-2
votes
0answers
6 views

Ito's lemma on $a=\int_0^t x(q) \mathrm{d}B(q)$,where $B=$brownian motion process. [on hold]

Can someone help me apply Ito's lemma on $a=\int_0^t x(q) \mathrm{d}B(q)$, where $B=$brownian motion process. I did this so far: $$\mathrm{d} a=\frac{\partial }{\partial B} \bigg[ \int_0^t x(q) ...
0
votes
1answer
22 views

$dX_t=-\mu X_tdt + \sigma dW_t$. Prove that $X_t = e^{-\mu t}X_0 + \sigma \int_0^t e^{-\mu(t-u)}dW_u $

So the solution says use Ito-s formula, taking $Y_t:= e^{\mu t}X_t$ to obtain $dY_t = [\mu e^{\mu t}X_t - e^{\mu t}\mu X_t + e^\mu t \sigma dW_t] $. As far as I can see though, Ito's formula says ...
3
votes
1answer
62 views

Integrating brownian motion times exponential function

I am trying to calculate $$\int_0^tB_se^{\lambda s}ds$$ but I am unsure of how to start the computation. The motivation behind this is that I read (and am now trying to prove) that ...
3
votes
0answers
99 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 ...
1
vote
0answers
24 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, ...
0
votes
0answers
27 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? ...
0
votes
0answers
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 \\ ...
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) - ...
2
votes
0answers
39 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 ...
1
vote
1answer
24 views

Clarification about a very simple stochastic integral

I'm studying stochastic integrals right now and I feel like this question is incredibly easy but I'm not sure. I want to evaluate $\int_0^t sdB_s$. Using Ito's formula I get $tB_t$ by setting ...
1
vote
1answer
24 views

Integrating the difference of brownian motion

I'm reading the solutions to an exercise where it is stated that $$\int_t^T\Big(W(u) - W(t)\Big)du = \int_t^T (T-u)dW(u).$$ But can someone enlighten me to what theorem/rule can be used to show this? ...
0
votes
0answers
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 ...
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 ...
0
votes
1answer
30 views

A few questions about Stochastic Processes and Numerical Methods

I am having a few problems understanding the Ornstein Uhlenbeck solutions, on wikipedia under solution (http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process) it described using variation of ...
1
vote
1answer
42 views

First hitting time in the one-dimensional case by solving a boundary value problem

If have a question about section 3.1 in the paper Kramers' law: Validity, derivations and generalisations by Nils Berglund. (See http://arxiv.org/abs/1106.5799 page 7 - 9) On page 8 it says, that ...
0
votes
0answers
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 ...
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 $e^{bt}$ ...
4
votes
1answer
60 views

Laws and Moments of two dimensional brownian motions

I am a bit rusty on this. So let us consider the following two dimensional standard Brownian motion issued from zero defined on the probability space $(\Omega, \mathcal{F},\mathbb{P})$ (note that, in ...
4
votes
0answers
39 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 ...
1
vote
1answer
74 views

Quadratic covariation of Itô processes

I haven't found any similar question in the forum, so I trust some of you will find this thought-provoking (at the very least). Perhaps you can help me. Let's consider first the two following ...
0
votes
1answer
67 views

expectations of Brownian motions

Let $B_t$ be a standard Brownian motion started at zero, and let $M_t$ be a stochastic process defined by $M_t=3\int_0^{t^{1/9}} s^4dB_s$ Compute $E\left[1+\int_0^t(1+M_s)^4 dM_s\right]$. Compute ...
0
votes
1answer
48 views

Show that a process is no semimartingale

_Hello everyone! I got a little question about how to show that the process $X_t:=|B_t|^{\frac{1}{3}}$ is NOT a semimartingale. So far I tried to apply Ito. Since if $X_t$ was a semimartingale so is ...
3
votes
1answer
180 views

Expectation of stochastic integrals related to Brownian Motion

I'm trying to solve a problem that's now doing my head in a bit. I'll share with you the question and let's see if somebody can shed some light into the matter: Let B be a standard Brownian Motion ...
-1
votes
1answer
44 views

$B(t)$ is brownian motion. I want Find $d(M(t))^2$,where $M(t)=e^{B(t)-\frac{t}{2}}$,

let $B(t)$ is brownian motion. Find $d(M(t))^2$,where $M(t)=e^{B(t)-\frac{t}{2}}$,
1
vote
2answers
41 views

I want Find $d(\frac{X(t)}{Y(t)})$ where $B(t)$ is a brownian motion and $X(t)=tB (t)$ and $Y(t)=e^{B(t)}$.

Let $B(t)$ is a brownian motion and $X(t)=tB (t)$ and $Y(t)=e^{B(t)}$. Find $d(\frac{X(t)}{Y(t)})$ Thanks for help.
1
vote
1answer
87 views

Exercise 3.3.25 of Karatzas and Shreve

This is the Exercise 3.25 of Karatzas and Shreve on page 163 Whith $W=\{W_t, \mathcal F_t; 0\leq t<\infty\}$ a standard, one-dimensional Brownian motion and $X$ a measurable, adapted 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 ...
3
votes
1answer
100 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, ...
0
votes
2answers
61 views

Variance of sum of two ito integrals

I don't really understand how to solve the following problem: Var(X) where X = $\int_0^2 2t dW(t) + \int_4^6 W(t) dW(t)$ If I use $E [(A+B)^2] = E(A^2) + E(B^2) + 2E(AB)$ I get to the point where I ...
1
vote
1answer
72 views

$\mathbb{E} \int_a^b W^3(t)\,dW(t)=?$

Is it true that $\mathbb{E} \int_a^b W^3(t)\,dW(t)=0$, for $a < b \in \mathbb{R}$ I know that for an adapted process $\Delta(t), t\geq 0$, the integral $\int_0^t \Delta(u)dW(u)$ is a ...
1
vote
2answers
63 views

How to show that $\mathbb{E}(\int_0^T t\mathrm \, dW_t) = 0 $?

I just want to know why $\mathbb{E}\left(\int_0^T t \,\mathrm dW_t\right)=0$. I know it's got something to do with the Gaussian distribution but I don't really know what.
1
vote
1answer
129 views

Application of Optional Sampling Theorem

Lets assume that Brownian Motion starts from some point $x$ for which $a<x<b$ holds. Let $\tau=\inf\{t:B_t\not\in [a,b]\}$ be a stopping time. Now I want to prove that for $\theta>0$ ,an ...
2
votes
3answers
129 views

Distribution of stochastic integral

Assume that $\mathrm{d}S = \sigma \, \mathrm{d}W$ with initial level $S(0)$ and where $\mathrm{d}W$ is usual Brownian motion. Now $A(T) = \frac{1}{T} \int_0^T S(t) \, \mathrm{d}t$ . What is the ...
0
votes
0answers
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 ...
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 $x_t$ ...
1
vote
0answers
24 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 ...
0
votes
1answer
44 views

Brownian Motion hitting random point

I got a problem that seems to be quite standard and easy, but I have lots of problems with it. I do already know that $T_a:=\inf\{t\geq 0: B_t=a\}$ is a stopping time for any $a\in\mathbb{R}$ fixed, ...
1
vote
1answer
91 views

hitting time of Brownian motion

I'm desperately trying to prove that for a standard BM $B_t$ the stopping time $T:=\inf\{t\geq0: B_t\geq\sqrt{1+t}\}$ is a.s. finite, i.e. $\mathbb{P}[T<\infty]=1$. I actually tried to play around ...
4
votes
1answer
145 views

Exponentials of stochastic processes and Brownian motions

This is my first time looking at problems in stochastic calculus, so please bare with the simplicity of the question. As always, any help is greatly appreciated. 1) Given $X_t=\int_0^ur_sds$ for a ...
0
votes
1answer
81 views

Question regarding Ito integral

I have a question regarding Ito integral, in particular, when I am trying to prove the normality of Ito integral, I encountered the following differential equation I need to solve: $$dX_{t} = ...
0
votes
1answer
175 views

Joint Distribution of two correlated ito integral

I have a question regarding finding the joint distribution of two process$$dX_{t}=a_{t}dB_{t}$$$$dY_{t}=b_{t}dW_{t}$$where $B_{t}$ and $W_{t}$ are two Brownian motions with correlated increments, in ...
2
votes
1answer
106 views

Question regarding Ito Process

I am new to Ito Process, so I have a following question. Consider a standard Ito Process, $$X_t=X_0+\int_0^t\mu_sds+\int_0^t\sigma_sdW_s$$ where W is the m-dimentional Brownian motion and X is a ...
0
votes
1answer
141 views

Mean and variance of a brownian bridge

I am trying to compute mean and variance of the stochastic process $X_t$, which is a Brownian bridge from x to y, in the time-interval $[t,T]$. $$X_t = y + ...
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
1answer
78 views

Computing expectation of a stochastic integral

I need to compute the expectation $$E\left[\int_0^tu \, dB_u \int_0^s u \, dB_u \right].$$ Being that is my first question, how can I initialize MathJax if I have it on my hard drive.
1
vote
1answer
308 views

How to integrate a Wiener process that freezes at a determined time?

I would like to calculate the expected variance of the average of a Wiener process from time $0$ to time $1$. The equation I believe I am trying to solve is: $$ \mathbb{E} \left[ \left( \int_0^1 W_t ...
2
votes
1answer
86 views

Limit of a stochastic integral

Let $W_t$ be a one-dimensional Brownian motion and I would like to prove $$\lim_{\beta\rightarrow+\infty}\sup_{0\leq t\leq T}\left|e^{-\beta t} \int_0^te^{\beta s}\mathrm dW_s\right|=0$$ This is an ...
1
vote
0answers
352 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 ...
4
votes
1answer
358 views

Brownian bridge

Let $W = (W_t;F_t)$, $t \leq 0$ be a standard Wiener process, and let $(X_t)_{0 \leq t \leq 1}$ satisfy the stochastic differential equation $$ dX_t =- \frac{X_t}{1-t}dt+dW_t,\quad 0 \leq t \leq ...
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 ...