This tag is used for questions about stochastic integrals - especially for calculations . For questions related to more theoretic aspects of stochastic integrals such as its construction. Stochastic-analysis may be a more appropriate tag.

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3
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0answers
20 views

What is the explicit obstruction to the failure of pointwise convergence in the stochastic integral?

Let $B(\omega,t)$ be a Brownian motion defined on some appropriately filtered probability space $(\Omega,\mathcal{F}_{t},\mathbb{P})$, and let $f(\omega,t)$ be a stochastic process defined on $\Omega$ ...
1
vote
1answer
18 views

Solution to truncated renewal function

Let's begin with some theory on the renewal process. In a renewal process $N(t)$, let $t$ denote the interarrival time, and $f(t)$ and $F(t)$ denote the PDF and CDF respectively. Let $M(t)=E[N(t)]$, ...
-1
votes
1answer
48 views

Solve the SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ [on hold]

Solve the following stochastic differential equations $ dX_t = \frac{1}{2 X_t} dt + dB_t$ or equivalently with a transformation $Y_t = X_t^2$ $ dY_t = dt + 2 \sqrt{Y_t} dB_t$ with $Y_0 = y_0 > ...
0
votes
1answer
32 views

Variance of integrated squared wiener process

So I'm trying to figure out the mean and variance of $X = \int_{0}^{1} W^2(t) dt $ where $W$ is the Wiener process. The mean I've worked out easily to be $\frac{\sigma^2}{2}$ but I'm having ...
0
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0answers
19 views

Gibbs Sampler integral computeable

here is an example of a changepoint in a poisson world with the gibbs sampler, it is an bayesian approach. the data are assumed to follow this distributions : $\begin{equation} \nonumber Y_i \sim ...
0
votes
0answers
13 views

Apply Ito's formula to Bessel prosess [closed]

Let $X_t=\sqrt{(B_t^1)^2+(B_t^2)^2+(B_t^3)^2}$ be a Bessel process (starting from 0) with respect to a 3-dimentional standard Brownian Motion $B_t=(B_t^1,B_t^2,B_t^3)$. How to apply Ito's formula to ...
0
votes
1answer
16 views

Ito integral's zero mean

My Sto Cal prof gave a long proof for the fact that $E[\int_{0}^{t} f_s dW_s] = 0$ where W is Brownian and f is Borel x $\mathscr{F}$-measurable, adapted and satisfies some integrability condition. ...
1
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1answer
30 views

Brownian motion on the circle and Itô processes

Consider the differential system \begin{cases} dX_t &=& -\frac{1}{2}X_t dt - Y_tdB_t, \\ dY_t &=& -\frac{1}{2}Y_tdt + X_tdB_t, \end{cases} $X_0 = 1$, $Y_0 = 0$. Let $X_t$ and $Y_t$ ...
1
vote
1answer
15 views

Basic question on application of Itô's formula to a stochastic process

I am working on a problem where I now find myself wanting to apply Itô's formula to: \begin{equation} X_t = \exp(W_t -W_0-\frac{t}{2}+\int\limits_0^tX_sds) \end{equation} where $W_t$ is 1D Brownian ...
0
votes
2answers
50 views

Verifying Property of Stochastic Integral

I am trying to verify this simple property for a stochastic integral. Given that f(t,w) is a bounded, nonanticipating function for a given Wiener process $W_t$ show that $E((\int_{0}^{T} f(s,w) ...
1
vote
0answers
108 views

Condition for a process to be a supermartingale

I am struggling in this question: Let $W$ denote a Brownian motion. Given that $ X_t = e^{- \lambda t} X_0 + \int_0^t \sigma e^{- \lambda (t-s)} \,dW_s$ solves the SDE \begin{equation} dX_t = - ...
1
vote
1answer
17 views

A variant of renewal function

Let's begin with some theory on the renewal process. In a renewal process $N(t)$, let $t$ denote the interarrival time, and $f(t)$ and $F(t)$ denote the PDF and CDF respectively. Let $M(t)=E[N(t)]$, ...
-1
votes
0answers
26 views

Solve SDE $dX_t = tX_tdt + e^{t^2/2}dW_t$

Solve $dX_t = tX_tdt + e^{t^2/2}dW_t, X_0 = \alpha$ by considering $X_t = a(t)(X_0 + \int_0^t b(s) dW_s)$, where a(t) and b(t) are not random. I don't think I quite understand stochastic ...
1
vote
1answer
32 views

Covariance of two geometric Brownian motions

Assume we have two geometric Brownian motions $$ dX_t = \mu X_t dt + \sigma X_t dW^1_t, \qquad \qquad dY_t = \mu Y_t dt + \sigma Y_t dW^2_t $$ where the Wiener processes are correlated with $E[dW^1_t ...
5
votes
0answers
118 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
2
votes
0answers
37 views

Why is the pathwise integral of $\alpha_s$ w.r.t the Lebesgue measure continuous?

My class notes on stochastic calculus say that the if $(\alpha_s(\omega))_{s\in \mathbb{R_+}}$ is progressive then $\int_0^t \alpha_s ds$ is a pathwise continuous process? How does the joint ...
1
vote
1answer
20 views

Proof of continuity of stochastic processes defined by Ito integrals

I'm currently trying to understand the proof of Theorem 4.6.2 in Kuo, Hui-Hsiung: Introduction to Stochastic Integration: Suppose $f \in L^2_{ad} ([a,b] \times \Omega )$, then the stochastic ...
0
votes
0answers
31 views

Girsanov's formula for an Ornstein-Uhlenbeck process

This is homework so no answers please. Question:If I know that for an OU process $X_t\stackrel{d}{=}e^{-t} B_{e^{2t}}$, can I use that for the Radon-Nikodym derivative of $X_t$? Context and Attempt ...
0
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0answers
27 views

Proof of equality in Expectation with the Help of a Brownian Motion (Put-Call-Symmetry)

Hey I want to reproduce a proof of Damien Lamberton; proof begins at page 14. Under some assumptions i want to show that \begin{align} \sup_{t\in \mathcal T_{0,T}}\mathbb ...
3
votes
1answer
64 views

conditional expectation of some solution of SDE

Let $(M_t)$ be a nonnegative martingale in a probability space $(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, \mathbb{P} )$ given by \begin{equation} dM_t = M_t \sigma_t dW_t \end{equation} for some ...
2
votes
0answers
13 views

Numerical integration scheme for stochastic system driven by colored noise (filtered white noise)

I have given quite a few hours to this problem, but I seem to be getting nowhere. Can anyone just give a hint or point towards a text on where to go looking for the concept and solution.
4
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1answer
90 views

Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

Consider the standard Brownian motion on $[0,1]$: $$ dB_t, \; B_0 = 0, $$ defined on the probability space $(\Omega, P)$. It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times ...
1
vote
0answers
70 views

Measurability of solution of diffusion equation in sub-sigma algebra

I want to solve the following problem: Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$. Let $a( x;. )$ and $f(x;.)$ be ...
0
votes
1answer
17 views

Prove that $\sigma (\cap_{i \in I} C_i)=\cap_{i \in I} \sigma (C_i)$

Do we have the following identity? $$\sigma (\cap_{i \in I} C_i)=\cap_{i \in I} \sigma (C_i)$$ Here $C_i$ is a subset of a set $\Omega$.
0
votes
2answers
57 views

What is an alternative book to oksendal's stochastic differential equation: An introduction?

What is an alternative book to oksendal's stochastic differential equation: An introduction? But also An alternative that is over 300 pages and at the same level? Some professor refer that book as a ...
1
vote
1answer
46 views

Will this well enough to serve as a prerequisite to oksendal's book?

Will this well enough to serve as a prerequisite to oksendal's stochastic differential equations: an introduction with applications book? I refer to shiryeav's probability, but i guess it still miss ...
1
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1answer
18 views

Prove that $B \in \Lambda_\text{loc}^2 $ if $B=(B_t)_{t \in \mathbb{R_+}}$ is a real valued B.M

I know that $\Lambda_\text{loc}^2=\{\phi $ is progressive $: \forall t \geq 0,\int_0^t \phi_s^2 \, ds < \infty\text{ a.s.} \}$ Since B.m $B_t$ is almost surely continuous and ...
1
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1answer
17 views

A question on the extension of of integrants from simple processes t0 $L^2$?

I have a question. While defining the Stochastic integral w.r.t to the Brownian Motion we begin with simple processes which are adapted and left continuous and then extend it to the square integrable ...
0
votes
1answer
20 views

Inequality regarding convex combination of random variables

In the appendix of notes on stochastic integration that i am reading, Mazur's Lemma is presented as following http://i.stack.imgur.com/GUyXN.png I have trouble understanding/proving the following ...
0
votes
1answer
51 views

Applying Ito's formula

This is probably an easy question but I am getting aquanted with Ito's formula and stuck on an exercise in my textbook. Let $X_{t}=W_{t}-a t/2$ where $a$ is a real number and $W_{t}$ is brownian ...
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0answers
39 views

Expectation of Exponential of Stochastic Integral

Let $z$ be the standard Brownian motion, $\omega$ an element of the sample space. Is it true that $$ \mathbf E\bigg[\exp\Big(\int_0^t f(\omega,s)\,\mathrm dz(s)\Big)\bigg] = \mathbf ...
0
votes
1answer
29 views

The independence between stochastic integral and sigma-algebra

Let $(\Omega, \mathcal{F}, \mathbb{P} )$ be the probability space, and {$W_t,0\leq t\leq T$} is a Brownian motion and $\mathcal{F_t}^W$ is the canonical filtration. For the $f(t)\in L^2([0, T])$(a ...
1
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0answers
27 views

Stochastic integral density of simple functions no1

I am trying to understand proposition 2.6 page p.134 from Karatza's book Brownian motion and stochastic calculus. If $M$ is continuous square integrable martingale on $(\Omega, \mathcal{F}, P)$ and ...
2
votes
0answers
47 views

Integral of a geometric Brownian motion [duplicate]

I would like to compute $G$ defined as follows $$G(t):= \exp(-\int _0^t h_s~ ds )$$ with $h$ being a geometric Brownian Motion. For that I would need first to compute $$\int_0^t ...
0
votes
3answers
50 views

Change of Variables Theorem

I am searching for a proof of the following theorem: THEOREM Suppose $(X_1, \ldots, X_n)$ is a random vector with joint density function $f_{X_1, \ldots, X_n}(x_1, \ldots , x_n)$ and $g$ is ...
0
votes
1answer
56 views

The ito integral is gaussian [duplicate]

Let $\Omega, F, P)$ be the classic setting. I saw that if $f$ is a function which satisfies some assumptions then the integral with respect to the brownian motion is Gaussian. Ie $\int_{0}^{t} f_u ...
1
vote
1answer
27 views

Resource on Pathwise Computations Involving Brownian Motion

Let $B_{t}(\omega)$ be a standard Brownian motion on $(\Omega,\mathcal{F},\mathbb{P})$. I read in a footnote recently that almost surely the quadratic variation ...
3
votes
3answers
105 views

1-dimentional stochastic differential equation

I would like to solve this SDE $$dX_{t}=\left(\sqrt{1+X^{2}}+\dfrac{1}{2}\right)dt+\sqrt{1+X^{2}} dB_{t}$$ I've tried to solve first the homogeneous equation ...
0
votes
0answers
25 views

Computational rules for expectations of functions of wiener processes.

What are some general rules that are helpful for computation/calculation of expectations such as $$ E(X_t | \mathcal{F_s} ), $$ where $X $ is a function of Brownian motions $W_t$ and $\mathcal{F}$ is ...
1
vote
1answer
36 views

Paley Wiener stochastic integral

Sorry for the stupid question, no answers necessary anymore! let $(B_t)_{t\in [0,1]}$ be a standard Brownian motion and $F\in C[0,1]$ differentiable. Then the sequence (which is an easy version of ...
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0answers
28 views

Convergence in $L^2(\Omega\times (0,T))$

Let $$f_i=\exp(\int_0^T h_i(s)\,{\rm d}W_s-1/2\int_0^T h^2_i(s)\,{\rm d}s)$$ where $W_s$ is a brownian motion in a probability space $(\Omega,F,P) $ and $h_i\in L^2(0,T) $. Suppose $F_n\to F$ in ...
2
votes
1answer
120 views

Hermite Polynomials and Brownian motion

I am asked to prove the following : Let $B_t$ be a standard brownian motion. The $n$th Hermite polynomial is $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
1
vote
1answer
61 views

Integral with respect to brownian motion

Let $f$ be a continuous function on $[0,\infty)$ and $B_t$ be a standard Brownian motion. Define $X_t=\int_0^t f(s) dB(s).$ a) Show that $X_t$ is Gaussian and computer its covariance $C(X_s, X_t)$ ...
0
votes
1answer
48 views

absolute continuity - Dirac measure with respect to gaussian measure [duplicate]

Let $a \in \mathbb{R}$ and Dirac measure $\delta_a (A) = 0$ if $a \notin A$ and $\delta_a(A) = 1$ if $a \in A$, and let $\mu_1$ be the one-dimensional gaussian measure. Let $\mu$ and $\nu$ be two ...
0
votes
1answer
65 views

Is this Stochastic integral a martingale ?

Let $(B_t)$ be a Brownian motion and set $X_t = \int_0^t B_t^2 dB_s$. Is $X_t$ martingale? My idea is to rewrite $X_t$ in terms of Ito's Formula $(f(x) = \frac{1}{3}x^3)$ $X_t = \int_0^t B_t^2 dB_s ...
0
votes
0answers
13 views

Determining Bounds of a Generating Function of a Stopping Time [duplicate]

Consider the diffusion process $$DX_t=b(X_t)dt+\sigma(X_t)dW_t$$ where $\sigma\sigma*$ is positive definite and $b, \sigma$ are smooth and bounded. Given a one-dimensional domain bounded from 1 side ...
1
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1answer
45 views

Variance process of stochastic integral and brownian motion

Let $(W_t)$ be a Brownian motion with respect to a filtration $(\mathcal{F}_t)$. For all $t \geq 0 $ set $$X_t = \int_0^t W_s^2 \mathrm{d} W_s,\qquad Y_t = W_t^7.$$ Find the covariance process ...
0
votes
2answers
39 views

simple stochastic differentiate

someone can help me to differentiate $$a(t-1)+bt+(1-t)\int_{0}^{t}\dfrac{dB_s}{1-s}?$$ I've tried but I really don't know how to do with the last part.. Thank you somuch for your help
2
votes
1answer
43 views

Kunita Watanabe Identity

I am looking for a proof of the following version of Kunita Watanabe Identity: "Let $M,N \in M_{c,loc}$ and $H$ be a locally bounded previsible process. Then $[H \cdot M, N ] = H \cdot [M,N]$" I ...
1
vote
2answers
69 views

Stratonovich integral

I'm having some troubles to calculate the Stratonovich integral $I(sin)(t)=\int_{0}^{t}\sin{B_{s}}dB_{s}$. I've tried with the limit of ...