Questions tagged [stochastic-integrals]

This tag is used for questions about stochastic integrals – especially for calculations. For questions related to more theoretical aspects of stochastic integrals such as their constructions, (stochastic-analysis) may be more appropriate.

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17 views

limit of Stochastic Integration

Suppose that $X_t:=\int_{0}^{t}b_sds+\int_{0}^{t}\sigma_tdB_t$ where $b,\sigma \in L^{\infty}(F)$. For $\pi:0=t_0<t_1<...<t_n=T$, denote $$S_L(\pi):=\sum_{i=0}^{n-1}X_{t_i}B_{t_i,t_{i+1}}$$. ...
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25 views

Expectation of an Ito Process at a specific time

Suppose we have the following dynamics: $$dX_t = a_t X_t dt + b_t X_t dW_t; \;\;\; X_0 = x.$$ Would the expectation of $X_t$ at $t=T$ be $$\mathbf{E} [X_T] = x + \int_0^T a_t X_t dt$$? I know that it ...
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11 views

SDE for a positive cadlag, bounded and non-increasing process

Giving a process $(D_{t})_{0\leq t\leq T}$ with the following properties: adapted, cadlag, non-increasing, $D_{0}<=1$ and $D_{T}=0$. Can I write the SDE associated to that process in one of the ...
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1answer
27 views

Can we state the distribution of this Brownian motion as follows?

Consider the following Brownian motion: $W(e^{2t})$ Where $t\in[0,\infty)$. What can we then say about how $W(t)$ is distributed? Could we say: $W(e^{2t})\sim \sqrt{e^{2t}}N(0,1)$?
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29 views

Summary of $\frac{dM(t)}{M(t)}$ using Ito's formula in dimension $d$

EDIT: This question has been solved! Consider a $d$-dimensional Brownian motion $W$ and the stochastic differential equation(s) $$ \begin{align} dX&=b(X)dt+\rho(X)dW\\ X_0&=x, \end{align} $$ ...
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12 views

Time-Dependent Moments of General SDEs

Context: In some lecture slides for a class I am taking, we are given the equations $$d\langle x \rangle = \langle dx \rangle$$ $$d\langle x^2 \rangle = \langle 2\,x\,dx \rangle + \langle \frac{1}{2}\...
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34 views

Stochastic integral as a mapping in the integrator

Let $f$ be measurable and consider the mapping that maps \begin{align*} D([0,T])\to \mathbb{R}, (\omega_t)_{t\in[0,T]}\mapsto \int_0^T f_s d\omega_s, \end{align*} where the integral is meant to be an ...
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15 views

Covariance of stochastic integral with brownian motion

Let's $X(t) = r(t)$ and $r(t)$ is interest rate which follows 'Vasicek model' I want to get the proof; how multiplication of two expectaion can be the integral with the interval which have minimum ...
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20 views

How to use Stochastic integration by parts

Hey I have problem with understanding this calculation (first, second and third equality) In what way we use here integration by parts?
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23 views

Why $\int_t^T \int_0^s f(s,u)dW(u)ds=\int_0^t \int_t^T f(s,u)dsdW(u)+\int_t^T \int_u^Tf(s,u)dsdW(u)$??

Maybe it's trivial but why can we write this integral as the sum of these two integrals? We use Fubini Theorem but I don't understand why there are such limits on the right hand side.
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32 views

Solution for $dX=dt+2\sqrt{X}dW$ with positive-valued $\sqrt{X}$

Consider a Brownian motion $B$, $x\in\mathbb{R}_+$ and the stochastic process $X:=(\sqrt{x}+B)^2$. I want to show, that $X$ is a solution of the SDE $$dX = dt +2\sqrt{X} dW$$ where $W$ is some ...
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18 views

Reference that simple $L^2$ stochastic processes are dense in the set of predictable $L^2$ processes

Let $W=\{W_t\}_{t\in[0;T]}$ be a Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets. Let $\{X_t\}_{t\in[0;T]}$ be a read-valued, predictable and ...
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1answer
44 views

L2 constructions of Brownian motion that fails the continuity

The basic idea of $L^2$ constructions of Brownian motions is as follows. Let $\{\xi_k\}$ be a sequence of iid Gaussian random variables. Let $\{\phi_k\}$ be a complete orthonormal basis system in $L^2(...
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1answer
36 views

Solve Ito integral for continuous contributions to stock portfolio

I'm following Milevsky & Posner (2003) to model the value of a portfolio, $P$, assuming GBM (2) and a continuous contribution rate of $1$ (i.e. dollar-cost averaging): $$ \mathrm d S_t/S_t = \mu \...
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22 views

Convolution of gaussian process

Let $\text{GP}(\mu(x),k(x,x^{\prime})$ be a gaussian process. Here, $\mu$ is the mean function. Typically, $\mu$ equals to $0$. $k$ is the kernel function. Can you define a convolution of the gaussian ...
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22 views

Construction of Ito integral in oksendal's book

In Oksendal's book of Stochastic Differential Equations, I have a problem assimilating the proof of the second step in the process of constructing Ito integral, this is the statement of the second ...
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1answer
58 views

Martingale & Ito Integral Help

I have seen a few textbooks that say $ \int_{0}^{t} W_{s}ds$ is NOT a martingale. I believe they are correct, but I'm confused what is wrong with the logic below that seems to show it is a martingale. ...
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20 views

Stochastic Product Rule Example

When given a function I wanted to know the how the application relates to: $Y_t = X_tdY_t + Y_tdX_t + (dX_t)(dY_t)$ (I am okay with using the 2 variable 2 equation function form). For example consider:...
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1answer
24 views

How to show the sum of the squares process is deterministic

Can I please get some feedback on my work for the following problem? Is there a more simple approach to arrive to the solutions for part a? Thank you for your time and consideration! Let $X$ be a ...
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28 views

Finding expectation and variance of stochastic integral from given SDE

I am given the following interest rate model with SDE: $dX_t=\alpha \,dt+\sigma \,dB_t$ where $α$ and $σ$ are constants, and $B_t$ is a Brownian motion. I am supposed to determine $$ E\left[ \int_0^T ...
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22 views

Equivalence of distribution of stochastic integrals

For $(X,B)$ and $(Y,W)$ where $B$, and $W$ are Wiener process and $X$, and $Y$ are chosen so that $X \cdot B$ and $Y \cdot W$ makes sense, where $X \cdot B$ etc. are stochastic integrals; I want to ...
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25 views

What is the infinitesimal generator?

This is probably a textbook example. What is the infinitesimal generator of the process? dX(t) =-aX(t)dt + bdB(t) dY(t) =dB(t) where a,b are constants and it is the same Brownian motion in both ...
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155 views

Expectation of $1/V(t)$ finite, $V(t)$ strictly positive stochastic process

How can we prove that the expectation of the stochastic process $1/V(t)$, $\forall t \in [0,T]$, is finite? \begin{eqnarray}\nonumber \mathbb{E} \left[ 1 / V(t) \right] &<& \infty, \end{...
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1answer
27 views

How can I interpret infinitesimal/differential notation in involved integrals? E.g. in the Chapman-Kolmogorov condition

In our class on stochastic processes I keep getting confronted with a problem w.r.t. notation. I am a mathematics major and attended all the prerequisite classes on standard analysis. In those ...
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46 views

About Ito's rule

I have a questions about Exercise3.3.25 at Karatzas&Shreve"Brownian Motion and Stochastic Integral". $W=\{W_t, F_t; 0\leq t<\infty\}$ is a standard, one-dimensional Brownian motion ...
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34 views

$\mathbb{E}[\int_0^TW^2_sdW_s | W_T=c]$

Let $(W_t)_{t\in [0,T]}$ a standard Brownian Motion and let $c\in\mathbb{R}$. How to compute $$\mathbb{E}\left[\int_0^TW^2_sdW_s\mid W_T=c\right]?$$ We thought about rewriting the stochastic integral ...
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46 views

Product of two stochastic integrals is a martingale?

Let $(\Omega,(\mathcal{F}_t),\mathbb{P})$ be a filtered probability space, and let $(W_t,\mathcal{F}_t)$ and $(B_t,\mathcal{F}_t)$ be two independent Brownian motions adapted to the filtration $\...
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1answer
30 views

martingale representation of two independent Brownian motion

Let $W^1_t$ and $W^2_t$ be two independent standard Brownian motions. Then $W^2_t$ is a martingale with respect to the its own filtration but not adapted to the filtration generated by $W^1_t$. I want ...
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1answer
44 views

Why this stochastic integral is not necessarily a martingale

I am studying Stochastic Integral theory. My professor said that given $M_t$ a continuous martingale and $p\in (1,\infty)$ the stochastic integral $$\int_0^t M_s^{2p-1} \mathrm{sgn}(M_s) \mathrm{d}M_s,...
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51 views

Integrating wrt a stochastic process

I am confronted with the following expectation $$E_t \left[\int_t^Tg(S_s)dS_s \right]$$ Where $S_t$ is a stochastic process. How would we go about computing this quantity? If we can't do so in the ...
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22 views

Writing down the proof of Proposition 1.1(ii)(1.12) in Ikeda and Watanabe

I am studying chapter 2 of the book "Stochastic Differential Equations and Diffusion Processes by Nobuyuki Ikeda and Shinzo Watanabe." I am struggling to write down proof for the following ...
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52 views

quadratic variation and local martingale

Consider a predictable process $H_t$ and a continuous local martingale $M_t$ for $t \in [0,T]$ I want to show that $$X_t:=\left(\int_0^t H_s dM_s\right)^2 - \int_0^t H^2_s d \langle M_s\rangle $$ is ...
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1answer
31 views

Cross Variation of Ito Integral with $L^2$ bounded continuous martingale

I am trying to prove the identity: $$ < I(K), N_{\cdot}>_t = \int_0^t K_s \ d < B,N_{\cdot}>_s $$ where $I(K)_t$ is by definition $ \int_0^t K_s \ d B_s $, and we take $K$ as a ...
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1answer
92 views

integrability of random variables w.r.t. Brownian motion

Consider random variables $Z_t>0 \ \forall t \in [0,T] $, where $E[Z_T]=1$ and $Z_t=E[Z_T|F_t]$, where $F_t$ is the Brownian filtration and $H$, which is integrable with respect to Brownian motion ...
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35 views

Convergence of Random Integrals

Let $g$ and $F$ be functions defined on $[0,1]$. The function $F$ is distribution function. There are estimators $\hat{g}_n$ and $\hat{F}_n$ such that $$ \int_0^1 |\hat{g}_n(t) - g(t)| dt, \text{ and }...
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1answer
44 views

Is this stochastic integral well-defined?

Let $x_t = \int_{s=0}^t dW_s$ where $W_t$ is the standard Weiner process. Now I define $$ f(x_t) = \frac{1}{(2-x_t)^4} $$ Using Ito's Lemma, $$ df(x_t) = \frac{4}{(2-x_t)^5}dW_t - \frac{10}{(2-x_t)^6}...
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1answer
26 views

Showing the Itô logarithm is the inverse of the Doléans-Dade exponential

Consider the stochastic exponential: $F[M] = e^{M(t)-\frac{1}{2}\langle M\rangle(t)}$ for an local martingale $M$. Define: $$M:= \log(L(0)) + \int_0^* \frac{1}{L} dL $$ where $L$ is a strictly ...
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1answer
39 views

Conditional expectation of a function of a stochastic process

I've been stuck on figuring out how this expression came to be, and I just can't seem to able to find out. Basically, given a stochastic process Y that satisfies the SDE: $dY_t = -rY_tdt - \theta ...
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39 views

Efficient numerical solution of brownian motion SDE with non-smooth diffusivity

I'm trying to numerically solve the 1-dimensional stochastic differential equation for brownian motion (Fokker-Planck) with rapidly varying diffusivity $K$, $$ \textrm{(Ito)}\quad dX_t = \frac{\...
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0answers
58 views

Getting a continuous modification in existence proof of SDE (Oksendal)

I am reading Øksendal, Stochastic Differential Equations (6th ed.), the existence and uniqueness result for SDEs. But I am stuck at the last step where he shows that the solution can be take as a ...
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1answer
37 views

Coordinate change of the solution of an SDE

Let $X_t$ be the solution to an $d-$dimensional SDE $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$ and let $A$ be an invertible $d \times d$ matrix. Let $Y_t=AX_t$. Which SDE does $Y$ solve? I feel like it is ...
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1answer
79 views

Can this stochastic integral exist?

Assume you have a filtered probability space with a Brownian motion $B_t$. Let $X_t$ be a progressively measurable process with $P(\int_0^T X_s^2 ds <\infty)=1.$ Assume that $\int_0^TX_sdB_s=0$ a.s....
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1answer
35 views

Does the converse of Girsanov theorem hold in this case?

Given an underlying filtered probability space $\left( \Omega,\left\{ \mathcal{F}_t \right\}_{t\in[0,T]},P \right)$ that satisfies the usual conditions, define an OU process with $$ dY(t)=-\kappa Y(t)\...
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1answer
32 views

integral w.r.t. Brownian motion is gaussian process?

Consider the stochastic term from the solution of the SDE of an Ornstein-Uhlenbeck process: $$X_t=\sigma \int_0^te^{-\lambda(t-s)} \, dBs, $$ where $\sigma>0$. How can be seen that $X_t$ is a ...
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2answers
132 views

Expected value of $S_t$ where $dS_t=(\mu S_t+a)dt + (\sigma S_t +b)dW_t$

I have the stochastic differential equation: $$dS_t=(\mu S_t+a)dt + (\sigma S_t +b)dW_t$$ where $W_t$ is a Wiener process with $S_0 > 0$ and $\mu, \sigma, a, b \in \mathbb{R}$. I have found the ...
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0answers
27 views

Proving associativity for a certain stochastic integral.

In this question I want to prove the property which is answered in this question: Why can we change the differentials?(Brownian stochastic integrals) Description of the problem: Assume you have a ...
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1answer
36 views

Why can we change the differentials?(Brownian stochastic integrals)

Assume you have a filtered probability space $(\Omega, \mathcal{F},\mathcal{F}_t,P)$, assume that $B_t$ is a Brownian motion with respect to this filtered probability space. Define $$\text{sign}(x)=1, ...
5
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1answer
96 views

Definition of Stratonovich Integral

I have a doubt in definition of the Stratonovich integral. In "Stochastic Calculus for Finance" by Steven Shreve, he defines it using the midpoint $\frac {(t_i+t_{i+1})}{2}$ of the ...
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0answers
20 views

Example of function having finite quadratic variation except Brownian motion [duplicate]

I am looking for an example of function having finite quadratic variation except Brownian motion. As a I know that for Brownian motion B(t) have finite quadratic variation and quadratic variation of B=...
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1answer
52 views

Norm in the space of square integrable martingales [closed]

I was learning stochastic integration and encountered two different norms used in the space of square integrable martingales They are as follows: 1.Let M be a square integrable martingale, then $|M|_t$...

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