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
0answers
27 views

Reference: Computing Martin Capacity

For Borel set $A$ the Martin Capacity is defined as: $\mathrm{Cap}_{M}(A)=[\inf\{\int \int \frac{G(x,y)}{G(0,y)}d\mu(x)d\mu(y):\mu \mbox{ probability measure on }A \}]^{-1}$ and Green's function ...
1
vote
0answers
19 views

Stability of simulation of brownian noise

As I understand, Brownian noise can be simulated by the process $$x_{n+1}=x_n+R_n$$ where $R\sim U[-a,a]$. The expected value for $x_n$ is then $x_0$. But $\text{Var} x_n\to\infty$ as $n\to\infty$ ...
1
vote
0answers
34 views

Brownian motion starts fresh variant

It is a standard result that if $W_t$ is a Brownian Motion and $S$ is a stopping time of the standard filtration $F_t$ then we have that $B_t = W_{S+t} - W_S$ is a Brownian Motion. I quote the ...
1
vote
0answers
17 views

Convergence of sampling from Brownian motion

For a standard linear Brownian motion $\{B(t)\mid\ 0\le t\le 1\}$, for natural $n\ge 0$ and natural $1\le k\le 2^n$, let $d(n,k)=B\left(k2^{-n}\right)-B\left((k-1)2^{-n}\right)$ be the differences of ...
1
vote
0answers
15 views

A conformal image of a Brownian motion is a time changed Brownian motion

I have read a paper which has stated the following: A conformal image of a Brownian motion is a time changed Brownian motion. The paper cites R. Durret, Brownian motion and martingales in ...
1
vote
0answers
24 views

Expression for $B_1$

I think that it is indeed the case that $$ B_1 = \int_0^1 \frac{B_1 - B_t}{1-t} dt, $$ where $B$ is a standard one-dimensional Brownian motion. Am I right? If so, how you we prove it? Thanks a lot ...
1
vote
0answers
19 views

Product of Geometric Brownian motions

Let $S,P$ be geometric BMs: $$dS_t=S_t(\mu dt + \sigma dW_t^1)$$ $$dP_t=P_t(\tau dt + \beta (\rho dW_t^1+ \sqrt{1-\rho^2}dW_t^2)$$ Where $W^1$ and $W^2$ are independent standard BM. I want to show ...
1
vote
0answers
54 views

Variance of Integrated Geometric Brownian Motion

I'm just asking for verification that my derivation is correct, as I can't seem to find this result elsewhere. I'd like to calculate $Var(\int_0^T X(t) dt)$ where $X(t) = X_0e^{(\mu - ...
1
vote
0answers
70 views

Donsker for randomly stopped processes

A question regarding Donsker's invariance principle. Donsker states that if $X_1, X_2, ...$ are independent and identically distributed with mean $0$ and variance $\sigma^2$ and if $S_t^n$ is the ...
1
vote
0answers
48 views

Ito's Lemma and Geometric Brownian Motion With Jumps

I have a price process: \begin{equation} dF_t = d\Pi_t - \mu_\pi \sigma_t F_t \gamma \, dt + \sigma_t F_t \, dz \end{equation} And wish to simulate the process $x_t = \ln(F_t)$ by Euler method, ...
1
vote
0answers
87 views

Tail of hitting times for Brownian motion on the circle

For $y\in \mathbb R/\mathbb Z$ and $\varphi\in C([0,\infty);\mathbb R/\mathbb Z)$ let $T_{y}(\varphi) \ := \ \inf\{t>0: \varphi_t = y \} \ \ \ $ (first time the path $\varphi$ hits $y$) ...
1
vote
0answers
24 views

Hitting time for a planar diffusion

Let $A$ be an open subset of $\Bbb R^2$, and let us consider a diffusion $\mathrm dX_t = f(X_t)\mathrm dt + g(X_t)\mathrm dW_t$ where $f$ and $g$ are globally Lipschitz continuous maps. Suppose I am ...
1
vote
0answers
61 views

Expectation of the infimum of a GBM

does somebody know a reference, where I can find the value of the expectation of the running infimum of a geometric Brownian motion, namely: Given a filtered probability space ...
1
vote
0answers
117 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 ...
1
vote
0answers
99 views

An application of the Dambis-Dubins-Schwarz theorem. Is my argument correct?

I attended a lecture today, in which the professor went through an example with a lot of tedious calculations to show something which I'd think would follow directly from the Dambis-Dubins-Schwarz ...
1
vote
0answers
54 views

Probability that the value at time T from one geometric Brownian motion process is greater than the value from another GBM

I am having a competition between $n$ people (starts at time $t$=0), each who accumulates points on a daily basis, which I assume is a geometric Brownian motion process with parameters $\mu_i$, ...
1
vote
0answers
265 views

Running maximum for Geometric Brownian Motion

Can anyone provide the expression and source for the running maximum $M_t$ for geometric Brownian motion $X_t$ as a function of the initial value $X_0$, drift $\mu$ and diffusion $\sigma$? $X_t$ ...
1
vote
0answers
86 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: ...
1
vote
0answers
26 views

Brownian motion minimisation problem

Let $B_t$ be a Brownian motion, let $\sigma > 0$ be fixed and let $X_t$ be a process with fixed beginning value $x_0$ that satisfies $dXt = u_tdt + \sigma X_tdB_t.$ Solve ...
1
vote
0answers
27 views

Probability of winding number in 2D Brownian motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
1
vote
0answers
37 views

Conditional covariance

Just a simple question when we have $v_{st} = \operatorname{cov}(B_s, B_t\mid Z)$, where $B_t$ is a brownian motion. I know that the answer is $\min(s, t) - E[B_s Z]E[B_t Z]/E[Z^2]$ but i don't know ...
1
vote
0answers
27 views

Finding asymptotic behaviour

I have a problem in finding the asymptotic behavior of this sum: $$\sum_{i=0}^{n-1} \bigl|B^2 (t_{i+1})-B^2 (t_i)\bigr|$$ over $[0,T]$ when $h= t_{i+1}-t_i \to 0$ and $B$ is Brownian motion. The ...
1
vote
0answers
54 views

Bounding the expectation of monotone function of stopping times of Brownian motion

Let $X_t$ be a standard Brownian motion and let $Y_t:=X_t + \epsilon B_t$ where $B_t$ is an independent standard Brownian motion and $\epsilon>0$ is small. Let f be a monotone increasing function. ...
1
vote
0answers
27 views

First hitting time for a brownian motion with two exponential boundaries

I asked a previous related question here: First hitting time for a brownian motion with a exponential boundary Now Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first ...
1
vote
0answers
123 views

Brownian motion hitting probability

Let $B_t$ be a brownian motion and $g(t)$ a function of the time $t$. $B_0=0$. Let $\Phi$ be the c.d.f. of a normal distribution. At time $t$, the probability that $B_t > g(t)$ equals ...
1
vote
0answers
50 views

Bounded Brownian Bridge

I am trying to calculate the following expectation value of $$ E[exp(-\int_{t_0}^{t_1} X_s ds)] $$ in which Xs is a bounded Brownian bridge, which means $X(t_0)=a$, $X(t_1)=b$ and $A<X(t)<B$. ...
1
vote
0answers
128 views

Brownian motion with drift

I need help with the following problem: Let us denote the water level in a dam at time $t$ by $X(t)$, where $t$ is measured in months. We will assume that, at least until the first time that the dam ...
1
vote
0answers
115 views

Conditional expectation with three random variables

We have $N_1, N_2, N_3$ normally distributed random variables with $µ_i =E[N_i]$, $σ_{ij}=Cov(N_i,N_j)$. We also have $\tilde{µ}_i=E[N_i|N_2 = x] $, $\tilde{σ}_{ij}=Cov(N_i,N_j|N_2 = x)$ and $v^2 ...
1
vote
0answers
48 views

Contradiction on equality with stochastic integrals

I want to compute $E[∫_0^tB_u \, du ∫_0^sB_u \, du]$ and I know from another source that should be equal to $ts^2/2$. But when I try to compute it like: $$\begin{align} & E\left[(tB_t- \int_0^tu ...
1
vote
0answers
132 views

Brownian motion, rate of large events

Given the most simple brownian motion: $$ \dot x(t) = \sigma \eta(t)$$ where $\langle \eta(t)\eta(t')\rangle=\delta(t-t')$, I define as large event in a time-frame $\tau$ a portion of the trace ...
1
vote
0answers
92 views

Probability a geometric Brownian motion stays within an interval.

Let $X_s$ be a $(\mu,\sigma)$ geometric Brownian motion with $X_0 = x$. For some positive numbers $c < x < d$ and time $t$, what is the probability $X_s \in [c,d]$ for all $s \in [0,t]$? In ...
1
vote
0answers
91 views

Leibniz Rule applied to Brownian integral

I am looking to take the partial derivative of an integral with respect to brownian motion. For Simplicity I will make it the same integral as in this post (don't have enough reputation to comment): ...
1
vote
0answers
538 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 ...
1
vote
0answers
156 views

Hitting probabilities for Brownian motion

Let $\mathbb D$ be the complex unit disk. Let $B$ be a standard complex Brownian motion started at $0\in \mathbb D$. Let $\tau = \inf\{ t : B_t \in \partial\mathbb D\}$. I am trying to show that if ...
1
vote
0answers
63 views

Rate of increase of maximum process of Brownian Motion

Suppose $M_t=\sup_{0\leq s\leq t}\{B_s\}$, where $\{B_t\}_0^{\infty}$ is a standard Brownian Motion. I would like to know if it is true that $M_t e^{-t}$ converges to 0 almost surely? Thanks!
1
vote
0answers
77 views

Expected value of brownian motion for all positive paths

I've got this question but I can't figure it out. Derive the expected value of $B(t_1)$ of all paths that are positive $t_1$ and calculate the expectation for $t_1=1$ and variance$=1$? Thanks
1
vote
0answers
43 views

Simulating of GBM

I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem: Given a GBM of the form $dS(t) = \mu S(t) dt + ...
1
vote
0answers
411 views

Show that this semimartingale is a local martingale

Let $B_t$ be a standard Wiener motion, $I_t=\int_0^t|B_s|^2\!\text{ds}\ $and $S_t=\max_{0\leq s\leq t}B_s$. Let also $F:\mathbb{R}^2_+\times\mathbb{R}\times\mathbb{R}_+\rightarrow\mathbb{R}$ a ...
1
vote
0answers
104 views

Is this geometric Brownian Motion?

The SDE for GBM is usually specified as: $$dX(t) = X(t)[\mu dt + \sigma dW(t)]$$ If we model diffusion as stochastic, is the following still GBM? $$dX(t) = X(t)[\mu dt + \sigma_t dW(t)]$$ ...
1
vote
0answers
78 views

Fractional Brownian motion, selfsimilar

Let $0<H<1$. A real-valued Gaussian process $\left(B_H(t)\right)_{t\geq 0}$ is called fractional Brownian motion (fBm) if $\ \mathbb{E}[B_H(t)]=0$ and ...
1
vote
0answers
55 views

What is the intuitive meaning of $K_1, K_2, K_3$ in regards to the conditional density formula derivation in Brownian motion.

In my text, there is a passage that says: "Suppose we require the conditional distribution of $X(s)$ given that $X(t) = B$, where $s < t$. The conditional density is: $$ \begin{align*} f_{s\mid ...
1
vote
0answers
190 views

Integral with respect to Wiener process.

Suppose that $\sigma(t,T)$ is a deterministic process, where $t$ varies and $T$ is a constant. We also have that $t \in [0,T]$. Also $W(t)$ is a Wiener process. My First Question What is ...
1
vote
0answers
646 views

Stochastic integral: Interchanging the order of expectation and integration

Let $B$ be a standard Brownian motion and $$ X_t=\int_0^t f_s ds+\int_0^t g_s dB_s, $$ where, $|f|$ and $|g|$ are both bounded, almost surely, by some positive constant $M$. Is it true that $$ ...
1
vote
0answers
221 views

Defining Brownian motion through Kolmogorov's extension theorem

In section 2.2. of Oksendal's book on Stochasic differential equations, he defines Brownian motion by specifying a family of probability measures $\nu_{t_1, \ldots, t_k}(F_1, \ldots, F_k)$ that ...
0
votes
0answers
13 views

Stopped supremum of the Brownian local time still $L^p$ bounded in space?

Let $B_t$ be a standard Brownian motion and $L_t^x$ its local time in $x$ at time $t$. For fixed $t$ and $p>1$, it holds that $$ \sup_{x \in \mathbb{R}} \operatorname{E} [ (L_t^x)^p ] < ...
0
votes
0answers
36 views

Intensity of fractional brownian noise

Having a White noise driven SDE $dX = f(X)dt + \sqrt{2D}dW$, the noise intensity is equal to D. What is the noise intensity, if I consider fractional brownian noise, instead of white one?
0
votes
0answers
18 views

Change of variable in stochastic integral

Let $B$ be a standard Bronwian motion. Can we do a change of variable in the sense $s=\theta+h$ $$\int_{0}^{t+h}X_sdB_s=\int_{-h}^{t}X_{\theta+h}dY_\theta.$$ In this case what is the process ...
0
votes
0answers
46 views

Expectation of e^(cX) if X is a geometric Brownian motion

(Edit:) The short version: Calculate $$E[e^{cY}]$$ if $c < 0$ and $Y$ is lognormally distributed, i.e. $\log(Y) \sim N(\tilde\mu, \tilde\sigma^2)$. The long version: I want to calculate ...
0
votes
0answers
13 views

Convergence of the distribution of a GBM at a random time when time converges in probability

I have got the following question. Let $(S_t)_{t\in[0,T] }$ be a geometric Browninan motion. Consider a sequence of bounded random variables $(\tau_n)_{n\in\mathbb N}$ such that $\tau_n\downarrow ...
0
votes
0answers
48 views

First hitting time Geometric Brownian motion

I have the following problem: My Process underlies the SDE $ d W_t = \mu W_t dt + \sigma W_t d B_t $ with $B_t$ being a standard Brownian motion, $\mu,\sigma >0$, i.e. $W_t = S_0 \exp\Big( ...