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 $t-s$.

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38 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 $A<X(t)<B$. ...
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99 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 ...
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114 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 ...
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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 ...
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130 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 ...
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80 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 ...
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77 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): ...
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350 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 ...
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125 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 ...
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58 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!
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76 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
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78 views

Independence of Brownian motion-related stopping times

Let $(B_t,\mathcal{F}_t)_{t \geq 0}$ a Brownian motion on a probability space $(\Omega,\mathcal{A},\mathbb{P})$. For $a \in \mathbb{R}$ define a stopping time $\tau_a$ by $$\tau_a := \tau(a) := ...
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39 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 + ...
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390 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 ...
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99 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? $$dX(t) = X(t)[\mu dt + \sigma_t dW(t)]$$ ...
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68 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 ...
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52 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 ...
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179 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 ...
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505 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 $$ ...
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212 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 ...
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11 views

Construct SDE with two uncorrelated Brownian motions

Using $Y(t) = wX_1(t) + \sqrt{1-w^2}X_2(t)$ as a model to construct a process where X1 and X2 are brownian motions with drifts and brownian increments $dX_1(t)= \mu_1dt + \sigma_1dW_1(t)$ ...
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27 views

Brownian Motion Conditional Probability Question

If $X_t$ is a standard Brownian motion, how does one calculate a conditional probability. Specifically, $P(X_2\gt 0 |X_1\gt0)$. I am thinking that the two are independent so I can just calculate ...
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18 views

Reflected borel sets are worse traps for Brownian paths.

This is for a research project. I am trying to prove that given a borel set, it's reflected version will have a lower Wiener measure of brownian paths intersecting it. In this paper they elaborate ...
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39 views

Probability calculations with integral of Geometric Brownian Motion

I am looking for the probability $Prob[A_t<A_0]$, where $A_t$$=$$A_0$$+$$\int_0^t$$S_0$$e^{(\mu-(1/2)\sigma^2)s+\sigma W_s}$$ds$$-$$c$$t$ The expression $S_0$$e^{(\mu-(1/2)\sigma^2)t+\sigma ...
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26 views

Cubed Brownian motion

I have to do the following exercise: Let $(W_t)$ be a Brownian motion. (a) Does X given by $X_t:=W_t^3$ have constant expectation? (b) Is it a martingale? (c) Does it have independent increments? ...
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14 views

2 2-dimensional Brownian motions are close to each other

Suppose $B^1$ is a standard 2 dimensional Brownian motion and $B^2$ is a 2 dimensional Brownian motion with mean zero and covariance matrix $\Gamma = \begin{pmatrix} a & b \\ b & a \\ ...
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20 views

How to understand this equation for brownian motion

I am reading this article from the notes 'an intro to SDE'. Here I dont know why in (1) he take that integral from - infinity to infinity. I mean why we do that? I just dont know what the physics or ...
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19 views

Show that $ \text{ess-sup}_\Omega g (x+ B_T) = \sup_{y \in \mathbb R ^d }g(y)$

Show that $$ \text{ess-sup}_\Omega g (x+ B_T) = \sup_{y \in \mathbb R ^d }g(y)$$ where $B$ is a d-dimentional brownian motion , $x \in \mathbb R ^d $ and g a Lipschitz bounded function of $\mathbb R ...
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33 views

Expected Value of the minimum stock price where stock price is an exponential brownian process

Hi I am trying to figure out what would be the solution to the following equation: $\tilde{E}[S_{min}]$ where $S_{min}$ is the minimum stock price and the stock price is of the form ...
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51 views

brownian motion and stopping time

I have an exercise about Brownian motion which I don't understand completely. Let $(B_s)_{s\geq0}$ be a standard real Brownian motion. For $t > 0$, we define the random times $g_t ...
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22 views

Convolution of Brownian function with characteristic function

Given Brownian function defined on the interval $[0,1]$. Our aim is to filter this function, one may use the low band pass filter. The idea is to cut the high frequency of its Fourier transform by ...
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36 views

given SDE how to find martingale measure

I've been stuck with the question how to find a measure to make a discounted price a martingale. I cannot use Girsanov because I am only given the SDE for which an unique strong solution exists but ...
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37 views

Trying to understand geometric brownian motion with example

I've been spending the past week trying to understand geometric brownian motion/log-normal distribution/geometric processes/ without too much success. The problem is all the summaries online begin by ...
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61 views

Indicator function of an infinitesimal set

While reading a paper related to functional of brownian motion I came across the following notation $1(B_t \in dx)$, where $1(A)$ is the indicator function of the set A, and $B_t$ is a standard ...
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34 views

Are my estimates of parameters of geometric brownian motion correct?

I wrote a simulation of a geometric Brownian motion which works like this: $t_i - t_{ i-1 } \sim Exp(\lambda )$ $Z_i \sim N(0,1)$ $Y_i \sim e^{ \sigma \sqrt { t_i - t_{ i-1 } } Z_i +\left( \mu ...
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57 views

Independence in Brownian Motion

I've read two times in different lecture notes that for a Brownian Motion $(B_s)_{s\ge 0}$ and $t<u$ the random variable $B^t_{u}:=B_{t+u}-B_t$ is independent from ${\cal F}_t:=\sigma\{B_s:s\le ...
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13 views

Distributions related to 1-dimensional Brownian Motion

Let $W_t$ be a one-dimensional Brownian Motion. Then how do we go about finding the distribution of $ \int_0^{1} W_t dt$ ? Moreover, assume $W_t = (W^1_t,W^2_t)$ is a 2-dimensional Brownian motion. ...
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120 views

Degradation Model in Matlab

I am trying (using MATLAB) to generate the following image from the Wu Tian Chen research article 'Condition-based Maintenance Optimization Using Neural Network-based Health Condition Prediction': ...
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20 views

Simulation of a Bidimensional Fractional Brownian motion

I would like to simulate and understand the simulation of a bidimensional fractional Brownian motion (I would like to try and use it to simulate terrain in a 3d game I am developing), but I cannot ...
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26 views

A brownian bridge evaluate at a particular random variable

I was wondering of someone could help with the following. I have a random variable given as $\lambda^{*}=\arg \max_{\lambda \in (0,1)} [B(\lambda)-\lambda B(1)]^{2}/\lambda(1-\lambda)$. I am now ...
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42 views

Ito's lemma and Backward evolution operator.

$\Phi$ is the backward evolution operator. $W=\theta+\phi+S$ $d\theta=\mu\theta dt+\sigma_1 dZ_1$ $d\phi=r\phi dt+\sigma_2 \phi dZ_2$ $dS=rSdt$ $dZ_1 dZ_2=\rho$ ...
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21 views

$W=\Phi+\theta+S$ where $\Phi, \theta, S$ are geometric brownian motions.

$W=\Phi+\theta+S$ where $\Phi, \theta, S$ are geometric brownian motions. $d\theta=\mu\theta dt+\sigma_1 dZ_1$ $d\Phi=r\Phi dt+\sigma_2 \Phi dZ_2$ $dS=rSdt$ For $t \le s \le T$, ...
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56 views

Expected value of the product of multiple Stratonovich Integrals

I want to calculate: $\mathbb{E}(J_1* J_{10}* J_{10} *J_{110})$ where $J_1=\int 1 dW$, $J_{10}=\int\int 1 dW dt$, $J_{110}= \int \int \int 1 dW dW dt$ are multiple Stratonovich Integrals over the ...
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43 views

Arbitrage with fractional Brownian motion

I need some help to understand L.C.G. Rogers' paper "Arbitrage with fractional Brownian motion" (1997). The fractional Brownian motion $(X_t)_{t\in \mathbb{R}}$ with self-similarity parameter ...
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66 views

Correlation function of Brownian motion. What am I doing wrong?

Can anyone tell me where I am going wrong here? (I am leaving out any random fluctuation forcings, because I don't think they are relevant to my problem.) 1: $\displaystyle \frac{dv(t)}{dt}=-\eta ...
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96 views

Moment generating function of two brownian motions

wondering if you can help me with this: Let $B_t$ and $W_s$ be standard brownian motions, Cov($B_t$,$W_s$)=$\rho$min(t,s), $\rho\neq0$. How do you find the moment generating function ...
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205 views

Geometric Brownian Motion

Consider asset price $S$ that evolves according to Geomtric Brownian Motion with constant $\mu$ and $\sigma$ $$dS = \mu Sdt + \sigma SdX$$ Show by the application of Itô's Lemma to function $\log S$ ...
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81 views

How to prove this inequality involving integration with respect to Brownian motion?

If $B_t$ is the Brownian Motion, I have to verify that $$E\left\lvert\int_s^t G(t,w)\,dB_t\right\rvert^6\leq 15^2\cdot (t-s)^2\cdot\int_s^t E\lvert G(t,w)\rvert^6\,dt$$
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65 views

Probability of a brownian motion leaving some area

Let $B_t$, $t\geq0$ be a standard $n$-dimensional Brownian motion, that is $B_t(\omega)\in\mathbb{R}^n$ and let $\Lambda\subset\mathbb{R}^n$ be some ball such that the Brownian motion starts within ...
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0answers
96 views

Explanation of the Girsanov's transformation

The Girsanov's theorem is making me all confused. In my course literature they explain it by some simple discrete examples of coin-tossing etc. Saying that $Z$ is the ratio of $\frac{P^a(A)}{P(A)}$ ...