MathOverflow

I am wondering if the following expression can be processed a bit analytically, $$E \left[ e^{aX} \int_0^X e^{bu}dW(u)\right],$$where $W_u$ is the normal Brownian motion (1D Wiener process), and $X$ is a random variable.
I know that $E \left[ \int_0^T e^{bu}dW(u)\right]$ is zero, so $E \left[ \int_0^X e^{bu}dW(u)\right]$ should also be zero. However $e^{aX}$ and $\int_0^X e^{bu}dW(u)$ are clearly correlated, so the expectation of their product is not trivial....

$\DeclareMathOperator\GL{GL}$Let $\GL(n,\mathbb{R})$ be the classical group of all invertible matrices with real entries. Its connected component containing identity is denoted by $\GL_{+} (n,\mathbb{R})$.
Is $G=\GL(n,\mathbb{R})$ the only Lie subgroup of $\GL(n,\mathbb{R})$ with the following property?...

A semi doubly stochastic matrix is a matrix with real entries whose all rows and columns sum to 1.(This is the classical notion of doubly stochastic matrix without the positivity assumption on entries).
A smooth manifold $M$ is called a semi stochastic manifold if it admit an atlas for which the derivative of every transition map is a semi double stochastic matrix....

The following relates to what is known as the Lebesgue inequality, which gives an error bound for approximating functions under certain linear mappings, and ways to generalize that inequality.
Let $X$ be a normed linear space and let $Y$ be a closed subspace of $X$ with a best approximation in norm from that subset to every function in $X$ (e.g., $Y$ is a finite-dimensional linear subspace of $X$). Let $L:X\to Y$ be a bounded and idempotent...