[b][u]RESEARCH[/u][/b] [b][u]Completed:[/u][/b] (a) Establishment of a solution of an elliptic boundary value problem on polygonal domains by Fredholm theory. (b) Existence results for various classes of quantum stochastic differential equations having solutions in a locally convex space endowed with a weak topology generated by a family of seminorms defined on the space of quantum stochastic processes. (c) Introduction and study of a number of numerical schemes for approximating weak solutions of a class of quantum stochastic differential equations. (d) Establishment of the existence and stability of strong solutions of Lipschitzian quantum stochastic differential equations. (e) Introduction and study of Kurzweil equations associated with a class of quantum stochastic differential equation in the framework of the Hudson-Parthasarathy formulation of quantum stochastic calculus. (f) Construction of approximate attainability sets for Lipschitzian quantum differential inclusions (LQDI). (g) Establishment of an exponential formula for the reachable sets of quantum stochastic differential inclusions. (h) Development and analysis of approximate solutions for LQDI and their error estimates. (i) Establishment and applications of the existence of continuous selections of solution sets of Lipschitzian quantum stochastic differential inclusions. [b][u]In Progress:[/u][/b] (a) Theoretical and numerical solutions of stochastic models of real life problems in Mathematical Economics, Finance and Biomedical Systems. This partly involves the development, analysis and implementations of numerical algorithms for stochastic partial differential equations by finite element and finite difference methods. (b) Investigation of the oscillatory properties of solutions of stochastic delay differential equations. (c) The study of stochastic price dynamics models for option prices especially in incomplete markets. (d) The development of numerical schemes for the solutions of jump - diffusion stochastic models of derivative pricing in complete and incomplete markets. (e) The development of numerical schemes for weak and strong solutions of quantum stochastic differential equations and inclusions of Lipschitzian, hypermaximal monotone or evolution types with Caratheodory conditions. (f) Study of topological properties of solutions of quantum stochastic differential inclusions of several types that are important for thorough understandings of physical systems that they model. (g) The search for the noncommutative generalizations to the various results that are well known in the numerical analysis of deterministic and classical stochastic differential equations.  These include results about the stability as well as consistency of certain numerical schemes. (h) Applications of approximate numerical schemes to the study of quantum fields and stochastic controls. (i) Investigation of stability properties of quantum stochastic differential equations and inclusions. (j) Solutions of inverse problems in classical and noncommutative quantum stochastic analysis. (k) Investigation of the existence of solution of quantum stochastic differential inclusions with one sided Lipschitz or upper semi continuous coefficients and the development of the approximation procedure for these classes of inclusions. (l) Investigation of stability properties of quantum stochastic inclusions with upper semi continuous or one sided Lipschitz coefficients. [b][u]Dissertations and Thesis:[/u][/b] (a) AYOOLA, E.O. (1987). Solution of an Elliptic Boundary Value Problem on Polygonal Domain by Fredholm Theory.  [b][u]M.Sc. Dissertation, Department of Mathematics, University of Ibadan[/u][/b]. (b) AYOOLA, E.O. (1999).  On Numerical Procedures for Solving Lipschitzian Quantum Stochastic Differential Equations. [b][u]Ph.D. Thesis, Department of Mathematics,  University of Ibadan.[/u][/b]   [b][u]Contribution to Scholarship:[/u][/b] My contribution to scholarship can broadly be classified  into two groups A and B. [b]Group A[/b] This concerns the development, analysis and applications of numerical solutions of quantum stochastic differential equations (QSDE) and inclusions (QSDI) in the framework of the Hudson - Parthasarathy formulation of quantum stochastic calculus. These equations and inclusions are noncommutative generalization of classical stochastic differential equations (SDE) and inclusions (SDI) driven by martingales. They involve unbounded linear operators on some Hilbert spaces and reduce to the classical case by considering commuting self adjoint operator processes on a simple Fock space.  On numerical procedures for solving QSDE, I have published a number of new results concerning new discrete schemes such as multistep schemes, one step schemes, Runge - Kutta and one step integral schemes when the coefficients are smooth enough and for the case when the coefficients satisfy general Caratheodory conditions. These schemes are extensions of the analogous schemes in the classical context to the present noncommutative case involving QSDE defined in infinite dimensional locally convex spaces Several benefits are associated with these schemes. An important benefit is the complete elimination of the need for additional burden of simulation of random increments of the driving process as obtained in the implementation of stochastic Taylor schemes for classical Ito SDE. There are no correction terms in the limit as in the case of Wong - Zakai approximation of classical SDE. As demonstrated in my publications, computation of the discrete solutions are done in the same way as for classical initial valueproblems My further contribution is the establishment of the Lagrangian quadrature schemes for computing weak solutions of QSDE. The theory of Chebyshev minimax best approximation was employed to establish the quadrature algorithm. Precise error estimate was given in the case when the zeroes of Chebyshev polynomials are chosen as discretization nodes Further contributions involved the establishment of numerical procedure for QSDI. Since our discrete schemes for QSDE require continuity of the coefficients at least in the state variables, numerical approximations for discontinuous QSDE are established via the QSDI since discontinuous QSDE can be reformulated in some sense as QSDI with regular coefficients. To this end, I have established some error estimates by employing the averaged modulus of continuity associated with the coefficients of the QSDI.  I have also established an exponential formula for the reachable sets. The formula has been found useful for approximation of the reachable sets.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               I have also contributed to scholarship by extending the classical theory of Kurzweil-Henstock integral calculus to quantum stochastic analysis. I have extended the well known equivalence relationship between Kurzweil - Henstock integrals andthe Lebesgue integrals to quantum stochastic integrals under somesuitable conditions in the framework of Hudson – Parthasarathy quantum stochastic calculus. The extension facilitates the proof of the equivalence of Lipschitzian QSDE and its associated Kurzweil equation. By using the associated Kurzweil equation, I established the numerical approximation of QSDE whose coefficients satisfy pure Caratheodory conditions. Furthermore, I have established the existence, uniqueness and stability of strong solutions of QSDE under a Lipschiz type condition. Extension of the results to the case of QSDE satisfying a general Lipschitz condition was undertaken jointly with Gbolagade, A. W. Our results show that QSDE with continuous coefficients are Lipschitzian in a general sense.  This idea has recently been extended to the establishment of the existence of solutions of Quantum stochastic differential inclusions QSDI that satisfy a general Lipschitz condition. The general Lipschitz condition considered here generalized Ekhaguere's approach in his pioneering paper of 1992 in this field. I successfully formulated and proved a theorem on the nonuniqueness of solutions and some bounds for the derivatives of the matrix elements of solutions for this case of the general Lipschitz condition. Another very important contribution is the establishment of a continuous selection of a multifunction associated with the set of solutions of Lipschitzian QSDI.  As a corollary to the main result, I proved that the solution set map and the reachable set admit some continuous representations. A follow up result has just been published. This concerns the establishment of a continuous selection of the set of solutions to Lipschitzian QSDI, defined directly on the locally convex space of stochastic processes with values in the space of adapted weakly absolutely continuous solutions.  As a corollary, I proved that the reachable set multifunction admits a continuous selection. Furthermore, jointly with John Adeyeye, we have recently established that given any finite set of trajectories of a Lipschitzian QSDI, there exists a continuous selection from the complex valued multifunction associated with the solution set of Lipschitzian QSDI interpolating the matrix elements of the given trajectories. This result extends my previous result that concerns only one matrix elements of solutions into a finite number of them.  In addition, some bounds in the seminorm of the locally convex space of solutions, for the difference of any two of such trajectories were also established.  In continuation of my research work on QSDI, I have established some topological properties of solution sets of QSDI. These properties have recently been published by a leading Mathematics Springer journal. A continuous mapping of the space of the matrix elements of an arbitrary nonempty set of quasi solutions of Lipschitzian QSDI into the space of the matrix elements of its solutions was established. This consequently facilitated the establishment of the space of the matrix elements of solutions as an absolute retract. It was further established that the space is contractible and connected in some sense. As a corollary, a generalization of my previous selection result was furnished by removing the requirement of compactness of the domain of the selection map.     [b]Group B[/b] My contributions in this group concern some qualitative and numerical aspects of classical stochastic differential equations driven by Brownian motions. Jointly with my former doctoral student A.O. Atonuje, we have published some results concerning the non contribution to the oscillatory behaviour of solutions of stochastic delay differential equations (SDDE). We have proved that even when non-oscillatory solutions exist in the corresponding deterministic delay differential equation, the presence of noise perturbation stimulates an oscillation subject to certain conditions on the delay terms. Furthermore, we have recently shown that in the absence of the noise term, non -oscillatory solutions can occur for the deterministic case. But with the presence of noise, all solutions of SDDE oscillate almost certainly whenever the feedback intensity is negative. Furthermore, jointly with my former doctoral student I. N. Njoseh, we have recently published some results on the finite element method for a strongly damped stochastic wave equation driven by a space - time noise. We provided some error estimates of optimal order for semidiscrete and fully discrete finite elements schemes by using L2-projections of the initial data as starting values.