Let be a separable Hilbert space and represents the norm induced by the inner product of , and let be a self-adjoint operator on bounded from below and denotes the domain of . Suppose is an eigenvalue with multiplicity 1 attaining the valu…

りかのおべんきょうでオオスズメバチを2ひきかい始めました。 どちらも女王で、とてもかわいいです。 たいせつにしたいとおもいます。I started to keep 2 Japanese giant hornets (Vespa mandarinia japonica), called "Osuzumebachi" in Japanese. They ar…

I make a note about the title because it took a hard time finding a proof. This topic relates to an abstract Wiener space in the Malliavin calculus.Proposition. Let be a real separable Banach space and be a real separable Hilbert space emb…

In this article, we introduce how to solve simultaneous linear equations with NISQ, which is the abbreviation for Noisy Intermediate Scaled Quantum, from Xu et al. [1].We consider the following linear equation: \begin{align*} Ax=b, \tag{1}…

りょうしじょうほうで出てきたので整理するよ～In this article, we introduce that any square matrix can be decomposed as the linear combination of the Pauli matrices.Denote by the set of all square matrices of order whose components value on …

In this article, some equalities are introduced for showing the Markov property of stochastic processes. After then, we demonstrate the Markov property of a geometric Brownian motion.In what follows, represents a probability space and the …

A solution of Exercise 3.14 in the Øksendal SDE book is given in this article. It seems this exercise to be complex and require careful observation. Let be the filtration generated by the Brownian motion .We see obviously that is -measurab…

The following might be nontrivial so I will write down a proof. Claim Assume that the filtration satisfies the usual conditions. Let be a Lévy process (with càdlàg sample paths) adapted to and be a Borel set in such that . Define the rando…