[nt]+1}(\omega)(nt-[nt])$ of $\sum_{k=1}^{n}{Y_k}(\omega)$ for r.v. ...
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217 views

Conditional expectation $\mathbb E\left(\exp\left(\int_0^tX_sdB_s\right) \mid \mathcal F_t^X\right)$

I have found a theorem (see below) in two papers an I try to figure how it could be proved. The result seems to be intuitive, but I'm not able to prove it in a rigorous way. Assumptions: Consider a ...
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1answer
32 views

Joint distribution of $(W(1),W(3),W(3)-W(2))$ for a Brownian motion $(W(t))_{t \geq 0}$

Let $(\Omega,\mathcal{F},P)$ be a probability space, $(W(t),t \ge 0)$ a Brownian motion and $(\mathcal{F}_t,t \ge 0)$ its natural filtration. What is the joint probability distribution of ...
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83 views

Moment generating function of $(W_T, \max W_t)$

Does there exist an explicit formula for the moment generating function $\psi(u, v) = E e^{u W_T + v M_T}$ of the pair $(W_T, M_T)$ where $M_T = \max_{0\leq t\leq T} W_t$? Using the well-known pdf of ...
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1answer
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Calculating $ \mathbb E \left[e^{-\mu W_T } 1_\left( {\min W_t \leq a} \right) \right]$ for a Wiener process

Let $W_t$ be a standard Wiener process, $a$ some real number, and $\chi (x)$ the indicator function. I am trying to calculate the following expectation: $$ \mathbb E \left[e^{-\mu W_T } \chi \left( ...
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1answer
23 views

Variance of an integral of Brownian Motion

Let $W(u)$ $(u \geq 0)$ be a Brownian motion on a probability space $(\Omega, \mathscr{F}, \mathbb{P})$. Let $I(T) = \int_0^T W(u) du$. One can use integration by parts to show that $I(T) = ...
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1answer
39 views

$ P(W_t - W_\tau > 0 \text{ and } \tau <t) = \frac{1}{2}P(\tau < t) $ for a stopping time $\tau$

Let $W_t$ be a standard Wiener process and $\tau = \min \lbrace t \geq 0 :W_t \geq a \rbrace$, the first time the process reaches level $a$. By symmetry of the Gaussian distribution we have $$ P(W_t ...
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Mean and variance regime-switching model

Suppose we have the following model for stock price: $$ X_{t}=X_{0}\exp\left(\int_{0}^{t}(r-\frac{1}{2}\sigma_{\epsilon(s)}^2)ds+\int_{0}^{t} \sigma_{\epsilon(s)}dW_{s}\right) $$ This follows a normal ...
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1answer
57 views

The quadratic variation of $B \cdot B$, where $B$ is a Brownian motion

Let $B$ be a standard, one-dimensional Brownian motion. Can I show that $[B \cdot B] = B^2 \cdot [B]$, using the "fundamental identity of stochastic integration", namely that $[H \cdot X, Y] = H \cdot ...
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23 views

The length of the set which will be covered by Brownian motion in a time $t$

I have the following question in mind which I wanted to answer: what is the measure of the set which will be covered by a standard Brownian motion $B(t)$ in a time $t$? Call this random variable ...
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1answer
48 views

Show that $(W_t)_{t \geq 0}$ and $(W_t^2-t)_{t \geq 0}$ are not uniformly integrable

I'm considering the following martingale $M_t:=W_t^2-t,\ t\geq 1$, where the $W_t$ is a Brownian motion. I want to prove that this martingale and the Brownian motion are not uniformly integrable. I ...
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63 views

$\sigma+$-field of a Brownian motion

For a standard Brownian motion define \begin{align}\mathcal{F}_{0+} &= \bigcap_{t>0} \mathcal{F_t},\\ \mathcal{F_t} &= \sigma(W_s, 0 \le s \le t)\end{align} Which of the following ...
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1answer
27 views

Expected Value and Variance of a GBM Function

What is the the expected value of the process $Y = X^{3}$, where X satises the SDE $$ dXt = −X_tdt + σX_tdB_t $$ $(σ > 0)$ and $X_0 = 1$ I have two different answers: 1) I know that $X_t$ is a ...
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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 ...
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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 ...
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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 ...
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1answer
41 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 ...
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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} ...
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1answer
15 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 ...
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1answer
99 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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3answers
64 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 ...
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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 ...
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1answer
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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 ...
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1answer
28 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 ...
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1answer
42 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 ...
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1answer
40 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 ...
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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 $$ ...
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2answers
50 views

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

Let $B_t$ a standard Brownian motion. Why $$\mathbb E[B_t^2]=t\implies B_t\sim\sqrt t\ \ \ ?$$
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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 $$p_x\{B_{T_{a,b}}=b\}=p_x\{\tau_a<\tau_b\}\ \ \ ?$$ ...
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2answers
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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 ...
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1answer
35 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 ...
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1answer
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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: $$p\{-x\leq B_t\leq x \mid B_{t_n}=\pm x_n,...,B_{t_1}=\pm x_1\}=p\{-x\leq B_t\leq x\mid B_{t_n}=x_n,...,B_{t_1}=x_1\}\ \ \ ?$$
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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 $0s\leq t\leq 1$, $$ ...
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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 ...
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83 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 $$ ...
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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 ...
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25 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 $(0,1]$ ...
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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 ...
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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 ...
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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], ...