Badri Mamporia

A sufficient condition is given for the existence of a solution to a stochastic differential equation in an arbitrary Banach space. The method is based on the concept of covariance operator and a special construction of the Itô stochastic integral in an arbitrary Banach space

The following results are announced: If a sequence of random elements tea normed space converges to zero in probability, then there exists a permutation of the given sequence, which is Cesáro convergent almost surely. Every bounded in L, sequence of random variables allows a permutation under which it satisfies the strong law of large numbers. A statement about Banach spaces with the Banach-Saks property is also given.

Ito’s formula for the generalized random processes and for the random processes with values in a separable Banach space is proved

A generalized stochastic integral of predictable generalized random processes (predictable random processes) with respect to a real Wiener process is defined. The question of the existence of the stochastic integral in a Banach space is reduced to the problem of decomposability of the generalized random element. The stochastic differential equation for generalized random processes is introduced and existence and uniqueness of solutions are developed. As a consequence the corresponding results on stochastic differential equations in a Banach space are given.

 The analysis of the definition of Wiener process in a Banach space is given. It considers the definitions of generalized Wiener process and Wiener process in a weak sense. The representations of them by the sums of identically distributed independent (weakly independent) Gaussian random elements are given

In this paper we concern with the question of weak independence of random elements. The case, where the random elements are Gaussian ones with values in a Banach space is considered. In this case the theory of covariance operator allows us to obtain certain (not as yet final) result

In this paper the stochastic differential equation in a Banach space is considered for the case when the Wienerprocess in the equation is Banach space valued and the integrand non-anticipating function is operator-valued. At first the stochastic differential equation for the generalized random process is introduced and developed existence and uniqueness of solutions as the generalized random process. The corresponding results for the stochastic differential equation in a Banach space is given. In [5] we consider the stochastic differential equation in a Banach space in the case, when the Wiener process is one dimensional and the integrand non-anticipating function is Banach space valued

Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral

in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.

Generalized stochastic integral from predictable operator-valued random process with respect to a cylindrical Wiener process in an arbitrary Banach space is defined. The question of existence of the stochastic integral in a Banach space is reduced to the problem of decomposability of the generalized random element. The sufficient condition of existence of the stochastic integral in terms of

p-absolutely summing operators is given. The stochastic differential equation for generalized random processes is considered and existence and uniqueness of the solution is developed. As a consequence, the corresponding results of the stochastic differential equations in an arbitrary Banach space are given

A linear stochastic differential equation in an arbitrary separable Banach space is considered. To solve this equation, the corresponding linear stochastic differential equation for generalized random processes is constructed and its solution is produced as a generalized Itô process. The conditions under which the received generalized random process is the Itô process in a Banach space are found, and thus the solution of the considered linear stochastic differential equation is obtained. The heart of this approach is the conversion of the main equation in a Banach space to the equation for generalized random processes, to find the generalized solution, and then to learn the conditions under which the obtained generalized random process is the random process with values in a Banach space.

The Dirichlet ordinary and generalized harmonic problems for some 3D finite domains are considered. The term “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. An algorithm of numerical solution by the method of probabilistic solution (MPS) is given, which in its turn is based on a computer simulation of the Wiener process. Since, in the case of 3D generalized problems there are none exact test problems, therefore, for such problems, the way of testing of our method is suggested. For examining and to illustrate the effectiveness and simplicity of the proposed method five numerical examples are considered on finding the electric field. In the role of domains are taken ellipsoidal, spherical and cylindrical domains and both the potential and strength of the field are calculated. Numerical results are presented

In development of stochastic analysis in a Banach space one of the main problem is to establish the existence of the stochastic integral from predictable Banach space valued (operator valued) random process. In the problem of representation of the Wiener functional as a stochastic integral we are faced with an inverse problem: we have the stochastic integral as a Banach space valued random element and we are looking for a suitable predictable integrand process. There are positive results only for a narrow class of Banach spaces with special geometry (UMD Banach spaces). We consider this problem in a general Banach space for a Gaussian functional

Some special cases of single-machine problems is considered. In these cases there are many optimal solutions. It is given the formulas of quantity of optimal solutions and calculated the probability of the event that an arbitrary schrdule is optimal; the sufficient conditions to increase the value of this probability is given and the corresponding optimal full completion time is calculated

The problem of representation of the Banach space-valued functionals of the one- dimensional Wiener process by the Ito stochastic integral is considered. Earlier, in [5] we have developed this problem in case the joint distribution of the Wiener process and its functional is Gaussian. In this article we consider the general case: rstly, for the weak second order Banach space-valued functional the generalized random process is found as an integrand. Further, for the one-dimensional functional of the Wiener process the sequence of adapted step functions converging to the integrand function, generalizing the corresponding result for the Gaussian case, is obtained (see [2]); the sequence of adapted step functions of generalized random elements converging to the integrand generalized random process is constructed for a Banach space-valued functional

A basic supply chain scheduling problem in which the orders released over time are to be delivered into the batches with unlimited capacity is considered. The delivery of each batch has a fixed cost D, whereas any order delivered after its release time yields an additional delay cost equal to the waiting time of that order in the system. The objective is to minimize the total delivery cost of the batches plus the total delay cost of the orders. A new algorithmic framework is proposed based on which fast algorithms for the solution of this problem are built. The framework can be extended to more general supply chain scheduling models and is based on a theoretical study of some useful properties of the offline version of the problem. An online scenario is considered as well, when at each assignment (order release) time the information on the next order released within the following T time units is known but no information on the orders that might be released after that time is known. For the online setting, it is shown that there is no benefit in waiting for more than D time units for incoming orders, i.e., potentially beneficial values for T are 0 < T < D, and three linear-time algorithms are proposed, which are optimal for both the offline and the online cases when T  D.

For the case 0 < T < D an important real-life scenario is studied. It addresses a typical situation when the same number of orders are released at each order release time and these times are evenly distributed within the scheduling horizon. An optimal algorithm which runs much faster than earlier known algorithms is proposed.

The integral type Wiener functional is considered and the stochastic integral representation formula of the Clark-Ocone type is established