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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A Feynman-Kac style derivation of a survival probability of a Compound Poisson process

Let $$R_t = u + \beta t - \sum^{N_t}_{i=1}U_i$$where $u\geq 0$, $\beta > 0$, $N_t$ is a Poisson counting process with intensity $\lambda$ and $U_i$ are jumps having a probability density $\nu(y)$ ...
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15 views

Using Ito theory to decide whether $M^f$ is martingale or a local martingale

I came across the following while reading Ikeda & Watanabe book Stochastic differential equations and Diffusion processes, in page 163-164 At first the sentence $$f(X_t)- f(X_0) - ...
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13 views

Is the space of all adapted processes with Càdlàg paths a Banach space?

Consider first the definition of a stochastic integral for simple predictable processes. $$I:\mathbb{S}\rightarrow\mathbb{D},\ H\mapsto I_X(H):=H_0X_0+\sum_{i=1}^nH_i(X_{T_{i+1}}-X_{T_i})$$ The ...
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23 views

Itō Integral multiplied by Riemann Integral

I was wondering whats the result of an Itō integral multiplied by a Riemann Integral. For example, what is $$\left(\int_0^T f(u)\ \mathsf dW_u\right)\left(\int_0^T g(v)\ \mathsf dv\right)$$ where $W$ ...
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1answer
37 views

Existence stochastic integral

I am trying to understand the prove of the existence of the stochastic integral for a local martinglale null at $0$ and continuous, $M\in \mathcal{M}^c_{0,\text{loc}}$, a predictable process $H\in ...
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18 views

Martingale and quadratic variation inequality

I have the following inequality $$\mathbb{E}(\mid[M^{\Pi^m},M^{\Pi^m}]_T^{1/2}-[M^{\Pi^n},M^{\Pi^n}]_T^{1/2}\mid^p)\leq \mathbb{E}([M^{\Pi^m}-M^{\Pi^n},M^{\Pi^m}-M^{\Pi^n}]_T^{p/2}),$$ where $M$ is a ...
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2answers
405 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 ...
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0answers
28 views

Burkholder-Davis-Gundy inequalities

I want to prove these inequalities, i.e.: For $p\geq 1\ \exists 0<c_p\leq C_p$ such that for any martingale $M$ we have the following inequality: $$c_p\mathbb{E}[[M,M]^{p/2}_\infty]\leq ...
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0answers
26 views

Ito's formula and Infinitesmal generator

Consider an Ito process $$ dX_t = \sigma_t dB_t $$ where $\sigma_t$ is a two-state continuous-time Markov chain with state space $\{ \sigma_1, \sigma_2 \}$ that switches state with Poisson ...
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0answers
23 views

Empirical Quantilfunction as Integral Bound

This is my first post, so please be nice ;) I'll try to outline my problem correctly and whilst keep it as short as possible! I have to deal (for my master thesis) with the integral ...
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1answer
260 views

What is the difference between Calculus and Analysis? In Stochastic processes?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
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0answers
17 views

Quadratic Variation of Stochastic Integral of Simple Predictable process

Take $H\in S$ to be a simple process defined as: $$H_t:=\sum_{i=1}^{n-1} H_i1_{(T_i,T_{i+1}]}(t),\ \ H_i\in \mathcal{F}_{T_i}, \ (T_1\leq...\leq T_n \ stopping\ times),$$ and $X$ a Good Integrator. ...
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2answers
51 views

Stochastic Exponential - Protter

I am trying to understand the proof of Theorem 37 at page 84 of the book Stochastic Integration and Differential Equations by P. Protter. In the proof there is the following statement, referred to ...
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0answers
17 views

Cross variation

I have a question about the following argument. I see in my book a claim that given 2 stochastic integrals : \begin{align}X_1&:=\int_{0}^{t}f_s\mathsf dM_s\\ X_2&:=\int_{0}^{t}g_s\mathsf ...
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1answer
75 views

Show that $E[X_t^2]<\infty$

Show that $E[X_t^2]<\infty$, where $$ X_t=e^{3W_t-\frac{3t}{2}}-3e^{W_t-\frac{t}{2}}\underbrace{\int_0^te^{2W_s-s}ds}_{A_t},\quad. t\geq0, $$ where $t$ is a fixed number and $W_t$ is Brownian ...
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29 views

Stochastic Integral of Simple Predictable Process is a Martingale

Take $H\in S$ to be a simple process defined as: $$H_t:=\sum_{i=1}^{n-1} H_i1_{(T_i,T_{i+1}]}(t),\ \ H_i\in \mathcal{F}_{T_i}, \ (T_1\leq...\leq T_n \ stopping\ times),$$ and $X$ a Martingale. I ...
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0answers
16 views

Weakening mean integral requirements of stochastic integrals.

Consider the Ito integral. It is well known that adapted and measurable processes $f(s,\omega)$ that satisfy \begin{align} E \Big[ \int_0^T \big| f(s,\cdot) \big|^2 ds \Big] < \infty \end{align} ...
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1answer
74 views

Proof of Itō's lemma for the Brownian motion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...
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26 views

The Itō integral $\sum_{i=1}^nH_{t_{i-1}}\left(B_{t_i}-B_{t_{i-1}}\right)$ of an simple process $H$ is independent of the choice of $(t_0,\ldots,t_n)$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
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0answers
14 views

limit of quadartic variation [duplicate]

I am trying to understand why : $[\int_{0}^{t}a_s dB_s]=\int_{0}^{t}a_s^2 ds$ [] is the 2-variation process, $B$ is brownian motion in the proof I have seen they used Riemman-sums to get an ...
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26 views

harmonic functions and ito formula

I am trying to prove the mean-value property for harmonic functions in $R^k$ by ito calculus. given $G$ bounded domain and $u$ harmonic function on $G$ then $u(a)=\int_{\partial B_r} u(y)ds(y)$ ...
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28 views

Computing an Ito Integral using the Definition

Let $B_t$ be a brownian motion adapted to $\mathcal F_t$. For general $\mathcal F_t$-adapted processes $X_t$ the Ito-integral could be defined as $$ \int_0^t X_s dB_s = \lim_{n\to \infty} \int_0^t ...
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1answer
35 views

Quadratic variation of the Brownian motion and Itō's lemma

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...
3
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1answer
42 views

Why is the solution of a stochastic differential equation wrt the Brownian motion suitable for a model of a disturbed time continuous process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be a Brownian motion on ...
3
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1answer
22 views

Notation in stochastic integrals

There are some notation I don't understand: Given $W_t$, $n$-dimensional Brownian motion, and a smooth function $u:R^n\to R$ my book asserts: $$E^x\left[u(W_0)\right]=u(x)$$ What is the notation ...
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1answer
23 views

Fubini's Theorem for Stochastic Integral

Probably a bit trivial, but I was curious about the validity of interchanging the following integrals (where $W_t$ is Brownian Motion): $\mathbb{E}[\int^{t}_{0} W^2_s ds] =? \int^{t}_{0} ...
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1answer
21 views

Expectation of Poisson integral for random integrand

Let $N$ be the Poisson random measure associated to a Levy process $X$, with intensity $\nu$. Furthermore, let $A$ be bounded from below and $f \in L^1(A,\nu)$ be measurable. It is well known that ...
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1answer
28 views

Motivation behind the definition of the Itô integral for elementary predictable processes

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

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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13 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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1answer
68 views

How do we apply Ito's lemma to a product of functions

In finance an optimal portfolio choice it is common to use some tools of stochastic calculus. Going through a book, I found the following statement, \begin{equation} a_t = \int_s N_t(s) P_t(s) ds ...
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1answer
40 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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16 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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23 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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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 ...
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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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2answers
64 views

Most General Theory of Stochastic Integration

I've learnt continuous stochastic integration using the classical books: - Revuz & Yor, - Karatzas & Shreve and - Oksendal. Now I want to learn general stochastic integration, i.e. possibly ...
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1answer
60 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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1answer
24 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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24 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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65 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
40 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
28 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
47 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 $$ ...
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1answer
26 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: $$dr_t = b(a-r_t)dt + \sigma dB_t$$ ...
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0answers
25 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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30 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
12 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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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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33 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 ...