A note on products of stochastic objects

In recent study of partial differential equations (PDEs) with random initial data and singular stochastic PDEs with random forcing, it is essential to study the regularity property of various stochastic objects. These stochastic objects are often given as products of simpler stochastic objects. As pointed out in Hairer(2014), by using a multiple stochastic integral representation, one may use Jensen's inequality to reduce an estimate on the product to those on simpler stochastic objects. In this note, we present a simple argument of the same estimate, based on Cauchy-Schwarz' inequality (without any reference to multiple stochastic integrals). We present an example on computing the regularity property of stochastic objects in the study of the dispersion-generalized nonlinear wave equations, and prove their local well-posedness with rough random initial data.