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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4
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40 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 ...
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0answers
5 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
votes
1answer
336 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
1
vote
0answers
49 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
2answers
40 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
40 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 ...
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 $\Omega$.
1
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1answer
16 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 ...
3
votes
1answer
289 views

Some basic questions about Stochastic Calculus

I have a transition function for a Markov process $X_t$. I want to find a density function for the stochastic process $Y_t := \int_0^t X_s \,ds$. Some questions about this: Is this the same as the ...
1
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1answer
275 views

Stochastic representation formula

Consider the following boundary value problem in the domain $[0,T]$ x $R$ for an unknown function F. $\frac{\partial F}{\partial t}(t,x) + \mu(t,x)\frac{\partial F}{\partial x}(t,x) + \frac ...
1
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1answer
16 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
602 views

Scalar product of Gaussian process

Assume that $n(t)$ is a White Gaussian Noise (WGN) process with $E[n(t)]=0$, $E[n(t)^2]=\sigma^2$ and $x(t)$ a deterministic function defined in $[0,T]$. How can I compute from first principles the ...
0
votes
1answer
19 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 ...
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
1answer
43 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
33 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 ...
1
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1answer
56 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
25 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
vote
1answer
74 views

Show that $Y=\int_0^1f(s)B_s \, ds$ is normal and find $\text{var}(Y)$.

$B_t$ is a standard Brownian motion, $f(t)$ is a continuous function on $[0,1]$. $Y=\int_0^1f(s)B_sds$. How to show $Y$ is normal. And what is the variance? I know I can use characteristic function ...
0
votes
1answer
54 views

Solution to the linear SDE $dX_t = \alpha X_t \, dt + \sqrt{2} dB_t$ using Itô calculus

So if I have the following generator and an initial condition: $$A(f)(x) = \alpha x f'(x) + f''(x) \\ X_0 = x \in \mathbb{R}^+$$ I've been asked to find $X_t$ and assume that $\alpha$ is a constant. ...
1
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1answer
25 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 ...
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0answers
24 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 ...
0
votes
1answer
29 views

A Property of the Ito Integral

Let $f,g \in \mathcal{V}(0,T)$ and let $0 \leq S < T.$ Then $E[\int^{T}_{S}f dB_t]=0$ Apparently this holds clearly for elementary functions, (Im not so sure), and can be obtained by taking ...
0
votes
1answer
60 views

Stochastic Integration and Ito Calculus

Before reading this I must not I think I am a little behind on some of the prereq for this topic but I really want to be able to understand it in a relatively meaningful way. I am having trouble ...
0
votes
3answers
47 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 ...
2
votes
1answer
127 views

Show that $dX_t=1_{X_t\not=0} dW_t$ does not have a pathwise unique solution.

Given the SDE : $$dX_t=1_{X_t\not=0} dW_t \qquad \text{with} \quad X_{0}=\xi $$ how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss Indeed Consider the ...
0
votes
1answer
48 views

Strong solution of stochastic differential equation

Consider the stochastic differenctial equation: $dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t$. How to find a strong solution?
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votes
2answers
81 views

How to solve the SDE $dX_t = \frac{b-X_t}{T-t} \,dt + dW_t$?

SDE: $$dX_t=\frac{b-X_t}{T-t}dt+dW_t,t<T, \qquad X_0 = a$$ Answer: Let $b(t)=\frac{-1}{T-t},c(t)=\frac{b}{T-t},\sigma(t)=1$, then $$\begin{align*} ...
4
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1answer
300 views

Ito's Lemma application

$Z(t) = \int_0^t g(s)\,dW(s)$, where $g$ is an adapted stochastic process. Find $dZ$ ?
2
votes
2answers
414 views

Explicit solution of a linear SDE

I'd like an explicit formula as a function of $W_t$ (standard Brownian motion) and $\lambda >0$ for the solution of the following SDE: $$\mathrm dX_t = \mathrm dW_t - \lambda X_t \,\mathrm dt$$ ...
0
votes
1answer
50 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 ...
3
votes
3answers
98 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
1answer
90 views

How to solve a linear stochastic differential equation?

I don't know how to find a solution of this stochastic differential equation: $dX_{t}=(1+\delta \mu X_{t})dt+\delta X_{t}dB_{t}$ Where $B_{t}$ is a standard Brownian motion and $\mu$ and $\delta$ ...
6
votes
1answer
136 views

Integral of Wiener Process and Central Limit Theorem

I am trying to solve the following exercise: (1) Given $W$ is a Wiener process, find a constant $M$ such that $\lim\limits_{t\to\infty} \frac{1}{t}\int_{0}^{t}\sin^2W_s ds=M$ (2) Then show ...
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0answers
21 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 ...
3
votes
1answer
83 views

Checking a solution for a SDE

I want to show that the process $Y(t) = e^t \int_0^t e^{-s}dW(s)$ satisfies the following SDE: $dX(t) = X(t)dt + dW(t), \ \ t\geq 0 , \quad X(0) = 0$ I think the right approach is to use Ito's ...
1
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1answer
32 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 ...
2
votes
1answer
80 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 $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
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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 ...
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1answer
56 views

Show independence of stochastic integral and stochastic process

Let $ M_t $ and $ N_t$ be two continuous local martingales with respect to a filtration $ \mathcal{F}_t $. Suppose that $ M_t $ and $ N_t$ are independent and set $X_t = \int_0^t M_s^4 \mathrm{d} M_s ...
0
votes
1answer
41 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 ...
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2answers
106 views

Representing a stochastic integral as product of a unknown random variable and a standard normal random variable

Consider a probability space $(\Omega,\mathcal F, (\mathcal F_t)_{t\geq0},\mathbb P)$ where $\mathbb F=(\mathcal F_t)_{t\geq0}$ is generated by $B=(B_t)_ { t \geq 0}$ a standard brownian motion ...
0
votes
1answer
53 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 ...
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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 ...
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1answer
42 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
34 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
63 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 ...
1
vote
1answer
39 views

Why can $\int_0^t f''(X_s) \, d\langle X \rangle_s$ not be a local martingale?

We know from Itos formula, if $X$ is a continuous local martingale and $f$ has two continuous derivatives, we can write $f(X_t)$ as $$ f(X_t) = \int_0^t f'(X_s) dX_s + \frac{1}{2} \int_0^t ...
0
votes
0answers
29 views

basic Stochastic differential equation

I'm sorry but I'm having some troubles to find a solution of this simple stochastic differential equation, $dX_{t}=2\sqrt{X_{t}}dB_{t}+2dt$ where $B_{t}$ is a Brownian motion, please can you help ...