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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9
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212 views

Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
6
votes
0answers
203 views

Does this random variable have a density?

I have a persistent problem, which I'm almost certain can be answered using elementary probabilistic arguments, but for some reason I've been stuck for some time. Here is the problem. Let $(B_s, s ...
6
votes
0answers
428 views

Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
5
votes
0answers
19 views

Can anyone solve a stochastic differential equation - related to neuroscience research?

I'm a neuroscience grad student, and I'm hoping one of ya'll could help me solve this problem regarding particle diffusion. It relates to my research on molecular-level neural plasticity, but I've ...
5
votes
0answers
127 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
5
votes
0answers
47 views

Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$ E[B_t B_s] = \min\{s,t \}, $$ which can ...
5
votes
0answers
38 views

The probability that a linear Brownian motion will hit a curve

Summary I am trying to estimate the probability that a standard linear Brownian motion will hit some curve. To make things a bit simple, I can assume that the curve is a graph of a function, that is ...
5
votes
0answers
271 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 ...
5
votes
0answers
146 views

Intuition for the optimality of bold play

There is a standard result (I think originally by Dubins and Savage) that if one wants to maximise the probability of winning a certain amount in an unfair game of chance then an optimal strategy is ...
4
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0answers
152 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
4
votes
0answers
55 views

2-D exponential functional brownian motion

I'm looking for the distribution of $X = \int_0^T e^{-W_t} dt \int_0^T e^{W_t}dt$ and $Y = \frac{\int_0^T e^{-W_t} dt}{ \int_0^T e^{W_t}dt}$ (where $W_t$ is a standard brownian motion) On most ...
4
votes
0answers
154 views

Time scaling of Brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion and $A_t$ be an increasing continuous process adapted to the filtration generated by the Brownian Motion and $A_0 = 0$. I am trying to prove ...
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
0answers
30 views

Brownian motion, modifications vs indistinguishablity

In Protters book Stochastic Integration and Differential Equations And in uncountable other sources, they mention the continuous sample paths of the brownian motion. That is: It holds that ...
3
votes
0answers
88 views

Brownian Motion and stochastic integration on the complete real line

I'm struggling to understand stochastic integration over intervals containing zero, i.e. integrals of the form $\int_{a}^{b} X_s \, d B_s$ where $-\infty \leq a < b \leq \infty$, $(X_t)_{t \in ...
3
votes
0answers
69 views

Law of iterated logarithms for BM

The law of iterated logarithms for the standard Brownian motion asserts that $(\ast) \limsup\limits_{h \downarrow 0} \frac{B(h)}{\sqrt{2h\log\log(\frac{1}{h})}} = 1$ I'm trying to prove the ...
3
votes
0answers
49 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
47 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
0answers
50 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
144 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
0answers
170 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
120 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
199 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
0answers
19 views

Intersection of two independent 1-d Brownian motions.

I am interested in the first intersection of two independent 1-d Brownian motions. More precisely, what is the joint distribution of the intersection point and intersection time? Any help is ...
2
votes
0answers
24 views

Strong markov property in two dimensional Brownian motion

I don't understand the following claim from my book: Let $(B_t)$ be a standard Brownian motion. Let $u:\Omega \rightarrow \mathbb{R}$ be a continuous function, where $\Omega$ is a domain and $B(x, ...
2
votes
0answers
43 views

Laplace transform of stopping times

I am nearly done with a question: Let $(B_t)$ be a Brownian motion on $\mathbb{R}$. For a fixed $x >0$, let $\tau$ be a stopping time defined by $$ \tau = \inf \{t \geq 0 : B_t \not \in (-x,x) ...
2
votes
0answers
43 views

Why is the pathwise integral of $\alpha_s$ w.r.t the Lebesgue measure continuous?

My class notes on stochastic calculus say that the if $(\alpha_s(\omega))_{s\in \mathbb{R_+}}$ is progressive then $\int_0^t \alpha_s ds$ is a pathwise continuous process? How does the joint ...
2
votes
0answers
35 views

How to calculate probability of an event in a stochastic setting?

Let $\left(\, B_{t}\,\right)_{t\ \geq\ 0}$ be a Brownian motion. Calculate the probability of the event: $$ E\equiv\left\{\,\exists\ \epsilon > 0 : \forall\ 0 < h < \epsilon, \max_{t\ \in\ ...
2
votes
0answers
159 views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
2
votes
0answers
27 views

Separation of variables and Fourier transformation

I know there's another question very similar to this argument. In the book "Probabilità e modelli aleatori" of Enzo Orsingher, at pag 134, it shows that the transiction function of an absorbing ...
2
votes
0answers
49 views

Independence of two processes

Suppose $X_t$ is the solution of the SDE $$dX=a(X)dt+b_1(X)dW_1+b_2(X)dW_2$$ $Y_t$ is the solution of the following SDE $$dY=p(Y)dt+q_1(Y)dW_1+q_2(Y)dW_2$$ Here, $W_1$ and $W_2$ are independent ...
2
votes
0answers
33 views

Finding a pre-visible process

Question: Let $W_t$ be a standard brownian motion under P with filtration $\mathscr F_t$. Let: $$ M_t=\mathbb E[W_T^2|\mathscr F_t] $$ Show that $M$ is a P martingale. This is simple enough using ...
2
votes
0answers
70 views

Defining the scale function of a diffusion process

My question has to do with correctly calculating the scale function of a diffusion process, but ultimately might only have to do with calculus. I'll briefly set-up my calculations, so you can quickly ...
2
votes
0answers
61 views

Hitting time of a maximum of random walk converges to that of Brownian motion

Suppose $S_n$ is a simple random walk; formally, $S_n=\sum_{i=1}^n X_i$ for $X_i\sim\mathcal{U}(-1,1)$, i.i.d.. Denote by $M_n$ the maximum of the random walk on $n$ steps; formally, $M_n=\max_{0\le ...
2
votes
0answers
70 views

Stopping times, open sets and Brownian Motion

Let $B_t$ be a brownian motion started at 0. I am trying to prove that $\tau$, defined as: $$ \tau = \inf\{t > 0 \mbox{ }|\mbox{ } \left|B_t\right| \geq \frac{1}{1+t} \} $$ is a stopping time with ...
2
votes
0answers
62 views

A right-inverse of Brownian motion local time at zero has stationary independent increments

Let $L_0^t$ be the local time for a standard Brownian motion at $0$ and define $$X_t=\sup\{s\ge0:L_0^s\le t\}, t\ge0. $$ I would like to show that $(X_t)$ has stationary independent increments. That ...
2
votes
0answers
37 views

Eigenfunctions of a 2D fractional Brownian motion covariance

The fractional Brownian motion is a centered Gaussian process with the following covariance function (covariogram): $E[B(t)B(s)]=C(\Vert t \Vert ^{2H}+\Vert s\Vert^{2H}-\Vert t-s\Vert^{2H})$ ...
2
votes
0answers
54 views

A Lemma in the book “ Mathematical Method for financial markets” (Chapter 5, Section 5.7)

In page 307, Section 5.7, Chapter 5 of the book "mathematical methods for financial markets" by Jeanblanc, Yor and Chesney, Lemma 5.7.1 is given as follows: Lemma 5.7.1.1 Let $W$ be a Brownian ...
2
votes
0answers
71 views

Why is a brownian motion conditioned to stay positive a Bessel-3

I am told this result long ago but I still don't know how to prove it. Is it because that this conditioning can be turned into a Girsanov probability change? Or is there any simpler ways to see it?
2
votes
0answers
100 views

Geometric Brownian motion - Volatility Interpretation

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
2
votes
0answers
51 views

Local time of fractional Brownian motion

For BM, there is a downcrossing representation of the local time at 0. Namely, $L_t(0)=\lim_2 (b_i-a_i)D(a_i,b_i,t)$, where $D$ is the number of downcrossing between level $b_i$ and $a_i$. I am ...
2
votes
0answers
184 views

Brownian motion conditional probability

If $B$ is the standard brownian motion and $a,b >0$ I want to show, using the reflection principle $$\mathbb{P}\left(B_t\geq a-b | \inf_{s\leq t} B_s \geq -b\right) = \frac{\mathbb P(|B_t+x|\leq ...
2
votes
0answers
219 views

Is the absolute value of Brownian motion a super martingale?Is it a sub martingale? Is it a Markov process?

I've just started to study random processes and I'm trying to solve the following problem: Let $W(t)$ be a Brownian motion with filtration $F(t)$ generated by $ W(t)$ (i.e., $F(t)=\sigma \left( ...
2
votes
0answers
154 views

Almost sure non differentiability of Brownian Motion

Problem: Let $t>0$, show that the standard Brownian motion is almost surely not differentiable a $t$ Now, through a Borel Cantelli argument I proved that, almost surely $$\limsup_{\epsilon ...
2
votes
0answers
99 views

Verifying a standard Brownian Motion?

Let $\{X_t, t\ge 0\}$ be a standard Brownian motion process. For a fixed positive number s and all $t\ge 0$, we define $Y_t = X_{t+s} - X_s$. Is $\{Y_t, t\ge0\}$ a standard Brownian motion? Attempt: ...
2
votes
0answers
49 views

Can anyone explain me this proof about a Brownian Motion?

Prove that the process $W_t=(1+t)U_{t/(1+t)}$ on $[0,\infty)$ is a Brownian motion. $\text{(b)}$ Clearly $Y_0=U_0=0$, and inherits continuity of sample paths from $U_t$ (and hence from $W_t$). ...
2
votes
0answers
769 views

Integrating deterministic function with respect to Brownian motion

I have looked everywhere for a satisfactory answer to this, including Shreve's textbooks, but I can't find one. If I want to integrate a some deterministic function f(t) with respect to brownian ...
2
votes
0answers
64 views

first hitting time probability for a Brownian motion with variable diffusion

I am looking for the first hitting time probability of the following Brownian motion: $dX=\mu X dt+ \sigma (X) X dW$ assuming $X(0)=X_0$ and $\sigma(X)= \sigma_1$ if $X>X_1$ and ...
2
votes
0answers
51 views

Exercise in brownian motion

Consider a system of n particles moving in three dimensional space under the action of an external force with $C^1$ potential V and coupled to a heat bath causing an external random effect. Then we ...
2
votes
0answers
117 views

Quadratic variation process of $G$–Brownian motion

I would like to prove the inequality $$\hat{\mathbb{E}}\left[\left(\int^T_0 \eta_t d \langle B \rangle_t \right)^2\right] \leq C \hat{\mathbb{E}}\left[ \int^T_0 \eta^2_t dt \right],$$ where $\langle B ...