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

1
vote
1answer
53 views

How to interpret the covariance matrix of Brownian motion

I'm reading Bernt Oksendal's "Stochastic Differential Equations". It says, Brownian motion $B_t$ is Gaussian Process, i.e. for all $0 \leq t1 \leq \cdots \leq t_k$ the random variable $Z = ...
1
vote
1answer
71 views

Distribution of Brownian motion before stoping time.

Let $B_{t}$ be a standard Brownian motion. Stopping time $\tau_{a} = \inf \{t \ge 0: |B_{t}| = a\}$. How to find $E[B_{\frac{\tau_{a}}{2}}]$? Or where is it possible to read about it? Thanks in ...
3
votes
0answers
80 views

Stochastic Integral and Ito Isometry

I am right now studying stochastic integral, and facing the following dilemma! I just want to check whether my understanding is right! The stochastic integral is defined by following: $I(t) ...
2
votes
1answer
87 views

Integral of square of Brownian motion with respect to Brownian Motion

While trying to compute $\int_0^TB_t^2\ dB_t$, $B$ being the standard Brownian motion, I got stuck at showing the following. $$\sum_{i=0}^{n-1}B_{t_i}(B_{t_{i+1}}-B_{t_i})^2 \rightarrow \int_0^TB_t\ ...
2
votes
2answers
64 views

Brownian Motion $dW_t \, dt=0$ proof!

I am facing a bit weird issue here. I am going through Shreeve book on stochastic calculus and faced the following theorem, while proving $dWdt=0$. $\sum_{j=0}^{n-1}(W(t_{j+1})-W(t_j))(t_{j+1}-t_j)$ ...
0
votes
1answer
194 views

Existence of a Continuous Modification of Fractional Brownian Motion

For a course on stochastic processes, I've been working on an exercise on fractional Brownian Motion. Showing that this process has a continuous modification is one of the final steps of the exercise, ...
0
votes
1answer
74 views

Radon-Nikodym Derivatives between Ito Processes

I am curious about the following problem: Let $B_t$ be a standard Brownian motion on $(\Omega, \mathcal F, \mathcal F_t, \mathbb P_a)$, where the filtration is generated by $B_t$. On a finite ...
3
votes
1answer
229 views

How to prove this Brownian motion convergence?

Let $W_t$ be a Brownian motion. How do I show the following? $$ \alpha > \frac{1}{2} \Rightarrow \lim_{t\rightarrow\infty} \frac{W_t}{t^{\alpha}} = 0 \text{ a.s.} $$ Showing convergence of this ...
0
votes
1answer
62 views

Explosion time of $dX_t=X_t(adW_1+bdW_2)$

I found in Karatzas & Shreve (1991), $dX=\sigma(X_t)dW_t$ cannot explode. But what about $dX_t=X_t(adW_1+bdW_2)$? Here $W_1$ and $W_2$ are independent. Feller's test for explosion seems to work ...
1
vote
0answers
40 views

What is the distribution of $\sup\limits_{t\geq 0}( B_t-xt)$

I would like to find the distribution of $\sup\limits_{t\geq 0}( B_t-xt)$, where $(B_t)_{t \geq 0}$ is a Wiener process and $x > 0$. I don't know how to begin. Any help is appreciated.
1
vote
0answers
50 views

Why does Brownian motion have finite $L^2$ norm?

The title might be a bit misleading. Sorry for that but here is the question. For predictable processes $X$, the $L^2$ norm over the set $[0,T]\times\Omega$ under the Doleans measure $\mu_M$, $M$ ...
1
vote
1answer
216 views

Strong Markov property of Bessel processes

I am thinking about the following: If $(B_t)_{t \geq 0}$ is a Brownian motion in $\mathbb{R}^3$, how can we show that the Bessel process (of order $3$) $(|B_t|)_{t \geq 0}$ has the strong Markov ...
1
vote
1answer
48 views

Proving that $X_t = W_t~ I (0<t\le T) + (2W_T - W_t) ~I(t > T)$ is a brownian motoin

The steps to showing that a process is a BM are as follows: (1)$X_0 = 0$ (2) $ \forall t ~~~X_t$ is continuous (3)$X_t \sim N(0,t)$ (4)$X_{t+s}-X_{s} \sim N(0,t)$ (5)$X_{t+s}-X_{s} \bot \mathscr ...
0
votes
1answer
92 views

Proving that a process is a Brownian motion by covariance and mean functions

The steps to showing that a process $(W_t)_{t \geq 0}$ is a Brownian motion (BM) are as follows: (1)$W_0 = 0$ (2) $ \forall t ~~~W_t$ is continuous (3)$W_t \sim N(0,t)$ (4)$W_{t+s}-W_{s} \sim ...
0
votes
1answer
33 views

Calculation of quadratic covariation of stopped processes

I am stuck in computing the quadratic covariation of the following two processes: Let $0< y <r$ and let $(B_t)$ be a Brownian motion started at $y$. Let $T_0 = \inf \{ t \geq 0 : B_t = 0 \}$ ...
3
votes
1answer
259 views

Conditional expectation of Wiener process

I want to calculate $E(W_t | W_1)$, $E(W_t^2 | W_1)$ and $E(W_t^2 | W_1, W_2)$, where $(W_t)_{t\geq0}$ is a Wiener process. For the first one I used the conditional distribution formula for the ...
1
vote
2answers
160 views

Is this stochastic process a martingale?

I have the following process: $X_t=tB_t-\int^{t}_{0}B_s \ ds$ where $B_t $ is a Brownian motion. Is this a Gauß-process and/or a martingale? Can someone help me with this? And how can I calculate ...
5
votes
1answer
66 views

Compute the distribution of $\int_0^1 B_t dt$

I need an help with the following: let $(B_t)_t$ a Brownian motion. Compute the distribution of $X:=\int_0^1 B_t dt$. Integrating by parts we have that: $$\int_0^1 B_t dt=B_1-\int_0^1 t dB_t.$$ Now, ...
1
vote
0answers
41 views

Is {Yt} a Brownian motion?

Suppose {B(t)} and {B˜(t)} are two independent standard Brownian motions and ρ is a constant, −1 < ρ < 1. The process Y(t) = ρB(t) + sqrt(1- ρ^2)*B˜(t) is distributed as a normal random variable ...
1
vote
0answers
37 views

Proving a process is a P Brownian Motion

Let $X_t = tW_{\frac{1}{t}} \forall t>0$ and $X_0 = 0$. I am trying to show that this process is a brownian motion under some measure P. I have shown that it is continuous and that it is ...
2
votes
0answers
57 views

Probabilities related to Brownian excursion

I am reading a paper that uses a fact about Brownian excursion which I don't understand. Let $(E_t)$ be a standard Brownian excursion, i.e. $E_t = X_t + i R_t$, where $X$ is a standard real Brownian ...
0
votes
1answer
39 views

Brownian Conditional Probability

In one of the question on forum $Pr (Z(1) < 0, Z(2) < 0)$ is calculated. I have a slightly similar question $Pr ( Z(1) < 0, Z(2) < 0, Z(3) < 0)$ ? $Z(1), Z(2) \; and \; Z(3)$ are ...
1
vote
1answer
150 views

Integration by parts formula for Wiener integral

Hi I need an help understanding "integration by parts" in Wiener integral. I've defined this integral as in the following: let $T=[0,t]\subset \mathbb R$ we want to define $\int_T f(s) dB_s$ where ...
4
votes
2answers
176 views

Is $\mathbb{E}\exp \left( k \int_0^T B_t^2 \, dt \right)<\infty$ for small $k>0$?

Suppose that $B$ is a Brownian motion. Does it hold that \begin{equation} \mathbb{E}\left[\exp\left(k\int_0^T[B(t)]^{2}\,dt\right)\right] <\infty\text{ ?} \end{equation} for some positive constant ...
2
votes
0answers
66 views

Pathwise integral of $W^{-a}$

Denote by $\tau(x) := \inf \{t \ge 0, W_t=x\},$ where $W_t$ is a Wiener process started at $W_0 = w_0 > 0$ and I would like to show that for any $a>1$ it almost surely holds that ...
4
votes
1answer
275 views

running maximum of brownian motion and reflected brownian motion

Hi I am learning the theory of Brownian Motion using Morters and Peres' book (http://www.stat.berkeley.edu/~peres/bmbook.pdf). Let $B$ be 1-dim standard Brownian motion and $M(t):=\max_{0\le s\le t} ...
1
vote
1answer
212 views

Proof that $p$-th total variation of a brownian motion is $0$ while $p>2$

The p-th total variation is defined as $$|f|_{p,TV}=\sup_{\Pi_n}\lim_{||\Pi_n||\to n}\sum^{n-1}_{i=0}|f(x_{i+1}-f(x_{i})|^p$$ And I know how to calculate the first total variation of the standard ...
2
votes
1answer
80 views

Brownian bridge with multiple possible end values

Brownian bridge $Z_t$ is a diffusion process distributed as Brownian motion $B_t$ conditioned on the event $B_1 = 0$. It is rather well-studied, and allows for a Markov-like SDE representation. I ...
2
votes
0answers
49 views

Lebesgue Measure of “excursions” of Brownian Motion

I know that the set $S$ where a standard Brownian motion $M:=B[\mathbb{R}]$ attains a strict local minimum is a.s. dense in $\mathbb{R}$. For every point $s \in S$, consider the interval $(s,t)$ such ...
1
vote
0answers
27 views

Probability of the maximum of a reflecting Brownian motion [duplicate]

Let $\{W_t\}_{t\geq 0}$ be a standard Brownian motion (starting at $0$). For $T$ large enough, I would like to prove that $P(\max_{t\in[0,T]} |W_t| \leq c T^{1/3})$ is bigger than a negative power of ...
-1
votes
1answer
64 views

increment of Brownian motion squared [closed]

$(W_t)_{t \geq 0}$ is Brownian motion, assume t>s, does $E[(W_t-W_s)^2W_s^2]=(t-s)s$ ? In other words, are $(W_t-W_s)^2$ and $W_s^2$ independent?
1
vote
1answer
70 views

Lower bound on the probability of the maximum of a reflecting Brownian motion

Let $\{W_t\}_{t\geq 0}$ be a standard Brownian motion (starting at $0$). For $T$ large enough, I would like to prove that $P(\max_{t\in[0,T]} |W_t| \leq c T^{1/3})$ is bigger than a negative power of ...
1
vote
0answers
94 views

Multiple absorbing boundaries

I am interested in the relation between absorbing boundaries and the trajectories of particles (evolving according to a Brownian motion). The probability to hit a boundary at a given time can be ...
0
votes
0answers
37 views

Brownian Motion covariance function - I can't the see steps in this proof.

The following is taken from Selfsimilar Processes by Paul Embrechts: I have only seen this before as: $\ E[B(t)B(s)]$ (without the ' prime next to the $\ B(s)$) The technique I have seen before ...
0
votes
2answers
93 views

Problem with a Brownian motion, Ito's formula and an indicator function

so I have done the first part of this question (it is at the bottom), but I have no clue how to do the second part. I think I understand the theory, but I do not know how to apply it. Any help would ...
2
votes
1answer
23 views

Calculating conditional expectation

I have to calculate $E\left(\int_1^4 W_t^3dt |\mathcal F_2\right)$ My solution: $E(\int_1^4 W_t^3dt |F_2)=E(\int_1^2 W_t^3dt |F_2)+E(\int_2^4 W_t^3dt |F_2)=\int_1^2 W_t^3dt+\int_2^4 E(W_t^3 |F_2)dt$ ...
3
votes
0answers
198 views

Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution? $$ \int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau} $$ where ...
0
votes
2answers
43 views

Brownian Motion independent increment computation

One can rather easily show that $E\left[\sum\limits_{i = 0}^{i = n - 1}W_{t_i}(W_{t_{i + 1}} - W_{t_i})\right] = -T + W_T^2$. What I'm confused about is why we can't simply say that for each $i$, ...
1
vote
0answers
26 views

A measure for the “typicalness” of a Brownian path

Suppose I have a continuous function $f:[0,1]\to\mathbb{R}$, and I wish to measure somehow how similar it is, in some sense, to a Brownian motion $\{B(t)\mid t\in[0,1]\}$ (with $B(0)=0$). I was ...
1
vote
0answers
48 views

Application of Girsanov theorem

Let $f(t)$, $t \geq 0$ be a smooth function with $f(0) = 0$ and let $B(t)$, $t \geq 0$ be a brownian motion. Let $P$ and $Q$ be two measures on $C[0,1]$ corresponding to respectively, $B(t)$, $t \geq ...
0
votes
1answer
108 views

Computing quadratic variation and criteria for Brownian motion

Let $f(t)$ be a nonrandom and continuously differentiable function and $B(s)$ be the brownian motion. a) Computer the quadratic variation of : $X(t) = f(t)B(t) - \int_0^t f'(s)B(s)ds$ b ) For ...
3
votes
1answer
55 views

What does $s$ and $t$ stand for in this definition of fractional brownian motion?

$$B_H(t_2,\omega)-B_H(t_1,\omega) = \frac{1}{\Gamma(H+1/2)}\Bigg\{\int_{-\infty}^{t_2}(t-s)^{H-1/2}dB(s,\omega)-\int_{-\infty}^{t_1}(t-s)^{H-1/2}dB(s,\omega)\Bigg\}$$ It's taken from Mandelbrot & ...
1
vote
1answer
163 views

Probability that Brownian Motion hits $t+1$ before $t-1$

Compute the probability that a brownian motion starting at $0$ hits the line $t+1$ before the line $t-1$. Here is what I did: I figured it has to do with optional stopping theorem. The ...
4
votes
0answers
214 views

Using Girsanov theorem to prove density of stopping time

Let $B$ be a standard Brownian motion and for $a>0$ and $b>0$, and set $$\sigma_{a,b} = \inf\{t\,:\, B_t + bt = a\}.$$ There are at least two ways to solve the following problem (the other one ...
5
votes
1answer
123 views

Prove identity in law for stochastic process driven by Brownian Motion

Let $B = (B_t)_{t\geq 0}$ be a standard brownian motion started at $0$. Consider the two following stochastic equations: \begin{equation} \begin{split} dX_t &=& (13 + 2X_t)\,dt + (6 + ...
1
vote
1answer
70 views

Sufficient and Necessary condition for the sum of brownian motions to be a brownian motion

The question has two parts : Part a): Let $(B_1(t), B_2(t), B_3(t))$ be standard a brownian motion in $R^3$. Write down necassary and sufficient condition for $\sum\limits_{i=1}^3 a_iB_i(t)$ to be a ...
0
votes
1answer
92 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
votes
1answer
70 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
votes
1answer
150 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?
1
vote
1answer
149 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 ...