From an Itô type calculus for Gaussian processes to integrals of log-normal processes increasing in the convex order
\textit{P. P. Carr, C.-O. Ewald} and \textit{Y. Xiao} [``On the qualitative effect of volatility and duration on prices of Asian options'', Finance Research Letters, 5, 162--171 (2008), \url{doi:10.2139/ssrn.1086927}] showed that the process \[ A_t=\frac1t\int_0^t\exp\bigg(B_s-\frac{s}2\bigg)\,ds,\quad t\geq0, \] where \((B_s,\;s\geq0)\) is a standard Brownian motion, is a process that is increasing in the convex order. The authors generalize this to a Gaussian framework. First, they present an Itô type formula for a Gaussian process that is well adapted to the study of processes that are increasing in the convex order in a Gaussian framework. Then, they show that some processes that are associated with \(A_t\) are increasing in the convex order. Finally, they construct Gaussian sheets allowing to exhibit martingales with the same one-marginals as the processes mentioned above.
scientific article; zbMATH DE number 5950726
scientific article