Sufficient conditions for the existence of solutions to stochastic inclusions $x_t - x_s ∈ ∫^t_s F_τ(x_τ)dτ + ∫^t_s G_τ(x_τ)dw_τ + ∫^t_s∫_{IRⁿ} H_{τ,z}(x_τ)ν̃ (dτ,dz)$ beloning to a given set K of n-dimensional cádlág processes are given.
3
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
Existence of strong and weak solutions to stochastic inclusions $x_{t} - x_{s} ∈ ∫^{t}_{s} F_{τ}(x_{τ})dτ + ∫^{t}_{s} G_{τ}(x_{τ})dw_{τ} + ∫^{t}_{s} ∫_{ℝ^{n}} H_{τ,z}(x_{τ})q(dτ,dz)$ and $x_{t} - x_{s} ∈ ∫^{t}_{s} F_{τ}(x_{τ})dτ + ∫^{t}_{s}G_{τ}(x_{τ})dw_{τ} + ∫^{t}_{s}∫_{|z|≤1} H_{τ,z}(x_{τ})q(dτ,dz) + ∫^{t}_{s}∫_{|z|>1} H_{τ,z}(x_{τ})p(dτ,dz)$, where p and q are certain random measures, is considered.
The paper is devoted to properties of generalized set-valued stochastic integrals defined in [10]. These integrals generalize set-valued stochastic integrals defined by E.J. Jung and J.H. Kim in the paper [4]. Up to now we were not able to construct any example of set-valued stochastic processes, different on a singleton, having integrably bounded set-valued integrals defined in [4]. It was shown by M. Michta (see [11]) that in the general case set-valued stochastic integrals defined by E.J. Jung and J.H. Kim, are not integrably bounded. Generalized set-valued stochastic integrals, considered in the paper, are in some non-trivial cases square integrably bounded and can be applied in the theory of stochastic differential equations with set-valued solutions.
The paper deals with integrably boundedness of Itô set-valued stochastic integrals defined by E.J. Jung and J.H. Kim in the paper [4], where has not been proved that this integral is integrably bounded. The problem of integrably boundedness of the above set-valued stochastic integrals has been considered in the paper [7] and the monograph [8], but the problem has not been solved there. The first positive results dealing with this problem due to M. Michta, who showed (see [11]) that there are bounded set-valued 𝔽-nonanticipative mappings having unbounded Itô set-valued stochastic integrals defined by E.J. Jung and J.H. Kim. The present paper contains some new conditions implying unboundedness of the above type set-valued stochastic integrals.
Some sufficient conditins for tightness of continuous stochastic processes is given. It is verified that in the classical tightness sufficient conditions for continuous stochastic processes it is possible to take a continuous nondecreasing stochastic process instead of a deterministic function one.
The paper contains new properties of set-valued stochastic integrals defined as multifunctions with subtrajectory integrals equal to closed decomposable hulls of functional set-valued integrals defined in the author paper [8]. In particular, it is proved that such defined integrals for set-valued predictable square integrably bounded processes having finite Castaing representations are square integrably bounded. Up to now this property has not been proved. Unfortunately, in the general case the above boundedness problem is still open.
The definition and some existence theorems for stochastic differential inclusion dZₜ ∈ F(Zₜ)dXₜ, where F and X are set valued stochastic processes, are given.