Lower functions for processes with stationary independent increments

Summary Let {X _t } be aR ¹-valued process with stationary independent increments and . In this paper we find a sufficient condition for there to exist nonnegative and nondecreasing functionh(t) such that lim infA _t /h(t)=C a.s. ast0 andt, for some positive finite constantC whenh(t) takes a particular form. Also two analytic conditions are considered as application.This research is partially supported by Korea Science & Engineering Foundation     

Stochastic integrators with stationary independent increments

Summary Lipschitzian stochastic differential equations driven by a process with stationary independent increments permit a priori growth and stability estimates up to any sure time, and the solutions depend differentiably on parameters provided the Levy measure of the driving term has suitable moments.Research partially supported by NSF grant Nr. MCS 8001779     

Self-similar processes with independent increments

Summary A stochastic process {X _t t 0} onR ^d is called wide-sense self-similar if, for eachc>0, there are a positive numbera and a functionb(t) such that {X _ct } and {aX _t +b(t)} have common finite-dimensional distributions. If {X _t } is widesense self-similar with independent increments, stochastically continuous, andX ₀=const, then, for everyt, the distribution ofX _t is of classL. Conversely, if is a distribution of classL, then, for everyH>0, there is a unique process {X _(H) ^t } selfsimilar with exponentH with independent increments such thatX ₁ has distribution . Consequences of this characterization are discussed. The properties (finitedimensional distributions, behaviors for small time, etc.) of the process {X _(H) ^t } (called the process of classL with exponentH induced by ) are compared with those of the Lévy process {Y _t } such thatY ₁ has distribution . Results are generalized to operator-self-similar processes and distributions of classOL. A process {X _t } onR ^d is called wide-sense operator-self-similar if, for eachc>0, there are a linear operatorA _c and a functionb _c (t) such that {X _ct } and {A _c X _t +b _c (t)} have common finite-dimensional distributions. It is proved that, if {X _t } is wide-sense operator-self-similar and stochastically continuous, then theA _c can be chosen asA _c =c ^Q with a linear operatorQ with some special spectral properties. This is an extension of a theorem of Hudson and Mason [4].     

Moderate deviations and functional limits for random processes with stationary and independent increments

We first give a functional moderate deviation principle for random processes with stationary and independent increments under the Ledoux's condition. Then we apply the result to the functional limits for increments of the processes and obtain some Csorgo-Revesz type functional laws of the iterated logarithm.     

Noncommutative stochastic processes with independent and stationary increments satisfy quantum stochastic differential equations

Summary The notion of a unitary noncommutative stochastic process with independent and stationary increments is introduced, and it is proved that such a process, under a continuity assumption, can be embedded into the solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy [8].This work was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 123, Stochastische Mathematische Modelle     

LARGE DEVIATIONS FOR FIELDS WITH STATIONARY INDEPENDENT INCREMENTS

ＬＡＲＧＥＤＥＶＩＡＴＩＯＮＳＦＯＲＦＩＥＬＤＳＷＩＴＨＳＴＡＴＩＯＮＡＲＹＩＮＤＥＰＥＮＤＥＮＴＩＮＣＲＥＭＥＮＴＳＧＡＯＦＵＱＩＮＧ（高付清）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｕｂｅｉＵｎｉｖｅｒｓｉｔｙ，Ｗｕｈａｎ４３００７２...     

Variations of random processes with independent increments

In the paper one considers random processes _{s ost} with independent increments, continuous in the mean (_P<). One establishes relations among multiple integrals, variations, i.e., the limits of sums of the form , and the Itô stochastic integrals.Translated from Zapiski Nauchnykh Seminarov Leningradskago Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 25–35, 1983.     

Fluctuations of homogeneous processes with independent increments

This is the author's abstract of a thesis presented for the academic degree of Doctor of Physicomathematical Sciences. The thesis has been defended on December 21, 1970 at the Council which awards academic degrees in mechanomathematical sciences at Novosibirsk State University. The official opponents were V. M. Zolotarev (Doctor of Physicomathematical Sciences), V. S. Korolyuk (Doctor of Physicomathematical Sciences and Corresponding member of the Academy of Sciences of the Ukrainian SSR), and Yu. V. Prokhorov (Doctor of Physicomathematical Sciences and Corresponding Member of the Academy of Sciences of the USSR).     

Pasting of two processes with independent increments

A terminating stochastically continuous strictly Markov process is obtained as a result of pasting two nonterminating homogeneous stochastically continuous Markov processes with independent increments, one of which is semicontinuous. It is shown that this process can be extended to a complete homogeneous stochastically continuous strictly Markov Feller process. Previously, this problem has been solved by the author under stronger restrictions-both pasting processes were semicontinuous.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 487–491, April, 1993.     

Continuation of semicontinuous processes with independent increments

We consider a process X _t ¹ with independent increments without positive jumps in the state space (–; +) VarX _t ¹ =+. For a stopped process in the space E⁰ we construct its continuation in E⁰ U {0}.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1269–1272, September, 1991.     

Connecting two semicontinuous processes with independent increments

We consider a stopped process X_t ⁰ in the phase space E⁰=(–, +)/{0} such that X_t ⁰=X_t ¹ if X_t ⁰ > 0 and X_t ⁰=X_t ² if X_t ⁰ < 0, where X_t ^j, j=1,2, are nonstopped stochastically continuous Markov processes with independent increments and with only negative jumps. We prove that there exists an extension of X_t ⁰ into a homogeneous, stochastically continuous, and strong Markov Feller process X_t in the phase space (–; +) and that the extension can be characterized by a measure N(dy) and three constants b, c₁ c₂.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 596–600, May, 1991.