Infinitesimal semigroups for one class of stochastic semigroups (Q792703)

A two parameter family of random variables \(X^ t_ s\), \((Y^ t_ s)\), \(0\leq s\leq t\leq T<\infty\), is called a multiplicative (additive) stochastic semigroup, if \(X^ t_ sX^ u_ t=X^ u_ s\), \(X^ s_ s=E\) (identity), \((Y^ t_ s+Y^ u_ t=Y^ u_ s\), \(Y^ s_ s=0)\), 0\(\leq s\leq t\leq u\leq T\). Under certain conditions of right continuity, the author establishes that \(\lim_{{\mathcal P}}\sum_{P}(X^{t_ i}_{t_{i-1}}-E)=Y^ t_ s\), where \(P=\{s=t_ 0<t_ 1<...<t_{n-1}<t_ n=t\}\) is a partition of [s,t], and \({\mathcal P}\) denotes the class of such partitions P of [s,t], meaning thereby that \(\{Y^ t_ s\}\) is infinitesimal (derivative) in relation to \(\{X^ t_ s\}\).