Suppose the following integrals \begin{equation} \int_t^T X_s \, ds\ \text{ and }\ \int_t^T Y_s \, ds \end{equation} are well-defined, where $X_s$ and $Y_s$ are continuous stochastic process. Do we have $$E\left[\int_t^T X_s \,ds \times \int_t^T Y_s \, ds\right]=\int_t^T E[X_sY_s] \, ds$$ in general?
2026-04-02 21:36:08.1775165768
Do we have the following?
70 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in STOCHASTIC-CALCULUS
- Interpreting stationary distribution $P_{\infty}(X,V)$ of a random process
- Reference request for a lemma on the expected value of Hermitian polynomials of Gaussian random variables.
- Why does there exists a random variable $x^n(t,\omega')$ such that $x_{k_r}^n$ converges to it
- Compute the covariance of $W_t$ and $B_t=\int_0^t\mathrm{sgn}(W)dW$, for a Brownian motion $W$
- Mean and variance of $X:=(k-3)^2$ for $k\in\{1,\ldots,6\}$.
- 4th moment of a Wiener stochastic integral?
- Unsure how to calculate $dY_{t}$
- What techniques for proving that a stopping time is finite almost surely?
- Optional Stopping Theorem for martingales
- $dX_t=\alpha X_t \,dt + \sqrt{X_t} \,dW_t, $ with $X_0=x_0,\,\alpha,\sigma>0.$ Compute $E[X_t] $ and $E[Y]$ for $Y=\lim_{t\to\infty}e^{-\alpha t}X_t$
Related Questions in STOCHASTIC-INTEGRALS
- Meaning of a double integral
- 4th moment of a Wiener stochastic integral?
- Cross Variation of stochatic integrals
- Stochastic proof variance
- Solving of enhanced Hull-White $dX_t = \frac{e^t-X_t}{t-2}dt + tdW_t$
- Calculating $E[exp(\int_0^T W_s dW_s)]$?
- Applying Ito's formula on a $C^1$ only differentiable function yielding a martingale
- what does it mean by those equations of random process?
- Why aren't the sample paths of this stochastic process defined?
- Is the solution to this (simple) Stochastic Differential Equation unique?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Note that \begin{align*} \int_t^T X_s\, ds \int_t^T Y_s \,ds = \int_t^T\int_t^T X_s Y_u \,ds \,du. \end{align*} Then, by assuming the Fubini, \begin{align*} E\left(\int_t^T X_s\, ds \int_t^T Y_s\, ds\right) = \int_t^T\int_t^T E\left( X_s Y_u\right)\, ds \,du, \end{align*} which is generally not equal to $\int_t^T E( X_s Y_s) \,ds$.