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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15 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 ...
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9 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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183 views

The Derivation of the Ito-Wentzell Formula

Is there a good derivation of the Ito-Wentzell Formula which is a generalization of the Ito's Lemma? Here are some unsatisfactory references to the Ito-Wentzell Formula: ...
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0answers
13 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
25 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
12 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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2answers
445 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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3answers
156 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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1answer
48 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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3answers
2k views

Variance of stochastic integral of brownian motion

How do i compute this integral? $ Var [\int_0^T W(t)dW(t)] $ I know the following $E [\int_0^T W(t)dW(t)]$ is 0 but i'm not sure how to apporch the above
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67 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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8 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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16 views

About analytic solution of Cox-Ingersoll-Ross

$$dr_t=k(\alpha- r_t)dt+\sigma \sqrt r_t dw_t$$ this is Cox-Ingersoll-Ross formula as we know. My question is: is there an analytic solution for this type of differential equation ?
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35 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 ...
2
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0answers
96 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
12 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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24 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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27 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
61 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 $$X_t-X_0=\int_0^t\mu X_s \, ds+\int_0^t \sigma X_s \, dZ_s, \tag{*}$$ ...
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19 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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22 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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17 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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1answer
72 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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2answers
69 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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42 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
48 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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35 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
29 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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28 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} ...
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0answers
80 views

Sufficient condition for martingale property

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space and $M=(M_t)_{t\geq 0}$ an $\mathcal{F}_t$-adapted stochastic process. If $$ \forall t<s, \ ...
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2answers
57 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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31 views

Can we apply the Itō formula to find an expression for ${\rm d}\eta_t(X_t)$ where ${\rm d}X_t=v_t(X_t){\rm d}t+\xi_t(X_t){\rm d}B_t$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(B_t)_{t\ge 0}$ be a $d$-dimensional $\mathcal F$-Brownian motion ...
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1answer
30 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 ...
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0answers
41 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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3answers
5k views

Expectation of geometric brownian motion

I was deriving the solution to the stochastic differential equation $$dX_t = \mu X_tdt + \sigma X_tdB_t$$ where $B_t$ is a brownian motion. After finding $$X_t = x_0\exp((\mu - \frac{\sigma^2}{2})t + ...
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1answer
29 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
25 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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8 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 ...
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1answer
21 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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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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25 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
17 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
32 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\}} ...
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16 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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27 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
9 views

How do one solve $ \int_t^T \exp[\int_0^u-\left( r-\delta_s\right)ds] dW_u $?

How to solve $ \int_t^T \exp[\int_0^u-\left( r-\delta_s\right)ds] dW_u $ ? $W_u$ is the wiener process, $\delta(t)$ is a general deterministic function, $r$ is constant.
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18 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
14 views

Problems with finding marginal density from joint density function

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
11 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 ...
5
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65 views

Ornstein-Uhlenbeck SDE solution

I'm following this solution of $$dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dW_t \tag1 $$ And the question is whether its solution $$X_t=\theta+e^{-\kappa(t-s)}(X_s-\theta)+\sigma\int_s^t ...