Question 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

3
votes
1answer
222 views

linear combination of two Wiener processes

I have a question concerning the linear combination of two Wiener processes (please see http://en.wikipedia.org/wiki/Wiener_process for a definition). Let $W$ and $\tilde{W}$ be two Wiener processes ...
3
votes
1answer
162 views

Fixed-Time Brownian Motion Exit Probabilities

A standard computation using martingale techniques allows us to compute probability that a Brownian motion started at zero exits the interval $[-a,b]$ ($a, b > 0$) at $-a$ or $b$. It appears to me ...
3
votes
1answer
149 views

convergence ito integral

It is easy to calculate the integral $\int_0^T B_t \, dB_t=\frac{1}{2}B_T^2-\frac{1}{2}T$ That means I showed that $\int_0^T S_n \, ...
3
votes
1answer
137 views

Brownian motion and hitting frequency

Suppose we have a Brownian motion $B_t$ with $B_0 = 0$ and $B_t - B_s \sim N(0,t-s)$. Every time $B_t$ hits $\pm h$, where $h$ is some "barrier" $>0$, I pay someone £1 and the brownian motion ...
3
votes
2answers
451 views

Show that this process is a martingale

Let $B_t$ be a Brownian motion and $M_t=\max_{0\leq s\leq t}B_s$. Show that: $$(M_t-B_t)^4-6t(M_t-B_t)^2+3t^2$$ is a martingale for $t\geq0$.
3
votes
1answer
220 views

Autocorrelation of scaled Wiener process?

If instead of a regular Wiener process $W_t$, we had a process of the form $X_t=g(t)W_{at}$ where $g$ is continuous and deterministic and $a$ is a deterministic scalar, then what is the ...
3
votes
1answer
172 views

expectation of a process of a multidimensional brownian motion

Let $B(t)=(B_{1}(t),B_{2}(t),B_{3}(t))$ be a standard three dimensional Brownian motion (i.e. it has independent components and starts at the origin). Now let $a=(a_{1},a_{2},a_{3})\neq(0,0,0)$ be a ...
3
votes
1answer
493 views

Partial Derivative of an Integral

If $f(t)$ is a deterministic function of $t$ and $B_{n}$ is a brownian motion and: $Z =\int^t_0 f(s)dB(s)$ How does one take the partial derivatives wrt to $t$ and $B_n$ on an integral like this? I ...
3
votes
1answer
134 views

Integrating a Brownian Bridge conditioned above a linear boundary

The Setup: A Brownian Bridge $B$ is a Brownian Motion on time interval $[0, 1]$ conditioned such that $B(0) = B(1) = 0$. I have a function $f(t) = mt+b$ with $m, b$ set such that $C(t) \le 0$ for $t ...
3
votes
1answer
168 views

Klenke's construction of Brownian motion

Why does Klenke's concise construction of Brownian motion via probability transition kernels satisfy the motion's characterizing properties, equations (14.17) and (14.18)? (results referenced in the ...
3
votes
1answer
321 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 ...
3
votes
0answers
41 views

Conditional expectation and coupled set of ODEs

How to find a coupled set of ODEs and initial conditions for the deterministic functions $a$ and $b$ such that $$\mathbb{E}\left[e^{-\int_{t}^{T} W^2(u)du} | \mathcal{F(t)}\right] = e^{-a(T-t) - ...
3
votes
0answers
40 views

When is a continuous path stochastic process be representable as diffusion or Ito process?

When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
3
votes
1answer
101 views

What is the probability a random walk hits x before it hits y?

This problem was motivated by my bitcoin trading and recalling some of my math education back in the day. I thought I'd ask people who know this much better than I... Suppose there is a continuous, ...
3
votes
0answers
42 views

Upper bounds on the sum in a Martingale process

My question is related the hitting time of not a random walk, but a more general martingale process. Suppose we start with an arbitrary $x_0=x$ with $0\leq x\leq 1$. We compute $x_{t+1}$ from $x_t$ ...
3
votes
0answers
151 views

How to prove Brownian motion is Gaussian Process?

I'm reading Bernt Oksendal's "Stochastic Differential Equations" and this is one of the proof that I'm totally lost. This is from Ch2.2, page 12-13 (sixth edition). First, Brownian motion is ...
3
votes
1answer
63 views

The first two moments of $\int_0^1 B_s^2 \, ds$

I was trying to solve the following problem from Continuous Martingales and Brownian Motion by Daniel Revuz and Marc Yor, but got my solution back as the answer for variance was wrong. I have already ...
3
votes
0answers
108 views

infinitesimal generator of reflecting Brownian motion

Suppose $f\in C_0^{\infty}([0,\infty))$ and $f'(0)=0$. I'm having trouble proving that $$\frac{1}{t}E_x[f(|W_t|)-f(x)]\to\frac{1}{2}f''(x)$$ uniformly on $[0,\infty)$ as $t\downarrow0$. Showing the ...
3
votes
2answers
289 views

Conditional distribution in Brownian motion

I need to prove the following: Let $X$ be a Brownian motion with drift $\mu$ and volatility $\sigma$. Pick three time points $s < u < t$. Then, the conditional distribution of $X_u$ given ...
3
votes
1answer
166 views

Optional sampling exercise

I came across the following exercise in Stochastic Calculus: Let $B=(B_t)_{t\geq0}$ be a standard Brownian motion. Let also $M$ be the following process: $M_t=B^4_t-6t(B^2_t-\dfrac{t}{3})$ for ...
3
votes
0answers
149 views

Show that $O_t$ is a Gaussian Process

Let $B_t$ be a Brownian motion process. Let $$O_t = e^{-\alpha t} \int^t_0 e^{\alpha s} dB_s$$ Find $\mathsf{E}[O_t]$ and show that $O_t$ is a Gaussian process. I think ...
3
votes
0answers
105 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, ...
3
votes
0answers
189 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. ...
2
votes
2answers
297 views

Expectation regarding Brownian Motion

This is a formula regarding getting expectation under the topic of Brownian Motion. \begin{align} E[W(s)W(t)] &= E[W(s)(W(t) - W(s)) + W(s)^2] \\ &= E[W (s)]E[W (t) - W (s)] + E[W(s)^2] \\ ...
2
votes
1answer
169 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 ...
2
votes
2answers
68 views

A Brownian motion $B$ that is discontinuous at an independent, uniformly distributed random variable $U(0,1)$

Suppose that $\left\{B\left(t\right): t \geq 0\right\}$ is a Brownian motion and $U$ is an independent random variable, which is uniformly distributed on $\left[0,1\right]$. Then the process ...
2
votes
2answers
61 views

Ito's Isometry for three factors

Ito's Isometry states the following: If $\{W_t\}_{t\ge0}$ is a Brownian motion and $\{\phi_t\}_{t\ge0},\{\psi_t\}_{t\ge0}$ are two non-anticipative piecewise-continous processes with $\mathbb ...
2
votes
1answer
38 views

Differentiability of paths of brownian motion

On a book I'm reading (Stochastic processes by Bass. R.F.) after he proves the law of iterated algorithm for a brownian motion $W$, namely that $$\limsup_{t\rightarrow \infty} ...
2
votes
1answer
78 views

Measure of $\{t:B_t\in E\}$ for some null set $E$.

I am wondering if the following result can be found in any textbook or if you have a proof of it. When $E$ is a null set and $B_t$ is the Brownian motion, we have almost surely : ...
2
votes
1answer
34 views

Motion of the centroid of $k$ Brownian particles?

Imagine we have $k$ Brownian particles diffusing in a three-dimensional solution, where each particle has the same diffusion coefficient $D$ (measured in $\mu^2/sec$). Now imagine that we have a ...
2
votes
1answer
156 views

How to show that $X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$ (“inverse brownian”) is a martingale?

Consider $$X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$$ where $ \left(B_{t }\right)_{t \geq 0}$ is a $ \mathcal F_t$- brownian motion in $\mathbb R ^3$, null at ...
2
votes
1answer
845 views

Expectation of Stopping Time w.r.t a Brownian Motion

How do you take the expectation of a stopping time with respect to a Brownian motion? The specific question is: $$ \tau = \inf\{ t \ge 0: B(t) \in \{-a, b\}\} $$ I understand the optional stopping ...
2
votes
2answers
96 views

Second order linear partial differential equation: $\partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0$

Is there a way to solve $$ \partial_t u(t,x)+\frac12 \partial_{x,x}u(t,x)+u(t,x)v(x)=0? $$ This appeared as a condition for $$ X_t=u(t,B_t)e^{\int_0^tv(B_s)ds} $$ to be a martingale. With $B$ a ...
2
votes
2answers
177 views

Independent increments?

The questions are simple: Does the process $ X(t) = \int_0^t B(s)ds$ have independent increments? What about $X(t) = \int_{t-r}^{t}B(s)ds$? Here $B$ denotes the standard Brownian motion. ...
2
votes
2answers
167 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
2
votes
1answer
134 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
721 views

Distribution of Brownian motion

How would I go about finding the distribution of $B(u) + B(u+v)$ where $u+v > u$? I know that both $B(u)$ and $B(u+v)$ are normal random variables. The sum of two normal random variables is also ...
2
votes
1answer
53 views

Is the graph of a Brownian motion over an interval measurable?

Let $n \in \mathbb{N}_1 := \{1, 2, \dots\}$ and let $B:\Omega \times [0, \infty) \rightarrow \mathbb{R}^n$ be a standard, $n$-dimensional Brownian motion over the probability space $(\Omega, ...
2
votes
1answer
53 views

Diffusion processes

I am trying to work out a problem to which I have not found similar solutions on the website. Perhaps you can help me out. Let $X = (X_t)_{t\geq0}$ be a non-negative diffusion process which solves ...
2
votes
1answer
54 views

Expectation of Integrals of Brownian Motion

Hello I am not a native english speaker so please let me know if something does not make sense. I am interested in computing the following: $$E\int_0^T(B_s(\omega,t))^4dt$$ Or at least showing it is ...
2
votes
1answer
49 views

How does the natural filtration of a Brownian motion look like?

I am trying to understand how the natural filtration for a Brownian motion might look like. Definitions: I will start with the definitions for reference. The definition of a natural filtration is ...
2
votes
1answer
86 views

Computing cross variation of independent brownian motions

I am familiar with computing the quadratic variation of Brownian motion, but was confused when the text I'm working through introduced cross variation of independent Brownian motions. the notation is ...
2
votes
2answers
51 views

Show that process satisfy given equation

I have to show that process (1) $$X_t=e^{-bt}X_0+\int_0^te^{-b(t-s)}\sigma dW_s$$ satisfies the following equation (2) $$dX_t=-bX_tdt+\sigma dW_t$$ My attempt: Multiply both sides of (1) by $e^{bt}$ ...
2
votes
1answer
58 views

I want to show $E(B(t)-B(s))^4=3(t-s)^2$

Let $B(t)$ and $B(s)$ are brownian-motion I want to show $$E(B(t)-B(s))^4=3(t-s)^2$$ thanks for help.
2
votes
1answer
37 views

Distribution of Difference of Independent Random Variables

Usually in the development of the theory of Brownian motion, one makes the assumption that $X_t$ (the coordinate functions on $(\mathbb{R}^*)^{[0,\infty)}$). have normal distributions with mean $0$ ...
2
votes
1answer
137 views

Probability of 2D Brownian motion passing through a particular point.

Let $B_t$ be a two-dimensional Brownian motion at time $t \in [0,\infty)$. Fix a point $p \in \mathbb{R}^2$. Is the probability that $B_t = p$ at some $t > 0$ equal to zero? If so, why?
2
votes
2answers
95 views

Progressive measurability of a specific set related to Brownian motion

Let $\{W_t: t \in R_+\} $ be a standard Brownian motion process on a given probability space. I am interested in assessing the progressive measurability of the following set: $Z(\omega) := \{t: ...
2
votes
1answer
98 views

Quadratic variation of $X(s)=W_{s+\epsilon}-W_{s}$

Let $W_s$ be a standard Wiener process. The quantity $W_{s+\epsilon}-W_{s}$ is another standard Wiener process when regarded as a function of $\epsilon$. Therefore, the quadratic variation of ...
2
votes
1answer
613 views

Distribution of Sum of Two Brownian Motions

How do we find the distribution of the sum of two Brownian Motions? The questions was asked here: Distribution of Brownian motion, and was answered with We can write ...
2
votes
1answer
67 views

Stochastic differential equation problem and applying ito formula

I am given that for $b,a,\sigma >0$ and $x \in (-a,b)$ and $\nu \in \mathbb{R}$, I have the following stochastic differential equation: $$ dZ_t = \nu \,dt + \sigma\, dW_t$$ $$ Z(0) = x$$ and ...