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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Applying the Multivariate Ito Formula

I want to show that the stochastic process $$ S_t^i = S_0^i \exp\left( \int_0^t \left(\mu_s^i - \frac{1}{2} \sum_{j=1}^m (\sigma_s)^{ij} \right)^2 d s + \sum_{j=1}^m \sigma_t^{ij} S_t^i dW_t^j ...
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9 views

Ito Isometry on Multivariable indicator function

The background of this question is a paper written by Morten O.Ravn and Harald Uhlig, titled "On Adjusting The HODRICK-PRESCOTT Filter For The Frequency of Observations" I will very much appreciate ...
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0answers
21 views

What is a stochastic differential equation of the form $dZ = f(Z_{prev}, X_{prev})dt + CdW_t$ called?

At every time step I can approximate the change in $Z$ using the following equation: $$ dZ = f(Z_{prev}, X_{prev})dt + CdW_t, \quad(1)$$ $$dW_t = r\sqrt{dt}$$ where $C$ is some constant, and $r$ is ...
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0answers
12 views

Mean and variance regime-switching model

Suppose we have the following model for stock price: $$ X_{t}=X_{0}\exp\left(\int_{0}^{t}(r-\frac{1}{2}\sigma_{\epsilon(s)}^2)ds+\int_{0}^{t} \sigma_{\epsilon(s)}dW_{s}\right) $$ This follows a normal ...
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0answers
27 views

covariance of a function of Wiener processes

Consider two independent Wiener processes, $W_1$ and $W_2$. The covariance of certain functions of Wiener processes is simple, for example ...
2
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0answers
17 views

Convergence of a stochastic integral [duplicate]

Let $(B_t)$ the standard Brownian Motion and $(H_t)$ be an adapted continuous process. Show that $$\frac{1}{B_t}\int _0^tH_sdB_s $$ converge in probability. I guess that the limit is $H_0$ but I ...
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1answer
23 views

Question regrading to a definition in Stochastic Calculus for Finance 2 by Shreve

I am confused with a definition in Shreve's Stochastic Caclulus for Finance 2 book. In page 129, Theorem 4.2.2, the Ito isometry theorem. It states that The Ito integral defined before satisfies ...
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1answer
58 views

$\sup_{t \in [0, \infty)} \left|[(H^{(n)} - H) \cdot X, Y]_t \right| \overset{P}{\rightarrow} 0$

1. Notation We start with establishing some (standard, I think) notation. Let $(\Omega, \mathcal{A}, P)$ be a given probability space. For any filtration $\mathcal{G} = (\mathcal{G}_t)_{t \in ...
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0answers
22 views

Discrete stochastic integral and optional sampling theorem

I want to prove the optional sampling theorem using the fact that discrete stochastic integrals for martingale integrators are still martingale. To prove: if $(M_t)_{t\in T}$ is a martingale and ...
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0answers
64 views

Problem including SDE

I have following problem. Let $Y_{t}$ be an exponential Lévy Process. That is: $$Y_{t} = Y_{0}e^{X_{t}}$$ Where $X_{t}$ is Lévy process. I have a function of $Y_{t}$, $f$ :$\mathbb{R}_{+} \times ...
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1answer
37 views

Is every continuous local martingale a uniform limit of step-processes?

The following question pertains to Wengenroth's textbook "Wahrscheinlichkeitstheorie", de Gruyter 2008 (in German). The covariance (aka compensator) of the continuous local martingales $X, Y \in ...
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1answer
25 views

Ito's Isometry using Brownian Motion

Let $B_t$ be standard Brownian Motion. Could someone please help me to show that $$E[(\int_{0}^{t}B_sdB_s)^2] = \int_{0}^{t}E[B_s^2]ds$$ I am sure that it has something to do with Ito's formula but ...
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1answer
39 views

Construction on Ito Integral with Brownian Motion

I have just started learning stochastic calculus and my professor posed the following as exercises to help understand how we construct the Ito Integral. Let $B$ be a standard Brownian motion. Fix ...
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1answer
42 views

Ito Integral Properties with Brownian Motion

I am working out some of the properties for the Ito integral with Brownian motion and I am trying to use the definition to verify that $$ \int _0 ^t s \, dB_s = tB_t - \int _0 ^t B_s\, ds $$ and $$ ...
2
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1answer
25 views

how to derive the stochastic differential equation of this process

How can I derive the SDE for the vasicek model : $$r_t = 0.1 + 0.1 e^{-t} + e^{-t}\int_0 ^t e^s dB_s$$ From observation, the SDE vasicek's model is such that: $$dr_t = b(a-r_t)dt + \sigma dB_t$$ ...
2
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0answers
21 views

Variance of Riemann integral of Stochastic integral

Let $f: \mathbb{R} \to \mathbb{R}$ be deterministic and let $W$ be a standard Brownian motion. Then by Ito's isometry we know $$ Var\left( \int_0^u f(s) dW(s) \right) = \int_0^u f^2(s) ds. $$ Now, ...
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0answers
26 views

Does Ito's Isometry hold if the integrand has a brownian motion in it?

I am wondering what is the distribution of: $$ \int_0^tW_sdW_s $$ Solution: (Thanks to @muaddib) Applying Ito's Formula to $W_t^2$ gives $d(W_t^2) = 2W_tdW_t +dt$, and so: $$ \int_0^tW_sdW_s= W_t^2 ...
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0answers
18 views

Can we integrate brownian motion with respect to a deterministic function

Let $B_t$ be brownian motion at time $t$, and $x$ be some random variable. For instance, I know that $$\int_0^T 1 dB_t = 1(B_T-B_0)$$ And that $$\int_0^T \cos(B_t) dB_t$$ cannot be directly ...
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Proof an equality related to Ito Integral

I have got a question: $h(u)$ is a continuous function, $h(u)$ also has bounded variation. I have to proof the equality that $\int_0^t h(s)dB(s)=h(t)B(t)-\int_0^tB(s)dh(s)$ I try to use $It\hat o$ ...
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32 views

Modified Stochastic Integral

Suppose one has a stochastic differential equation: $$dX_t = f(X_t) dt + g(X_t)d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process: $$d\eta(t) = \lambda \eta(t) dt + \sigma dW(t)$$ Suppose ...
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1answer
43 views

Deriving a closed form expression for stochastic integral (to show its a martingale)

I have $B_s = $ brownian motion at time $s$. $$ \int_0 ^t B_s \, dB_s$$ $$0 \leq t \leq T$$ And want to check if it is a martingale, first from its closed form expression, and then via conditions on ...
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1answer
33 views

How to check if integral wrt Brownian motion is a martingale

As in title, I have a process $$X_{t}=\int_{0}^{t}s^{2}dB_{s}$$ I found here a sufficient condition for such integral to be a martingale on the interval. But I am asked if it is a martingale, not ...
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1answer
72 views

Reading list to master Numerical Analysis' research literature

As of lately I have been going through many research papers in my current job, and even though I have a Mathematics background at Masters level in Mathematical Finance, I sometimes struggle to follow ...
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0answers
11 views

Levy property of transformed subordinator

Let $Z$ be a subordinator Levy process (which has a Levy density). Let $\rho<0$ and consider $$M_t = \sum_{s\leq t} (e^{\rho \Delta Z_{s} }-1)$$ I want to establish the Levy property of $M$. ...
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27 views

Why aren't these two sets of stochastic processes equal?

I'm learning about stochastic integrals now, and I don't understand the following: If $S$ and $L$ are two classes of processes where: $S=\{f(s,\omega) |f $ is progressively measurable and ...
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0answers
17 views

Expectation of Modified Stochastic Integral

Suppose one has a stochastic differential equation: $$dX_t = f(X_t) dt + g(X_t)d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process: $$d\eta(t) = \lambda \eta(t) dt + \sigma dW(t)$$ Suppose ...
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1answer
41 views

Calculate a differenciation [closed]

$$a>0,$$ $$b>0,$$ $$\sigma >0$$ $X$ is the solution of : $$dX_t=aX_t(b-X_t)\,dt+\sigma X_t \, dB_t,\quad X_{0}=1 $$ I have also shown before that $$L_t=e^{(ab-\sigma^2/2)t+\sigma B_t}$$ Now ...
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0answers
25 views

Girsanov theorem [duplicate]

I work on an exercice and I have to calculate: $$E(W_{t}^2e^{(\int_{0}^{T}\theta_{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta_{s}^2ds)})$$ $$\theta$$ is deterministic function I don't know how to ...
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0answers
29 views

Stochastic Differential Equation for Time Integral of Stochastic Process

Let $X(t)$ denote standard Brownian motion $dX(t) = a X dt + X dW(t)$ with solution $X(t) = e^{a t + W(t)}$. I want to consider the time-integrated process \begin{equation} Y(t) := \int_0^t d\tau~ ...
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1answer
33 views

positiv Martingale process

I would to like to prove that the process: $$e^{\int_{0}^{T}\theta _{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2ds}$$ is a martingale which is positiv and has a mean=1 $$\theta is continuous ...
2
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0answers
35 views

Integral of Brownian Motion with respect to an independent Brownian motion

I have this seemingly simple problem which I haven't been able to solve. I have two standard Brownian motions, $B$ and $W$, on the same probability space and under the same filtration (I am not so ...
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1answer
21 views

Conditions for Expectation of Ito Integral to have Expectation 0

Consider the Ito stochastic process $$X_t = X_0 + \int_{0}^{t} a_s ds + \int_{0}^{t} b_s dW_s$$ What conditions are necessary or sufficient (besides adaptability/measurability) to show that $$ E ...
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2answers
36 views

Two-dimensional Brownian motion

Let $B_1$ and $B_2$ be two $\mathbb{R}$-valued Brownian motions with $$\langle B_1,B_2\rangle=\int_0^t\rho_s ds,$$ where $\rho$ is progressively measurable with values in $(-1,1)$. We define ...
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0answers
8 views

Doleans measure for local martingales

I came across the following question in my textbook and something in it doesn't quite make sense to me. Let $M$ be a local $L^2$ martingale. Then $X,Y \in \mathcal{L}(M,\mathcal{P})$ are ...
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1answer
27 views

Density of stochastic integral

I am working on finding the PDF of $X_t^2$, where $X_t = \int_0^t A(u) \,dW_u$, a Wiener integral, i.e., $W_t$ is Brownian motion and $A(t)$ is a deterministic function. Here, would like to ask that ...
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1answer
71 views

An application of Itô's lemma

I found this question in a past exam for a course on Financial Economics. Given the function $f(t,x)$, let $F(t,x)$ be a function such that $∂F/∂x = f$. (a) By writing Itô’s formula in ...
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1answer
36 views

Stochastic Integral basics

As far as I understand, the stochastic integral is defined so that we can make sense of something like this: \begin{equation*} X_t = x_0 + \int_0^t g(s) ds + \int_0^t f(s) dW(s) \end{equation*} ...
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2answers
40 views

Total Differential / Ito dynamics

I found this process in a scientific paper: $M_t = \int_{0}^t e^{-(t-u)} \frac{dS_u}{S_u}$ where $dS_t = S_t (\phi M_t + (1-\phi)\mu_t) dt + \sigma S_t dW_t$ and I want to compute the ...
4
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1answer
58 views

Expectation of an Itô integral

I'm interested in computing the following expectation: $$\mathbb{E}\left[W_T\cdot\int_0^T f(s)\mathrm{d}W_s\right].$$ Here $\{W_t\}_{t\ge 0}$ is a standard $\mathbb{R}$-valued Brownian motion and ...
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0answers
9 views

is daily return with general stochastic volatility model stationary?

In order to estimate the parameter, we need to know whether this model will result a stationary daily return or not. And yes, actually there is an estimator for estimating the variance of this daily ...
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0answers
53 views

How to do integration by parts with brownian motion?

I am not sure how to perform integration by parts in the following expression: $$ \left(1-t\right)\left(B_t - B_s + \int_s^t \frac{r}{1-r} \mathrm{d} B_r \right) $$ Can anyone help me to solve this ...
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0answers
14 views

Markov Semigroups worked example

I have been reading this excellent paper on Markov semigroups, in which the assertion is made that a markov semigroup $\mathcal{P: L^1 \longrightarrow L^1}$ is defined by $\frac {d\mu}{dm}$ for some ...
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1answer
92 views

Application of the Burkholder Davis Gundy inequality

The proof of the Feynman-Kac formula uses a lemma which I need to proof, but I can not figure it out. The lemma is the folllowing: Let $X$ be a weak solution of ...
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2answers
33 views

Distribuiton of stochastic integral

If $(W_t)_{t\geq 0}$ is a Wiener process, $X_0=0$ and for all $t$, $t>0$ and $\alpha>0$. $X_t=\int_0^t\frac{u^\alpha}{t}dW_u$. I have want to answer 2 questions: What is the distribution of ...
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0answers
58 views

Ornstein-Uhlenbeck a Markov process

Consider the Ornstein-Uhlenbeck process defined by $$ X_t = e^{- \alpha t} X_0 + \sigma \int_0^t e^{ \alpha (s-t)} d W_s$$ with $\sigma,\alpha>0$. In many literature I have found they considered ...
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1answer
32 views

Closure of the set of elementary predictable stochastic processes

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}=(\mathcal{F}_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal{A})$ $H=(H_t)_{t\ge 0}$ be a real-valued stochastic ...
1
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1answer
36 views

$\sin(W_T)$ and Ito / Martingale Representation Theorem

I've been solving some exercises which require a function to be represented as an adapted stochastic process such that $$ X = \mathbb{E}[X] + \int_0^T \Theta(s)\,dW(s) $$ For example, $X = W(T)$ ...
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2answers
30 views

How to show stochastic differential equation is given by an equation

I I tried using substitution and I got an extra integral at the end and do not know how to proceed. Can anyone help me to break this down?
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1answer
45 views

How to solve Stochastic differential equation?

I do not have a clue on how to solve out this type of question, and how to deal with integration with a combination of brownian motion and linear function. Can anyone help me out please?
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1answer
35 views

Applying Picard-Lindelöf iteration to a stochastic integral equation

Suppose we have the following stochastic integral equation (we can make it an SDE) where $W$ is a standard Brownian motion $$ X_t = 1 + \int_0^t X_s d W_s. $$ I want to show that by using Picard ...