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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1answer
30 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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0answers
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
60 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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0answers
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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0answers
15 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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0answers
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 ...
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0answers
98 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 ...
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0answers
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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0answers
39 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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1answer
71 views

$dX_t/X_t=\mu+\sigma \, dZ_t$, 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 $$X_t-X_0=\int_0^t\mu X_s \, ds+\int_0^t \sigma X_s \, dZ_s, \tag{*}$$ ...
2
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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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0answers
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) \, ...
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0answers
20 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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0answers
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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0answers
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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0answers
37 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} ...
1
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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 ...
3
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1answer
79 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 ...
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0answers
45 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 ...
2
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2answers
59 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 $$ ...
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0answers
84 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
45 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 ...
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0answers
33 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 ...
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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
28 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\\ ...
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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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0answers
32 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 ...
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0answers
21 views

Fractional moments of stochastic integrals

I want to bound the moments of stochastic integrals as $$E\left|\int_0^1 f(s)d L_s\right|^\alpha,\alpha\in[0,1],$$ where $(L_s)_{s\ge0}$ is a Lévy process with Gaussian part $\sigma^2$ and Lévy ...
1
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1answer
44 views

Non-linear SDE: how to?

$$ \newcommand{\mcl}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\avg}[1]{\langle#1 \rangle} \newcommand{\pth}[1]{\left( #1 \right)} \newcommand{\bck}[1]{\left\{ #1 \right\}} ...
2
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0answers
39 views

Integrability condition of stochastic Fubini's theorem

This is a special case of stochastic Fubini's theorem for deterministic integrands: Let $f : [0,t] \times [0,t] \to\mathbb{R}$ be measurable. Assume that $$ \int_0^t \left( \int_0^t |f(r,s)|^2 dr ...
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0answers
32 views

Ito integral via simple process when the integrand is C^1

I have the following problem. Let $H_t$ be an adapted process with trajectories a.s. of class $C^1$ on $\mathbb{R}_{+}$. Compute using simple process $\int_o^t H_s d B_s$. My idea is to firstly set ...
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0answers
26 views

Stochastic calculus for continuous time Markov chains

I have absolved a course on stochastic analysis, i.e. integrals with respect to the brownian motion. Now I know that there is a theory of stochastic calculus for diskrete matringales, however I was ...
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0answers
19 views

Problems with finding marginal density from joint density function [closed]

For two absolute continuos stochastic variables I have that the joint density function is 8y if 0 I now have to calculate/ show what the marginal density functions are. I got the right answer for y ...
1
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1answer
14 views

Find E(X^-1) for stochastic variable

Let $X$ be a stochastic variable with density function: $f(x)=x\exp(-x)$ if $x>0$ and $0$ otherwise. Show that $E(X^{-1} )=1$. I believe I have to integrate but is it simple $x\exp(-x)$ I ...
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1answer
34 views

Check process is a martingale

I have such stochastic process with which I struggle all day, finally I found 2 mistakes, however answer is still unsatisfying. $$X_t = atW_t^2 - \int_0^t(W_s^2+s)ds,$$ I need to check if it is a ...
1
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1answer
59 views

Two Ito processes : are they a 2-dim Brownian motion?

I am stuck with the following problem : I have a Brownian motion $B_t$ and an Ito process $$X_t:=\int_0^t sgn(B_s)\ d B_s,$$ where $sgn(x)=1$ when $x \geq 0$ and $sgn(x)=-1$ when $x<0$. I have ...
1
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1answer
89 views

Use Ito's Formula to prove following identity

Again, I am not sure how the following works; Could someone please give me an almost stupidly detailed explanation of why/what is happening in the part below. First, the question itself; Q. $B_t$ ...
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0answers
20 views

What process does this SDE weakly converge to?

So my question is motivated by the following: Note that the ODE $$ dy_t = 2sgn(y_t)\sqrt{|y_t|}$$ $$y_0 = 0$$$$ has no unique solution. However, consider the SDE as follows: $$ dy_t = ...
2
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0answers
35 views

Versions of Tanaka's SDE

Consider the following versions: $$dX_t=x_0+sgn(X_t)dW_t \tag1$$ $$dX_t=x_0+1_{(0,+\infty)}(X_t)dW_t \tag2$$ $$dX_t=x_0+1_{(-\infty,0]}(X_t)dW_t \tag3$$ SDE (1) is a classical example of SDE with ...
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0answers
38 views

Variance of a simple Ito integral

I am trying to apply Ito's lemma to compute variance of the following integral $X(t) = \int_{0}^t W(s)dW(s),$ where $W(t)$ is a Wienner process. Could you please check my calculations? $$E(X(t)) = ...
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3answers
203 views

Stochastic Integrals are confusing me; Please explain how to compute $\int W_sdW_s$ for example

I have been trying hard to understand this topic, but only failing.Reading through my lecture notes and online videos about stochastic integration but I just can't wrap my head around it. The main ...
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0answers
22 views

Left limit poisson process (stochastic analysis)

Let $N_t$ denote a Poisson process with intensity λ > 0, and let $M_t = N_t − λt$ be the compensated martingale of N . How could I show that $\int_{0}^{t} N_{s-} dN_s =1/2 (N_t^2-N_t)$ Thank you
2
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0answers
36 views

Martingale (stochastic analysis)

Let $N_t$ denote a Poisson process with intensity λ > 0, and let $M_t = N_t − λt$ be the compensated martingale of N . I want to verify that the process Y given by $Y_t = \int_{0}^{t} N_{s-} dM_s$ is ...
0
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1answer
15 views

Can I swap conditional expectation and limit

My problem is the following : let $B_t$ be a standard Brownian motion and $H_t$ a progressive measurable process such that $\mathbb{E}\left(\int_0^{+\infty} H_t^2\ dt \right)<+\infty$. Denote ...
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1answer
29 views

integral of square of Brownian motion

What is expectation of $$\int_0^t B(s)^2ds$$ where $B(s) is standard Brownian motion. Is the integral a well known random variable?
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0answers
65 views

Integral inequality of transformed integrand with second order stochastic dominance flavor

Let $f,g : [0,1] \rightarrow [0,1]$ be two functions such that for all $x \in [0,1]$ $\int_0^x f(t) dt \geq \int_0^x g(t) dt$ and $\int_0^1 f(t) dt = \int_0^1 g(t) dt.$ Can I conclude that ...