Functional limit theorems for stochastic integrals with applications to risk processes and to self-financing strategies in a multidimensional market. I. (Ukrainian, English) Zbl 1224.60061

Mishura, Yu. S.; Shevchenko, G. M.; Yukhnovs’kyj, Yu. V.

Teor. Jmovirn. Mat. Stat. 81, 114-127 (2009); translation in Theory Probab. Math. Stat. 81, 131-146 (2010).

The authors propose sufficient conditions for the weak convergence of stochastic integrals \(\int_0^b\xi_n(t)dX_n(t)\) with respect to processes \(\{X_n,\;n\geq1\}\) of bounded variation, martingales, and semimartingales. The convergence of the corresponding probability measures is investigated in the Skorokhod space \(D[0, b]\), \(b>1\). The corresponding result for the convergence of stochastic integrals with respect to semimartingales in terms of the canonical decomposition is as follows. Consider a sequence \((\Omega^n,{\mathcal F}^n, ({\mathcal F}^n_t)_{t\geq0},\text{P}^n),\) \(n\geq0,\) of stochastic bases, and let \(\{X_n(t),{\mathcal F}^n_t,\;t\in{\mathbb R}_+,\;n\in\mathbb Z_+\}\) be a sequence of semimartingales that admit the decomposition \(X_n(t)=X_n^0+M_n(t)+B_n(t)\), where \(\{M_n(t),{\mathcal F}^n_t,\;t\in{\mathbb R}_+,\;n\in\mathbb Z_+\}\) is a sequence of square integrable martingales whose trajectories do not have discontinuities of the second kind and are right continuous; and \(\{B_n(t),{\mathcal F}^n_t,\;t\in{\mathbb R}_+,\;n\in\mathbb Z_+\}\) is a sequence of processes of bounded variation whose trajectories do not have discontinuities of the second kind and are right continuous. Let \(\mu_n(t):=\langle{M_n}\rangle(t)\) be the square characteristics of the corresponding martingales. Consider also a sequence \(\{\xi_n(t),{\mathcal F}^n_t,\;t\in{\mathbb R}_+,\;n\in\mathbb Z_+\}\) of \({\mathcal F}^n_{\cdot}\)-predictable processes. Define \(\int_0^b\xi_n(s)dX_n(s):= \int_0^b\xi_n(s)dM_n(s)+\int_0^b\xi_n(s) dB_n(s)\), where \(\int_0^b\xi_n(s) dB_n(s)\) is the Riemann-Stieltjes integral. The authors prove that (under some conditions) the family of stochastic integrals \(\int_0^b\xi_n(s) dX_n(s)\) weakly converges to \(\int_0^b\xi_0(s)dX_0(s)\) in the Skorokhod space \(D[0, b]\) if \((\xi_n(t),M_n(t),B_n(t),\mu_n(t))\) converges to \((\xi_0(t),M_0(t),B_0(t),\mu_0(t))\).
A semimartingale theorem is extended to the multidimensional case. An example of possible applications of the theorem to the convergence of stochastic integrals with respect to processes of bounded variation is presented in the case of risk processes. The “inverse” problem for the weak convergence that occurs in financial mathematics is considered. This problem concerns the behaviour of a certain class of strategies if the capitals converge.
For related results and references, see [Yu. S. Mishura, Theory Probab. Math. Stat. 22, 115–129 (1981; Zbl 0459.60037); Y. Mishura and D. S. Silvestrov, Theory Stoch. Process. 10(26), No. 1–2, 122–140 (2004; Zbl 1064.60008)].

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60F17	Functional limit theorems; invariance principles
60H05	Stochastic integrals
91B30	Risk theory, insurance (MSC2010)
91G80	Financial applications of other theories

Keywords:

stochastic integrals; functional limit theorems; weak convergence; semimartingales

Citations:

Zbl 0459.60037; Zbl 1064.60008