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
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 ...
3
votes
0answers
36 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 ...
2
votes
1answer
39 views

independence two stochastic processes

being $X, Y$ two continuous processes, $\theta \in R$ $U_t=\sin{(\theta)}X_t+\cos{(\theta)}Y_t$ $V_t=\cos{(\theta)}X_t-\sin{(\theta)}Y_t$ I have to show that U and V are independent brownian ...
0
votes
2answers
24 views

Deriving Geometric Brownian Motion's solution?

The Black Scholes model assumes the following underlying dynamics, known as Geometric Brownian Motion: $$dS_t=S_t(\mu dt+\sigma dW_t)$$ Then the solution is given: ...
1
vote
0answers
33 views

The completed natural filtration of brownian motion is right-continuous, proof?

I have a question concerning a claim in J.F. LeGall's book Mouvement brownien, martingales et calcul stochastique. Let $(\mathcal{F_{t}})$ be the canonical completed filtration on $\Omega$ of a real ...
2
votes
1answer
49 views

Ito formula applied to $\frac{1}{t}\int_0^t W_s ds $

I got this expression and I have to calculate its differential by the Ito formula, $W_t$ denotes the Brownian motion: $$\frac{1}{t}\int_0^t W_s ds $$ I calculate the derivative of ...
0
votes
1answer
16 views

Expectations of certain Brownian motion equations

$B_t$ is Brownian motion. It is assumed that motion starts at $0$. I do not understand how the highlighted equalities hold true. Is the first one equivalent to ...
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( ...
2
votes
0answers
25 views

generator of a function (stochastic) [closed]

How do I find a generator of $$g(Y_t)=Y_t^2-10Y_t+25 \, ,$$ where $Y_t$ is a geometric BM: $$dY_t=-1Y_tdt+2Y_tdW_t \, ,$$ and $W_t$ is white noise
0
votes
1answer
47 views

Solve Itô integral with power

$$\int_0^t e^{Ws} W_s^r dW_s$$ where $W_s$ is Wiener process and r> in $\mathbb{Z}$ My first approach would be to use Ito's lemma, however, coming up with the function $g(t,x)$ is difficult The ...
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 ...
0
votes
0answers
93 views

Property of G- Stochastic Calculus

i have maybe a stupid question about an equation. It is said that \begin{equation} \inf\limits_{P \in \mathcal{P}}\mathbb{E}_{P} \left[\int_0^T \varphi_{x}(t,X_{t})X_{t}\pi^{T}_{t}\,\mathrm ...
1
vote
1answer
48 views

(Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$

The purpose of this question is to complete my personal exposition on the rigorous proof of Ito's lemma. I have consulted more than half a dozen mathematical finance texts and not a single one, for ...
1
vote
1answer
95 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
1
vote
1answer
59 views

SDE and Stochastic calculus

$W_t$ is 1 dimension Brownian morion. $X_t=(cosW_t,sinW_t)$ Write SDE about $X_t$ I thought that $f(t,x)=(cosx,sinx)$, but I can't how "$t$" is expressed. I heard that the hint of this question is ...
2
votes
1answer
35 views

Question regarding Notes on Strong Markov Property

I wrote the following notes from a lecture a couple of weeks ago and I don't understand a particular line. Suppose $B_t$ is a Brownian Motion. Now look at $B^x_t = x + B_t$ which is a BM starting ...
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 ...
2
votes
1answer
63 views

Showing that a certain stochastic process does not have normal distributed increments

Edit: Question Resolved. See below. As a part of my bachelor thesis, I have to work through a paper about fake Brownian motion by Oleszkiewicz. In this paper he defines a stochastic process. Let ...
2
votes
1answer
36 views

Proving a Self Financing Portfolio

Question: Let $(S_t)_{t\ge 0}$ be a stock price process. Assume $u(.,.)$ satisfies the Black Scholes PDE with short rate $r=0$. Assume that under a risk neutral measure P: $$ dS_t=\sigma_tS_tdW_t $$ ...
4
votes
1answer
163 views

Lookback option with floating strike: boundary condition

I am trying to make sense of one of the boundary conditions of a look-back option with floating strike. Some notation first: let $v(t,x,y)$ denote the price at time $t$ of the option under the ...
0
votes
2answers
48 views

Using Ito's Lemma with more than one brownian motion term

Question : Let $$ dY_t=c_tdt+d_tdW^1_t+e_tdW^2_t $$ Where $W^1_t,~~W^2_t$ are standard independent brownian motions. I am trying to apply Ito's formula to this, say for example trying to find ...
0
votes
2answers
64 views

Is the following Itô-Integral not zero?

is the following statement true: $$\int_0^T t \, dW(t) \neq 0$$ I need it for a counter-example, that one can not change the order of integration between $dW$ and $dP(\omega)$. I thought of taking ...
4
votes
1answer
46 views

Expectation of $e^{-4B_\tau}$, where $\tau$ is an extended stopping time

This is an specific example so with a bit of luck I can get some general methodology from your answers. I have this stopping time: $$ \tau = \inf\{t \geq 0; B_t < t-2 \} $$ This is a clear ...
2
votes
1answer
44 views

The law of the iterated logarithm for BM and boundedness of stopping times

My question is regarding the usefulness of the law of the iterated logarithm, and its connection to stopping times. In many answers of this forum, I understand that some people often claim that some ...
5
votes
1answer
93 views

Is $t^{-\frac{1}{2}}B_{t^2}$ a Brownian Motion?

I think the title says it all. Let $X_t = t^{-\frac{1}{2}}B_{t^2}$, with $B_t$ being a brownian motion started at $0$. I think I have proved continuity at $0$ by doing the following: $$ X_t = ...
0
votes
1answer
51 views

Expectation of product of stochastic integral and brownian motion

Find the covariance: $$ COV((\int_t^T(T-s)dW_s), W_t) $$ I used the covariance formula: COV(X,Y) = E(XY) - E(X)E(Y) = E(XY) as E(X)=E(Y)=0 But I am stuck on figuring out the expectation of the ...
1
vote
1answer
42 views

Stochastic Integral Help

Let W(t) be a Brownian Motion. Show that the integral: $$ \int_t^T W(s)ds $$ can be written in terms of the stochastic integral: $$ \int_t^T (T-s)dW(S) $$ Is there an error with this question? I ...
2
votes
0answers
58 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 ...
4
votes
1answer
101 views

Ornstein-Uhlenbeck process: increments

I'm new to the forum so I hope this first question goes well. Let the Ornstein-Uhlenbeck process be defined as: $$ dV_t = - \beta V_t dt + \sigma dW_t $$ with $V_0 = v$, where $W_t$ is a Wiener ...
2
votes
0answers
52 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 ...
3
votes
1answer
52 views

Extended (or augmented) stopping times

I am trying to prove that $\tau$, defined as: $$ \tau = inf\{t > 0 \mbox{ }|\mbox{ } B_t < t-1 \} $$ is a stopping time with respect to the filtration $(\mathscr{F}_{t+}^B)_{t\geq 0}$ where ...
1
vote
1answer
68 views

Mean and Variance of Gaussian Process

Let $B = (B_t : t \geq 0)$ be a standard Brownian Motion. Fix $0 \leq s \leq t$. How can I prove that, conditionally on $\{B_s = x, B_t = z\}$, the intermediate value $$B_{\frac{t+s}{2}}$$ has ...
1
vote
1answer
60 views

Brownian motion transition density question

Let $Y_t = M_t - W_t$ where $M_t$ is the running maximum of brownian motion and $W_t$ is brownian motion. I want to show that $P^0[Y_{t+s} \in dy| Y_t = x] = p(s,x,y)+p(s,x,-y)$ where $p$ is the ...
0
votes
1answer
77 views

Ornstein-Uhlenbeck process and Markov property

There isn't a similar question in the forum, so here it goes. Firstly, let the O-U velocity process be defined as $$ dV_t = -\beta V_t dt + \sigma dB_t $$ with $V_0 = v$, and $B = (B_t), t \geq 0$ a ...
1
vote
1answer
129 views

Running average of Brownian motion

Question : Let us define the cumulative sum (Brownian motion): $$x_k = \sum_{i=1}^k y_i$$ and the running average : $$ \overline{x_k} =\frac{1}{W}\sum_{i=k-W+1}^k x_i$$ for $ k>W $, $W$ ...
2
votes
1answer
35 views

$\mathbb{E}[B_t-B_s], \mathbb{E}[\exp(\sigma(B_t-B_s))]$ etc.

This may be a duplicate but I cannot find the corresponding question. I have been asked to show: $\mathbb{E}[\exp(\sigma(B_t-B_s))] = \exp\left(-\dfrac{\sigma^2}{2}(s-t)\right)$ As a side note I ...
0
votes
1answer
41 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
91 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 ...
4
votes
0answers
132 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
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, ...
0
votes
0answers
92 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
15 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
44 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
62 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
41 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
30 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
20 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
65 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
42 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 ...