I guess $\int_0^t s^2 B(s) \, ds$ is a normal distribution as $\int_0^t B(s) \, ds$ but I don't know how to argue that.
2026-05-05 15:24:48.1777994688
Distribution of $\int_0^t s^2 B(s) \, ds$ where $B$ is a standard Brownian motion
137 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 BROWNIAN-MOTION
- Compute the covariance of $W_t$ and $B_t=\int_0^t\mathrm{sgn}(W)dW$, for a Brownian motion $W$
- Why has $\sup_{s \in (0,t)} B_s$ the same distribution as $\sup_{s \in (0,t)} B_s-B_t$ for a Brownian motion $(B_t)_{t \geq 0}$?
- Identity related to Brownian motion
- 4th moment of a Wiener stochastic integral?
- Optional Stopping Theorem for martingales
- Discontinuous Brownian Motion
- Sample path of Brownian motion Hölder continuous?
- Polar Brownian motion not recovering polar Laplacian?
- Uniqueness of the parameters of an Ito process, given initial and terminal conditions
- $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 $Y=\lim_{t\to\infty}e^{-\alpha t}X_t$
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?
Too long for a comment.
From $$ \frac{t^3}{3}B_t=\int_0^ts^2\,B_s\,ds+\int_0^t\frac{s^3}{3}\,dB_s $$ it follows that $$ \int_0^ts^2\,B_s\,ds=\int_0^t\frac{t^3-s^3}{3}\,dB_s $$ which is Gaussian with mean zero and variance $$ \int_0^t\frac{(t^3-s^3)^2}{9}\,ds\,. $$ Can you proceed?