Stochastic processes with finite semivariation in Banach spaces and their stochastic integral

In this Paper we study a new class of Banach-valued Processes which are summable: the Processes with integrable semivariation. One can define the Stochastic Integral for such processes, which can be computed pathwise, as a Stieltjes integral with respect to a function with finite semivariation (rather than finite variation).     

Comonotonic stochastic processes and generalized mean-square stochastic integral with applications

The concept of monotonic stochastic processes was introduced by Skowroński [Aequationes mathematicae 44 (1992) 249–258]. In this paper, we intro     

Refinements of mean-square stochastic integral inequalities on convex stochastic processes

Recently, in the class of convex stochastic processes, Kotrys (Aequat Math 83:143–151, 2012; Aequat Math 86:91–98, 2013) proposed upper and lower bounds of mean-square stochastic integrals by using Hermite–Hadamard inequality. This paper shows that these bounds can be refined. Our results extend and refine the corresponding ones in the literature. Finally, an open problem for further investigations is given.     

Skorohod integral of a product of two stochastic processes

In this note we provide sufficient conditions for the product of a predictable and a backward predictable process to be Skorohod integrable.Supported in part by NSF No. INT-9401109, NSA No. MDR-90494H2094, and ONR No. N-00014-96-1-0262.     

On free stochastic processes and their derivatives

We study a family of free stochastic processes whose covariance kernels K$K$ may be derived as a transform of a tempered measure σ$\sigma$. These processes arise, for example, in consideration of non-commutative analysis involving free probability. Hence our use of semi-circle distributions, as opposed to Gaussians. In this setting we find an orthonormal basis in the corresponding non-commutative L²$L^2$ of sample-space. We define a stochastic integral for our family of free processes.     

Continuity of the occupation density for anticipating stochastic integral processes

In this paper we prove the existence of a continuous local time for an anticipating process which is composed of an indefinite Skorohod integral and an absolutely continuous term.The work of P. Imkeller was done during his visit to the CRM of Barcelona.Partially supported by the DGICYT grant number PB90-0452.     

Occupation densities for stochastic integral processes in the second Wiener chaos

Summary The two-point distributions of Skorohod integral processes in the second Wiener chaos are mainly described by a Hilbert-Schmidt operatorT giving the mutual interaction of infinitely many Gaussian components and by simple multiplication operators. So are the Fourier transforms of their occupation measures. This enables us to use the well known Fourier analytic criterion discovered and elaborated by Berman to derive integral conditions for the existence of their occupation densities in terms of associated Hilbert-Schmidt operators. IfT is a trace class operator, we get a necessary and sufficient criterion, if it is not, still a sufficient one. In a case in which the interaction is particularly simple, we verify the appropriate integral condition and show that the results are essentially beyond the reach of enlargement of filtrations techniques of semimartingale theory.     

Additive comparisons of stopping values and supremum values for finite stage multiparameter stochastic processes

This paper concerns the optimal stopping problem for discrete time multiparameter stochastic processes with the index set Nd. In the classical optimal stopping problems, the comparisons between the expected reward of a player with complete foresight and the expected reward of a player using nonanticipating stop rules, known as prophet inequalities, have been studied by many authors. Ratio comparisons between these values in the case of multiparameter optimal stopping problems are studied by Krengel and Sucheston (1981) [9] and Tanaka (2007, 2006) [14] and [15]. In this paper an additive comparison in the case of finite stage multiparameter optimal stopping problems is given.     

Curved exponential families of stochastic processes and their envelope families

Exponential families of stochastic processes are usually curved. The full exponential families generated by the finite sample exponential families are called the envelope families to emphasize that their interpretation as stochastic process models is not straightforward. A general result on how to calculate the envelope families is given, and the interpretation of these families as stochastic process models is considered. For Markov processes rather explicit answers are given. Three examples are considered some in detail: Gaussian autoregressions, the pure birth process and the Ornstein-Uhlenbeck process. Finally, a goodness-of-fit test for censored data is discussed.     

Construction of branching diffusion processes and their optimal stochastic control

In this work we construct a branching diffusion process whose individuals are interdependent, as the unique solution of a martingale problem. As an application we propose and solve a closed loop, finite horizon optimal control problem.     

Nonminimal sets,their projections and integral representations of stable processes

New criteria are provided for determining whether an integral representation of a stable process is minimal. These criteria are based on various nonminimal sets and their projections, and have several advantages over and shed light on already available criteria. In particular, they naturally lead from a nonminimal representation to the one which is minimal. Several known examples are considered to illustrate the main results. The general approach is also adapted to show that the so-called mixed moving averages have a minimal integral representation of the mixed moving average type.     

On stochastic integral representation of stable processes with sample paths in Banach spaces

Certain path properties of a symmetric α-stable process X(t) = ∫_Sh(t, s) dM(s), t T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double α-stable integral. Also, necessary and sufficient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to[13], Ann. Probab.9, 624–632) is extensively used and the relationship between these two representations is discussed.