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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57 views

Itos formula on a transformation of bessel Processes

Let $W$ be a Brownian motion and $z,\kappa>0$. Let $X_t(z)$ be a solution to the SDE $$dX_t(z)=dW_t+2/(\kappa X_t(z))dt.\quad X_0(z)=z.$$ The solution is well-defined on $t<\tau(z)$ where $\...
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2answers
141 views

The stochastic integral $\int W_t dW_t$

I'm reading an introduction to Stochastic Calculus. I'm at the point where Ito integrals are developed and constrasted with the Stratonovich integral. Below is a calculation of $\int_0^T W_t d W_t$. ...
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1answer
36 views

Itô integral probability distribution

I know in general this must not have an analytical expression in terms of common functions, but how do you (at least in theory) get the probability distribution of $X_t$ for a given $t$ in the ...
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1answer
20 views

Conserved quantity for system of Stochastic Differential Equations

I'm considering the set of SDEs (in the sense of Ito) $\begin{align*} \mathrm d x &= -yx \mathrm d t+ x^2 \mathrm d B_t \\ \mathrm d y &= -y^2 \mathrm d t + xy \mathrm d B_t\end{align*}$ ...
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37 views

Wick renormalization of stochastic integral

I am trying to understand a paper that summarizes some results concerning Wick renormalization of some stochastic integral. In the last few lines of the paper the authors say: In Euclidean ...
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44 views

The limit of the ratio of two stochastic integrals

I am just wondering how to calculate the limit of stochastic integrals. Here is one example: $$ \lim\limits_{N \rightarrow \infty}\dfrac{\int_{0}^{N}B(s)dB(s)}{\int_{0}^{N}B^2(s)ds}$$ where $B(s)$ is ...
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1answer
26 views

I want to simplify the stochastic integral by change variable

Let $f:[0,t]\rightarrow \mathbb{R^+}$ be a deterministic and integrable and $(B_t)_{t\geq 0}$ is a standard Brownian motion. If $X_t=\int_o^tf(s)dB_s$, we know that $X_t$ has normal distribution with ...
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1answer
49 views

Check that an Ito integral is a martingale.

Before presenting my problem I will introduce some notation. Time index $t\in [0,T]$. $$C_t = \begin{cases} Z_n = B_{t_{n-1}}, & \text{if $t=T$} \\[2ex] Z_i = B_{t_{i-1}} , & \text{if $t_{i-...
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1answer
37 views

Fubini's theorem for Stochastic Integral, with sum

I am struggling here with part (2), . In usual instances, I've had the question phrased like this but I'm not sure how to deal with the summation?
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87 views

Multiplication of two stochastic integrals

I was wondering if someone can help me with the concept of stochastic integral multiplication. Consider multiplication of two stochastic integrals $$(\int^T_0f(u)dW_u)(\int^T_0g(s)dW_s)$$ where $W_u$ ...
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0answers
27 views

Construction of Ito Integral: Doubt from Oksendal, Chapter $3$, Page-$27$

In the book "Stochastic Differential Equations" by Oksendal, at the page $27$, in the last few lines he has written Define $g_{n}(t,\omega) := \int_{0}^{t}\psi_{n}(s-t)h(s,\omega)ds$. Then, $g_{n}(...
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1answer
21 views

Non-linear SDE: About the noise time-step

This is a follow-up on my previous post about stochastic differential equations. In the answer from @LuztL, and in the literature, I read commonly that the time-step of the noise should be somewhat ...
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0answers
24 views

Construction of Ito Integral: doubts from Kuo

In the book "Introduction to Stochastic Integration" by Kuo, at page $46$, he has written: $\int_{a}^{b} E(|f(t)-g_{n}(t)|^{2})dt \\ \leq \int_{a}^{b} \int_{0}^{\infty} e^{-\tau }E(|f(t)-f(t-\frac{...
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1answer
72 views

Stochastic Integration with respect to Cauchy Process?

I'm interested in a one-dimensional stochastic process: $$dX_t = f(X_t)dt + g(X_t) dZ_t$$ where $Z_t$ is a Cauchy process and $f,g$ are nice polynomials (I'm looking at an ODE that gets perturbed by ...
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0answers
25 views

To estimate the probability that a diffusion reaches a certain value

I have a diffusion process define by the following equation: \begin{equation} dX_t=X_t[\beta(N-X_t)-\alpha]dt+\sqrt{X_t(\beta(N-X_t)+\alpha}) { }dB_t \end{equation} and I proved that the solution ...
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1answer
44 views

How to show that this is a martingale?

Let $H_s$ be a predictable and bounded process. How can I show that $$M_t = \int_0^t H_s \, dW_s$$ is a martingale? Clearly since $H_s \in L^2_\text{loc} (W)$ we have that $M_t$ is a local martingale,...
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1answer
37 views

A computation using the Ito integral

I was assigned this exercise by my Stochastic Analysis Professor. Exercise. Let $B$ be a one-dimensional Brownian Motion, and consider the following processes: $X_t=\int_0^tB_sds\quad Y_t=\int_0^...
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1answer
30 views

A clarification on $L_{loc}^2$ process and stochastic exponential

In the book by A. Pascucci (PDE and Martingale Methods in Option Pricing) I have found the following definition of $\mathbb{L}^2_{\text{loc}}$ process. Later (pp. 329-330) for a process $\lambda\...
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18 views

Different definitions of local p integrability for local martingales

When talking about cadlag (but not continuous) martingales and local martingales in the context of stochastic integration one can come across different definitions depending on the author. These are: ...
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1answer
34 views

local martingales/ Ito formula

I have a problem with following task. Find $(A_t)_{t\ge0}$ a process of bounded variation on bounded intervals, such that $A_0=0$ and process $M_t=W_tsin(\int^t_0W_s^3dW_s)-A_t$ is a local martingale. ...
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2answers
34 views

derive integration by parts for a stochastic integral

The question is to show the following identity: $\int_{0}^{T}tdW(t) = TW(T)-\int_{0}^{T}W(t)dt$ This can be done quite easily with ito's however the question explicitly says to show the identity ...
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17 views

Calculate expectation of stochastic integrals

I am trying to calculate $$\mathbb{E}\left[\int^t_0 e^{\lambda s}dB_s \int^{t+h}_0 e^{\lambda s}dB_s \right], $$ where $(B_t)_{t\geq 0}$ is a brownian motion, $h>0$ and $\lambda > 0$ is some ...
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1answer
61 views

Why is the drift of an Itō process considered to be a Riemann integral even when it's not even Riemann integrable?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(B_t)_{t\ge 0}$ be a real-valued Brownian motion with respect to $\mathcal ...
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77 views

Why is linearity a requirement of a integral

I was reading Philip Protter's Stochastic Integration and Differential Equations textbook. He mentions that an operator, $I_X$, induced by $X$ should be linear to be called an integral. I have a ...
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16 views

What does Karhunen-Loève expansion have to do with cosine-sine basis expansion?

According to my research, Karhunen-Loève(KL) expansion is a version of Fourier series for stochastic processes and states that under some conditions, a stochastic process $X\left(\omega, t\right)$ can ...
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18 views

Can a Brownian motion be defined for negative time?

I was just looking at fractional brownian motions on this page. The definition of $B_H(t)$ requires integrating on a negative time domain on $dB(t)$ where $B(t)$ is a Brownian motion! Could you please ...
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0answers
48 views

Show that for every $p >0$, $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$

I am trying to prove that $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$ where $\rho(n)$ is a solution of the following Stochastic differential equation \begin{equation} \rho_n^2(t)=2 \int_0^...
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100 views

How can we desribe a particle whose motion is perturbed by a random forcing using a stochastic partial differential equation?

Let $d\in\left\{2,3\right\}$ and $\mathcal V_t$ be the bounded set occupied by a fluid at time $t\ge 0$. Let $x_0\in\mathcal V_0$ be a particle and $$[0,\infty)\to\mathbb R^d\;,\;\;\;t\mapsto X_t(x_0)\...
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32 views

Why $d\langle X \rangle_t = d X_t dX_t$ if $X_t$ is a semimartingale?

Following this question, proving the equivalence between equation $(1)$ and $(2)$, I deduced that $$d\langle X \rangle_t = d X_t dX_t$$ (where $X_t$ was an Ito's process, hence a semimartingale). I ...
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41 views

How can I prove the equivalence of these two Ito's lemma notations?

Let $X_t=(X_1, \dots , X_T), t \in [0,T] $ be a continuous semimartingale and $f$ a function of class $C^{1,2}$ (continuous and differentiable). Then, $f(t,X)$ is a semimartingale and we have, $\...
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$dX_t/X_t=\mu+\sigma \, dZ_t$, does this notation make sense?

I understand that the notation $$dX_t=\mu X_t \,dt + \sigma X_t \,dZ_t,$$ where $Z_t$ is Brownian Motion, is a shortcut to $$X_t-X_0=\int_0^t\mu X_s \, ds+\int_0^t \sigma X_s \, dZ_s, \tag{*}$$ ...
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0answers
27 views

Hilbert-Schmidt operator - converging norm series - Cylindrical brownian motion

I am reading about cylindrical brownian motion in the monograph of Prato and Zabczyk. For this construction a Hilbert-Schmidt operator is used, between to separable Hilbert spaces $U$ and $U_1.$ Let $...
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24 views

Does Ito isometry hold pointwise?

It is known that the stochastic integral satisfies the following property $$ \mathbb{E}\left[\left\langle \int_0^{\cdot}X(s)\,dM(s) \right\rangle_t\right]= \mathbb{E}\left[ \int_0^t X^2(s) \, d\left\...
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0answers
22 views

An application of Ito's formula

I am reading a proof in which I don't understand how to use Ito's rule to derive the following: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space such that $M^{(i)}$ and $M^{(k)}$ are ...
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2answers
75 views

Linear non-homogenous SDE

I'm struggling to understand how to resolve the following SDE: $$dX(t)=(\sin(t)-2X(t)) dt + (1+X(t))dB(t)$$ I understand that I should use the Ito formula but I have no idea how the $F(X(t),t)$ should ...
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46 views

How do I rearrange $E[\log p(X, Y|\Theta)|X, \Theta^{(i - 1)}]$ to $\int_{y \in \Upsilon} \log p(X, y|\Theta)f(y|X, \Theta^{(i - 1)})dy$?

Equation (2) from here. Is there a formula for this? Also what does it mean if only the bottom part of the integral is specified, and how does $y \in \Upsilon$ even work in an integral? Thanks in ...
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50 views

Visit probability as a function of continuous time

I am working on a project aiming to model visit probabilities in spacetime prisms. On a given location, I know the visit probability at any time (within the prism boundaries), i.e. the visit ...
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1answer
33 views

Proving a simple equality involving integrals and a brownian motion

I'm trying to prove the following equality $$ \int_0^T W(t) dt = \int_0^T (T-t) dW(t) $$ where $W(t)$ is a standard brownian motion. I'm been trying to make use of the fact, that $dt = dW(t) dW(t)$ (...
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38 views

Fokker-Planck derivation. Path integral?

I am trying to understand the development of Fokker-Planck equation as is described here. Unfortunately, I cannot understand how the first equation on page 4, \begin{multline} \frac{1}{2} \int_0^...
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1answer
43 views

Solving equation with Wiener process

I want to show that if $E(f(X_{t}))=E(f(W_{t})e^{\lambda W_{t}-0.5*\lambda^2*t})$, where $W_{t}$ is a Wiener Process, then $X_{t}\sim N(\lambda t,t)$. Does anyone have a clue how to solve this problem?...
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1answer
80 views

Showing martingale for a Brownian motion $(W_t)_{t \geq 0}$

I want to show that $\dfrac{e^{W_{t}^2/(1+2t)}}{\sqrt{1+2t}}$ is a martingale with respect to $F_{t}$. We can use that $$E(e^{\alpha X^2/\sigma^2})=\dfrac{\dfrac{\mu^2\alpha}{e^{\sigma^2(1-2\alpha)}}}{...
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46 views

Show that the solution to a stochastic differential equation is satisfied by the following

I am confused on how to get from the first statement to the second. Getting from the second statement to the third would just a simple case of substituting s=0. The solution sheet says to use ...
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2answers
60 views

Why is the integral $\int_0^1t\,dW_t$ a normal random variable?

Consider the random variable $X=\int_0^1t\,dW_t$, where $W_t$ is a Wiener process. The expectation and variance of $X$ are $$E[X]=E\left[\int_0^1t\,dW_t\right]=0,$$ and $$ Var[X]=E\left[\left(\int_0^...
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93 views

Pathwise definition of stochastic integral consistent with the Ito isometry

My definition of the stochastic integral is that it it is the image of the Ito isometry. Now we also prove Ito's formula and then apply it pathwise and get a pathwise definition in some cases. But in ...
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1answer
46 views

Advanced statistics book

I have a good background of statistics but during my researches I realized that I don't have a sound and proper knowledge of some advanced statistics topics such as: hypothesis tests like chi-square,...
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34 views

In Itô's lemma, if you wish to take expectations, when can you ignore the stochastic integral term?

Fix $d,k \in \mathbb{N}$. Let $\,b\colon \mathbb{R}^d \to \mathbb{R}^d\,$ and $\,\sigma\colon \mathbb{R}^d \to \mathbb{R}^{d \times k}\,$ be locally Lipschitz functions such that the Itô SDE $$dX_t=...
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0answers
10 views

Diffusion Process Expectation Smoothness Condition

Consider a diffusion process on a sample space $\Omega$ $$dx_t = \mu(\omega,t)dt+\sigma(\omega,t)dB_t,\, \forall\omega\in\Omega$$ where $B_t$ is the standard Brownian motion on the filtration $\...
3
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1answer
29 views

Strong solution SDE - independence of initial conditiion

I am currently studying the existence and uniqueness of strong solutions of SDEs of type $$\left[\begin{array}{l} \, dX_t=\mu(t,X_t)\,\mathrm{d}t+\sigma(t,X_t)\,\mathrm{d}W_t\\ X_0=\xi\end{array}\...
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0answers
47 views

Does this make sense?

Can I write this? Let $W_s$ be a Wiener process and let $x_s$ be a stochastic square integrable process adapted to the filtration generated by $W$. Is such an expectation nonsensical? And if not, how ...
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33 views

When does convergence in quadratic variation imply a uniform convergence or vice versa?

Given a sequence $\Pi=\{\pi_n\}$ of partitions of an interval $[0,T]$ the quadratic variation of a path $x\colon [0,T]\to \mathbb{R}$ is defined by $$ [x]=\lim_{n\to +\infty}\sum_{\pi_n}|x(t_{i+1})-x(...