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

0
votes
0answers
7 views

Augmented Filtration

In class, we showed that Brownian Motion is a martingale with respect to the filtration $F_t = \sigma(B(s): 0\leq s \leq t) $. For a HW assignment, I need to show it's a martingale with respect to a ...
1
vote
1answer
8 views

Proving that a process is a positive martingale

Let $X$ be the strong solution to the SDE $$ dX_t = \tanh X_t \,dt + dW_t, $$ where $W$ is a scalar Brownian motion defined on a probability space $(\Omega, \mathcal{F} ,\mathbb{P})$. (Such solution ...
2
votes
1answer
24 views

Pathwise solution of a stochastic integral equation, without stochastic calculus

Let $f$ be a Lipschitz continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $W$ be a standard Brownian motion. I don't know any stochastic calculus (nothing about stochastic integrals, nothing ...
2
votes
1answer
16 views

Density of running supremum of Brownian motion until a stopping time

I am stuck on an exercise in my book: The question relies on the following fact: Let $M$ be a continuous, non-negative local martingale such that $M_0=1$ and $M_t \rightarrow 0$ almost surely as ...
0
votes
0answers
13 views

What is the resulting stochastic process of divided Geometric Brownian motions

Let $W_{1,t},W_{2,t},...,W_{n,t}$ be $n$ independent geometric Brownian motions. Now let's say I construct the following processes: $$ X_1 = \frac{W_1}{\sum_i^n W_{i,t}} $$ $$ X_2 = ...
3
votes
2answers
48 views

A variation of Lévy's characterization of Brownian motion

It is shown here, without using stochastic calculus, that if $W_t$ is a standard Brownian motion, then $$ f(W_t)-\frac{1}{2}\int_0^t f''(W_s)ds $$ is a martingale, where $f\in C^2$ and compactly ...
0
votes
0answers
23 views

Levy process of argument in the complex plane

I am stuck on this question: Let $B$ be a Brownian motion in $\mathbb{C}$ started at $1$. Let $\theta_t$ be a continuous determination of the argument of $B_t$, i.e. $B_t = |B_t| e^{i \theta_t}.$ ...
2
votes
2answers
28 views

integral of exponential of Brownian motion

I am currently reading a proof that uses the following fact without proof: If $B$ is a scalar standard Brownian motion, then $\int_0^\infty e^{B_s} \,ds = + \infty$ a.s.. How can we justify this ...
1
vote
0answers
15 views

Extension of Law of Iterated Logarithms

Suppose I have a stochastic differential equation ($X_t$ is a vector) $dX_t = f(X_t) dt + \sigma g(X_t) d\eta(t)$ and define $V = \sum_{i=1}^{n} x_i$. Here, $\eta(t)$ is an Ornstein-Uhlenbeck process. ...
0
votes
1answer
27 views

Geometric brownian motion - Ito's lemma

I have a question about geometric brownian motion. dS = uSdt + /sigma/SdW and then we do log(S) and we want to found dlog(S). So we use Ito's lemma en I get the dt part of the lemma but I don't see ...
0
votes
0answers
15 views

Variance estimation of a diffusion process

The framework of this question is a 1 dimensional diffusion process, defined ny the following equation: $dx_t=adt+bdw_t$ Where $w_t$ is a standard berownian motion and and $a$ is a constant drif ...
1
vote
1answer
23 views

Examples of Wiener Martingales

$(X_t,\mathcal{F}_t)$ is called a Weiner martignale if i) $X_t$ is a Wiener Process ii) $(X_t,\mathcal{F}_t)$ is a martingale. (Here $\mathcal{F}_t$ is an increasing $\sigma$-field family). Let ...
2
votes
1answer
21 views

Finding $b$ such that $e^{5B_t - bt}$ is a martingale

I have $X_t = e^{5B_t}$ and Where $B_t$ is brownian motion at time $t$. $M_t = X_t \cdot e^{-bt}$ I need to find a value for $b$ such that $M_t$ is a martingale. I am encountering difficulty, ...
1
vote
0answers
14 views

Scaled distribution of Brownian motion

If I have $X = 5(B_t - B_s)$ Does this have a distribution of $\sim \text{N}(0,25(t-s))$ ? Since $B_t - B_s$ has distribution $\sim \text{N}(0,t-s)$ Then $X = \mu \cdot 0 + \sigma_1 Z$ where $Z ...
1
vote
2answers
44 views

Differential and Differential Equation - Difference in meaning?

I am a little confused, an exercise by a teacher has been set which says: For $X_t = 2e^{B_t}$ Where $B_t$ is brownian motion at time $t$. a) Find the stochastic differential $d(X_t)$ b) Find the ...
7
votes
1answer
89 views

Show that $f(W_t)-\frac{1}{2} \int_0^t f''(W_s) \, ds$ is a martingale without using Itô's formula

I'm learning the basics about Brownian motion (I know nothing about stochastic calculus), and I've shown that if $W(t)$ is a standard Brownian motion, then $W(t)^2-t$ is a martingale. Now I'm trying ...
0
votes
0answers
12 views

Showing that $X_t = \int^{1/[X]_t}_0 f_u dW_u$ is a Brownian motion

Assume we have an Ito process $$ X_t = \int^t_0 f_u d W_u $$ where $f_u$ is a deterministic function of $u$ and $W_u$ is a Brownian motion adapted to $\lbrace \mathcal F_t \rbrace$. I want to show ...
3
votes
1answer
43 views

How to compute stochastic integral: $\int_0^t d(B_s^2)$

Here, $B_t$ is Brownian motion at time $t$ What property is used to compute the integreal $\int_0^t d(B_s^2)$? Shouldn't there be some other variable attached with the differential $d$ ?
1
vote
1answer
31 views

Is Brownian Motion increasing?

Given a process $Y_t = e^{B_t}$ We know that since Brownian motion is continuous for $t \geq 0$. Since $B_t$ is a completely random motion, it is true that we cannot say whether it is monotone ...
3
votes
1answer
46 views

How to calculate $\mathbb{E}((B_3-B_2)(B_4-B_{\pi}) \mid B_1)$ for a Brownian motion $(B_t)_{t \geq 0}$

I want to find the best predictor of $(B_3-B_2)(B_4-B_{\pi})$ given an observation of $B_1$ Where $B_t$ is brownian motion for time $t \geq 0$. I am not sure how to approach this. I know it will be ...
3
votes
0answers
23 views

Best predictor of Brownian motion

Let $B_t$ be brownian motion at time $ t \geq 0$. Then I want to find the best predictor of $B_8 + 4$ given that there are observations of brownian motion up to time $t = 1$. Approach: Essentially, ...
2
votes
1answer
13 views

Conditional expectation and brownian motion - check my answer please

$X = \frac{ B_1+ B_3 - B_2}{\sqrt{2}}$ and $Y = \frac{B_1 - B_3+ B_2}{\sqrt{2}}$ Where $B_t$ Is brownian motion at time $t\geq0$ I want to find $\mathbb{E} [Y + 3X | X]$ It is known to me that $X, ...
2
votes
1answer
19 views

Independence of two random variables derived from a Brownian motion

If $X = B_1 + B_3 - B_2$ and $Y = B_1 - B_3 + B_2$ Where $B_t$ is Brownian Motion for $t \geq 0$ And I want to state with certainty whether $X$ and $Y$ are indep or not, do I simply just ...
3
votes
1answer
34 views

Distribution of Brownian Motion help

If $X = \frac{B_1 - B_3 + B_2}{\sqrt{2}}$ Where $B_t$ is brownian motion at time $t$. And I want to find the the distribution of $X$, how would I do so? $E[X] = 0$ is fairly straight forward. For ...
2
votes
1answer
50 views

Girsanov's theorem and absolutely continuous restrictions

Let $W$ be a Brownian motion on some probability space $(\Omega, \mathcal{F}, P)$. Let $\mathbb{F}^W$ be the filtration generated by $W$ and let $X$ be a process that is progressively measurable ...
1
vote
0answers
11 views

Reference request for conditional and unconditional covariance of n-times integrated Brownian motion

I'm working through an old Diaconis paper on Bayesian numerical analysis, and am currently calculating the details behind his brief comments on using $n$-times integrated Brownian motion as a function ...
2
votes
2answers
82 views

Martingale representation theorem application

Let $X = \exp(W_{T/2}+W_T)$. I try to figure the adapted process $g(s)$ such that according to the MRT we have $$X = \mathbb{E}[X]+\int^T_0 g_s dW_s.$$ I can figure out $X = \exp(2W_{T/2}+W_{T-T/2})$ ...
1
vote
1answer
26 views

proving independence of stochastic integrals

Does anyone know how to show that the stochastic integrals \begin{equation} \bigg\{ \int_0^1 \cos \Big[ (n- \frac{1}{2}) \pi t \Big] \,dW_t \bigg\}_{n \in \mathbb{N}} \end{equation} are ...
0
votes
2answers
24 views

Distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion

I am trying to find the distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion. One (very hand-wavey) way is to assume a priori that it is Normally distributed. Then one can ...
1
vote
1answer
28 views

Properties of brownian motion

I was doing some revision and had an admittedly elementary question. My lecture notes say, the following are properties of Brownian Motion {$B_t$} (Normal or Gaussian increments) For all $s < t, ...
0
votes
1answer
43 views

Convergence of exponential Brownian martingale to zero almost surely

Define the exponential Brownian martingale as $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ which is a martingale with respect to the natural filtration of $W$ which stands for a standard Brownian ...
0
votes
0answers
14 views

Integral of Constant Parameter Martingale

What is the $\int_{1}^{t}W_1W_sdW_s$. This is the question solved by Kuo in his paper an extension of the Ito's Integral (2008) but there limit runs from $0$ instead of $1$.
0
votes
1answer
41 views

Distribution of a transformed Brownian motion

Let $W$ be a standard Brownian motion. From an earlier proven result I know that $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ defines a martingale on the natural filtration of $W$ for all $a \in ...
2
votes
1answer
22 views

$n$ times integrated Brownian motion

I have an identity that expresses the $n$ times integrated Brownian motion and I would like to prove that. First, I define what I mean by $n$ times integrated Brownian motion. $$V_1(t) = \int_0^tB_s\, ...
2
votes
0answers
30 views

Measurability of the event that Brownian motion hits a given set

Let $W$ be a Brownian motion in $\mathbb{R}^{2}$ on a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ . Let us assume $\mathcal{F}$ is the sigma-algebra on the path space ...
-3
votes
0answers
13 views

Argmax distribution of Brwonian motion plus linear drift

I want to know the the density function or the tail of the density funcion of the following random variables: $$\underset{{t\in [0,+\infty]}}{\arg \max} \quad {W_t-t}.$$ Thank you very much
-1
votes
0answers
17 views

Defining a stochastic process indexed by angle

I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin ...
1
vote
0answers
7 views

Derive Laplace Equation through Random Walk

I am looking for the solution of this problem: Consider a bounded domain $\Omega\subset\mathbb{R}^2$ and let $u(x,y)$ be the probability of exiting $\Omega$ starting at $p=(x,y)$, assuming that the ...
1
vote
0answers
22 views

Convergence in distribution of BM started in (x,y) to BM started in (0,0)

Let $B$ be a Brownian motion in $\mathbb{R}^{2}$ . Let $\mathbb{P}_{(x,y)}$ denote a probability measure under which $B$ is started at $(x,y)$ . Is it true in general that, for measurable set ...
1
vote
1answer
49 views

Proof of martingale representation theorem monotone class argument

Martingale representation theorem for reference: Theorem: (Martingale Representation) Let $M$ be a square integrable Brownian martingale with $M_0 = 0$.Then there exists a process $X$ which is ...
0
votes
0answers
26 views

Prove independent increments

The question I have is below; I have found one version of an answer for this so far here. Is the following a Wiener process? My question that I am asking is, "How would you show that the increments ...
1
vote
0answers
27 views

Two-parameters Wiener process

Two-parameters Wiener process $W(r, u), r \in [0, 1], u \in [0,1]$ is a stochastic process with a covariate kernel $\mathbb{E}\left[W(r_1, u_1) W(r_2, u_2)\right] = \min(r_1, r_2) \min(u_1, u_2)$. ...
0
votes
0answers
20 views

Basic Stochastic Calculus

Let $B_t$ be brownian motion. Then if I need to calculate $\mathbb{E}[2(B_2-B_0)+(B_2+B_1)(B_3-B_2)]$ is this simply $0$ as independence results in: $\mathbb{E}[2(B_2-B_0)] + \mathbb{E}[B_2+B_1] ...
1
vote
0answers
15 views

Functions of Brownian Motion and Time

Sorry, this will be a little long. I'm currently working on a problem where I basically have an SDE logistic equation: $$dX_t = diag(x_1,\cdots, x_n)[b+Ax-\lambda \eta(t)] dt + diag(x_1,\cdots, ...
2
votes
1answer
44 views

Law of Iterated Logarithms

I know the Law of Iterated Logarithms states the following almost surely: $$\limsup_{t\to\infty} \frac{B(t)}{\sqrt{2t\log\log t}} = 1 $$ I was wondering if there are similar ones. For example, if I ...
1
vote
1answer
76 views

Wiener process - proof of independent increments

I have defined the Wiener process to be a stochastic process $X_t$ with values in $\mathbb{R}$ such that $X_0=0$, the paths $t \mapsto X_t$ are continuous, and for any times $0<t_1<\dots<t_n$ ...
1
vote
0answers
31 views

covariance of two correlated integrated brownian motions

Assume we have two integrated Brownian Motions $\int_0^tf(t)dW_t$ and $\int_0^tg(t)dY_t$ where the $d$-dimensional Brownian Motions $W$ and $Y$ are correlated according to the positive semi-definite ...
2
votes
1answer
55 views

Determine for which values of some parameters a stochastic integral is a Brownian motion

Let $W_t$ be a Brownian motion on $(\Omega, F, (F_t)_t, P)$. Find all values of $a$ and $b$ such that the stochastic integral $$X_t=\int_0^t a+\frac{bu}{t} \;dW_u$$ is a Brownian motion. 1)So I need ...
0
votes
0answers
29 views

Transition density of a Geometric Brownian-motion

The solution to SDE $$dS(t)=\sigma S(t)dW_t$$ is $$S(t)=S(0)\exp(-\frac{1}{2}\sigma^2t+\sigma W_t)$$ the transition density for this martingale is $$p(S(t),t;S(0),0)=\frac{1}{S(t)\sigma \sqrt{2\pi ...
0
votes
1answer
43 views

Distribution of Integral involving wiener process [closed]

Given $W(t)$ as a standard Wiener process, i.e. $W(t) \sim \mathcal{N}(0,t)$. Prove the following statement: $$\int_{0}^{1}tW(t)dt \sim \mathcal{N}(0,\frac{2}{15})$$