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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2
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1answer
156 views

One correlated Stochastic Integral

If $${\rm Cov}[dW_t,dB_t]=\rho dt$$ then what is $$\mathbb{E} \left[\int_0^t\sigma_{1s}dW_s \int_0^t\sigma_{2s}dB_s\right]$$ where $\sigma_{1s}$ and $\sigma_{2s}$ are two deterministic functions ...
1
vote
1answer
453 views

calculate the conditional distribution of brownian motion

Suppose $W=(W_t)$ is a Brownian Motion with respect to a filtration $(\mathcal{F}_t)$. How can I compute the conditional distribution of $W_{t+h}$ given $\mathcal{F}_t$. I started like this: ...
4
votes
1answer
1k views

Expectation value of a product of an Ito integral and a function of a Brownian motion

this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated. Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
1
vote
2answers
250 views

question about the bracket process of brownian motion

Suppose I have a multidimensional brownian motion $W=\{W_t\}$. Why is the following true: $$\langle W^k,W^l\rangle_t = \delta_{k,l}t$$ where $W^k$ denotes the k-th coordinate, $\langle ...
1
vote
1answer
1k views

Questions and Solutions in Brownian Motion and Stochastic Calculus?

I am currently studying Brownian Motion and Stochastic Calculus. I believe the best way to understand any subject well is to do as many questions as possible. Unfortunately, I haven't been able to ...
0
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3answers
1k views

Simulation of 2-dimensional Brownian motion

I am trying to simulate (for the first time) a 2-dimensional SDE, in Matlab. $$X(t)=F(t,X(t))\,dt + \sigma(t,X(t))\,dBt$$ I have no problem using the Euler-Maruyama method in the one dimensional ...
2
votes
1answer
211 views

The joint distribution of zeros of brownian motion

Let $\gamma_t$ be the last zero of brownian motion before $t$ and $\beta_t$ be the first zero after $t$. I need to calculate the joint distribution of $\gamma_t$ and $\beta_t$, i.e. $P(\gamma_t<x, ...
0
votes
2answers
721 views

Lottery “Sum” forecasting

I was wondering if anyone can provide some mathematical insights to forecasting the "SUM" in this link as a time series. It is an oscillatory, range bound and poisson distribution. How can Monte Carlo ...
0
votes
0answers
80 views

A Brownian motion starting from 0, it becomes -1 once reach -1, what is its expectation?

$X_t$ is a Brownian Motion, it reaches becomes -1 forever once it reaches -1. Mathematically, $T = \inf\{t: W_t = -1\}$ is a stopping time. When $t<T$, $X_t = W_t$ ,while $t\geq T$, $X_t = -1$. ...
9
votes
2answers
1k views

Hitting time of Brownian Motion with a drift

Let $X_t =x+bt+\sqrt{2}W_t$, where $W_t$ is a standard Brownian motion. Let $T=\inf\{t: |X_t|=1\}$. I am trying to find $\mathbb{E}[T]$ for the case $b\neq0$. Firstly, I am going to apply Girsanov to ...
1
vote
2answers
430 views

Independent increments in squared brownian motion

I'm trying to prove that the squared Brownian motion $(W_t^2)$ doesn't have independent increments, I tried using the covariance and it doesn't quite work, can anybody give me any pointer to how to ...
3
votes
1answer
412 views

$\mathcal{F_t}$-martingales with Itô's formula?

I need a little help with a problem. I am given some stochastic processes and supposed to show that they are $\mathcal{F_t}-$martingales. The first one is this, and they all look similar: ...
1
vote
1answer
379 views

Derivation of an alternative representation of the Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck process can be defined as: $X_t = e^{-\lambda t} \left( X_0 + \int_0^t e^{\lambda s} dB_s \right)$ where $\lambda > 0$ and $\{ B_t \}_{t \geq 0}$ is the standard Brownian ...
1
vote
1answer
109 views

Independence of $B(t)-B(s)$ with respect to $\mathcal{F}_s$

Assume that $X,\{Y_\alpha\}$ (where $\alpha \in A$, $A$ can be uncountable) are random variables. If $X$ and $Y_\alpha$ are independent for all $\alpha \in A$, i.e., $\sigma(X)$ and ...
3
votes
0answers
119 views

A question regarding the strong Markov property

In our lecture on Brownian motion & stochastic calculus we proved: If $ X $ is a canonical RCLL process having the strong Markov property and $ \tau $ is a stopping time with $ \tau < + \infty, ...
5
votes
2answers
631 views

Dominated convergence problems with Wald's identity for the Brownian Motion

In the course of proving Wald's second identity $E(B^2_T)=E(T)$, where $(B_t)_{t\geq0}$ is the Brownian motion and $T$ is a stopping time with $E(T)<\infty$, I got stuck with the following problem. ...
2
votes
1answer
212 views

$\mathbb{E}[e^{B_t}|F_s]$: expectation of some Brownian motion

I am supposed to find the following ($B_t$ is a Brownian motion, and $\mathcal{F}_s$ the generated filtration): $$\mathbb{E}[e^{B_t}|\mathcal{F}_s]=?$$ I tried this: shifting by $s$ to the left to ...
11
votes
3answers
2k views

Can I apply the Girsanov theorem to an Ornstein-Uhlenbeck process?

Let $W_t$ be a standard Brownian motion, and $X_t$ a measurable adapted process. Girsanov's theorem says that under certain conditions, the Brownian motion with drift $Y_t = W_t - \int_0^t X_s\,ds$ ...
0
votes
1answer
79 views

Probability that a Brownian Particle will Exit a Certain Part of the Slit Open Unit Disk

The exact statement of the problem is: For each z in the slit open unit disk, find the probability that a Brownian particle will exit the region through the circular part of the boundary. So this is ...
4
votes
1answer
866 views

Variance of product of Brownian motions

Let $\{B_{t}\}_{t\geq0}$ be Brownian motion. What is the variance of $B_{t}B_{s}$?
0
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1answer
334 views

$d$-Dimensional Brownian Motion Martingales

Let $d > 1$ and let $W_t$ denote a standard $d$-dimensional Brownian motion starting at $x\neq 0$. Let $M_t = \log|W_t|$ for $d = 2$, and $M_t= |W_t|^{2-d}$ for $d > 2$. Show that $M_t$ is a ...
4
votes
2answers
279 views

Question about an exercise in Revuz/Yor

I'm solving exercise 2.28 in Revuz/Yor. I was able to prove 1). Unfortunately at 2) I got stuck. I have to show: Let $B$ be a d-dimensional Brownian motion and $A\in \mathcal{A}:=\cap_t ...
5
votes
3answers
1k views

On hitting time of Brownian motion and Ito's lemma

I have two possibly related questions. Let $\tau:=\min\{t\geq0:B_t=1\}$, where $B_t$ is a standard Brownian motion. I am supposed to derive the fact that $\mathbf{E}\tau=\infty$ by applying some ...
1
vote
1answer
732 views

Independent increments of Brownian Motion

Suppose we have the $(W_t)$ Brownian Motion and the filtration $F=(\mathcal{F}_t)$, where $\mathcal{F}_t:=\sigma(W_s;s\le t)$. I know that for any $n\in \mathbb{N}$ and $0\le ...
15
votes
2answers
949 views

Brownian bridge expression for a Brownian motion

Let $B_t$ be a standard Brownian motion in $\mathbb R$, then the Brownian bridge on $[0,1]$ is defined as $$ Y_t = a(1-t)+bt+(1-t)\int\limits_0^t\frac{\mathrm dB_s}{1-s} $$ for $0\leq t<1$. Here ...
0
votes
0answers
38 views

Model stochastic processes

I'd like to model something similar to Brownian motion, where particles behave according to a microscopic local law. I'd like to examine the effective macroscopic behaviour. What sort of mathematical ...
1
vote
1answer
377 views

Continuity of the Ito integral

Let $B_t$ be a 1-dimensional Brownian motion. I am following "Stochastic Differential Equations" by Bernt Øksendal. On the page 32 (it is displayed in the link I've put) there is a proof of existence ...
1
vote
2answers
184 views

Distributions of Stochastic Processes

I learned that Brownian motion is any stochastic process $(W_{t})_{t\geq0}$ that satisfies four well-known properties. But I still don't understand how these four properties uniquely determine ...
3
votes
0answers
197 views

Expected time spent in the set

An exercise 2.14 from Bernt Øksendal's "Stochastic Differential Equations": Let $B_t$ be $n$-dimensional Brownian motion and let $K\subset \mathbb R^n$ have zero $n$-dimensional Lebesgue measure. ...
7
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1answer
322 views

Definition of the Brownian motion

The way I understood the definition of a Brownian motion $B_t$ in $\mathbb R$ is that it consists of two parts: We first define the finite-dimensional distributions $$ ...
3
votes
1answer
245 views

Stopping time on Wiener Process

Let $W_t$ be a Wiener process and for $a\geq0$ $$\tau_a:=\inf \left\{ t\geq0: |W_t|=\sqrt{at+7} \right\}.$$ Is $\tau_a<\infty$ almost everywhere? What about $E(\tau_a)$ then?
4
votes
1answer
126 views

maximum of a brownian motion and its integral

Let $W_{t}$ be a brownian motion and $$ W^{*}_{t} = \max_{s<t} W_{s} $$ Then can you please explain why we have this: $$ (W^{*}_{t} - W_{t})dW^{*}_{t} = 0 $$
3
votes
1answer
375 views

minimum of hitting time of a brownian motion

Let $Y$ be an exponential random variable with rate parameter $\lambda$. Let $T_{a}$ be the first hitting time of a Brownian Motion. I want to find $$ P(\min(T_{a}, T_{-a}) < Y) $$ In order to ...
1
vote
1answer
511 views

Convergence of the exponential martingale

How can we show that this martingale $$ e^{aW_{t} - \frac{1}{2}a^2t}$$ converges to $0$ as $ t \rightarrow \infty$ using law of iterated logarithm, for $a \neq 0$.
5
votes
3answers
1k views

expected value of brownian motion

How can you find this expected value? $$ \mathbb{E}[|W_{t}^2 - t|] $$ where $W_{t}$ is a brownian motion.
10
votes
2answers
272 views

Problem about partial sum of exponential random variable

Let $X_1, X_2, \dots,X_n, X_{n+1}$ be independent random variable of exponential distribution, and the mean is 1. Let $S_i = X_1 + \dots + X_i$ I want to know ...
2
votes
1answer
137 views

Interpretation of Notation of Laplacian and Brownian Motion

Let $f: \mathbb{R}^d \to \mathbb{R}$ be a twice differentiable function. In particular, $\Delta f$ is well defined. Let $W := (W_t)_{t \geq 0}$ be a $d$-dimensional standard Brownian Motion. ...
2
votes
1answer
456 views

Difference between a Brownian Motion and the root of its square

Let $W_{t}$ be a Wiener Process (a Brownian Motion starting at $W_{0} = 0$). What is the difference between $W_{t}$ and $\sqrt{W_{t}^{2}}$? Using the Ito formula (in differential notation), ...
2
votes
1answer
145 views

Length of Wiener Sausage

I am deriving a formula for a volume of Wiener sausage in one dimension. $$\mathbb{E}[\operatorname{vol}(W(t))] = 2r+\sqrt{\frac{8t}{\pi}}$$ where $W(t) = \bigcup_{s\leq ...
2
votes
1answer
601 views

Absolute value of Brownian motion

I need to show that $$R_t=\frac{1}{|B_t|}$$ is bounded in $\mathcal{L^2}$ for $(t \ge 1)$, where $B_t$ is a 3-dimensional standard Brownian motion. I am trying to find a bound for ...
0
votes
0answers
344 views

Brownian Bridge. Law of a process

Let $(B_t , 0 ≤ t ≤ 1)$ be a standard Brownian motion in 1 dimension. We let $(Z_t^y = yt + (B_t − tB_1 ), 0 ≤ t ≤ 1)$ for any $y \in R$ and call it the Brownian bridge from $0$ to $y$. Let $W_0^y$ be ...
0
votes
0answers
367 views

Min and Max of Geometric Brownian motion

I am trying to derive the distribution of $M_X(t) = \max\limits_{0\leq s\leq t}X(s)$ and $m_X(t) = \min\limits_{0\leq s\leq t}X(s)$, where $dX(t)=\mu X(t) dt+\sigma X(t)dB(t)$ and $B(t)$ is standard ...
2
votes
1answer
330 views

Expectation of an integral of the minimum of a Brownian motion and a constant

I would like to compute the expectation of the following expectation $\mathbb{E}[\int_a^\infty e^{-rt}\min(x_t,c)\,dt]\,$ where a, r, c are constants, $dx_t = \mu x_t dt + \sigma x_t dW_t$ is a ...
4
votes
0answers
160 views

Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion

Let $ 0 < r < 1$, fix $x > 1$ and consider the integral $$ I_{r}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$ In the investigation of ...
3
votes
1answer
101 views

$\varlimsup_{t\rightarrow\infty} \frac{B_t}{\sqrt{t}}>0$

I am trying to prove the following statement about the standard Brownian Motion: $\varlimsup_{t\rightarrow\infty} \frac{B_t}{\sqrt{t}}>0$. I know that it is trivial to prove the above statement by ...
5
votes
1answer
901 views

Quadratic Variation of Brownian Motion

Quadratic Variation of a Brownian motion $B$ over the interval $[0,t]$ is defined as the limit in probability of any sequence of partitions $\Pi_n([0,t])=\{0=t^n_0<\cdots<t^n_{k(n)}=t\}$ of the ...
1
vote
1answer
700 views

confusion about the multi-dimensional Brownian motion

I am confused on the multi-dimensional Brownian motion. $B_t$ is a standard Brownian motion based on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})$ if ...
4
votes
1answer
340 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
2
votes
0answers
112 views

Ruin probability

Let $X_t$ be a solution of the stochastic differential equation $$ dX_t= -\frac{c-1}{2 X_t}dt+ dB_t, \, \qquad X_0=x_0$$ where $c$ is a real constant and $B_t$ is a Brownian motion. Can you give me ...
8
votes
2answers
549 views

Sobolev meets Wiener

Even though the Wiener process (Brownian motion) is continuous, it has no derivative at any point. Does it at least have weak derivatives?