On tightness of solutions of stochastic integral equations driven by \(p\)-semimartingales
Tightness in the Skorokhod space \(D(\mathbb R_+,\mathbb R^d)\) of càdlàg functions of a sequence of laws of processes \((X^n)\) satisfying \[ X^n_t=H^n_t+\int_0^tf(X^n_{s-})\,dZ^n_s+\int_0^tg(X^n_{s-})\,dA^n_s,\qquad t\geq 0 \] is investigated. Here, \((H^n)\) is a sequence of càdlàg processes playing the role of initial conditions, \((Z^n)\) is a sequence of semimartingales, \((A^n)\) is a sequence of processes of locally bounded \(p\)-variation, \(f\) is an at most linearly growing either continuous or Lipschitz function, \(g\) is a Hölder continuous function, the first integral is the classical stochastic integral and the second one is the refinement Riemann-Stieltjes integral defined pathwise. The main result of the paper lists a number of technical conditions on the entries that imply tightness of \((X^n)\) in \(D(\mathbb R_+,\mathbb R^d)\). This result is applied, first, on the Euler approximation scheme, i.e. given processes \(H\), \(Z\) and \(A\) are approximated by particular step processes \(H^n\), \(Z^n\) and \(A^n\), and the corresponding solutions \((X^n)\) are tight. Secondly, the sufficient conditions for tightness of \((X^n)\) are adapted and simplified in the case when \(H^n\to H\), \(Z^n\to Z\) and \(A^n\to A\) converge in law in \(D(\mathbb R_+,\mathbb R^d)\). Finally, the case when \((Z^n)\) and \((A^n)\) approximate, in the sense of the Donsker theorem, a Wiener process and a fractional Brownian motion with the Hurst parameter \(H\in(\frac 12,1)\), respectively, is discussed and it is indicated that the main result is well applicable.
scientific article; zbMATH DE number 5626713