Professor. E.O. Ayoola

AYOOLA  OLUSOLA EZEKIEL

 Academic and Professional Qualifications

                                                                                                                B.Sc.,M.Sc., Ph.D. (Ibadan)

 PROFESSOR 

Area of Specialisation

Numerical Analysis, Stochastic Differential Equations

Office:

Department of Mathematics, University of Ibadan, Ibadan, Nigeria

Tel:  00 234 (0) 8075458906

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RESEARCH Completed:

(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.

In Progress:

(a) Theoretical and numerical solutions of stochastic models of real life problemsn 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 differ-

ence methods.

(b) Investigation of the oscillatory properties of solutions of stochastic delay dif-

ferential equations.

(c) The study of stochastic price dynamics(c) 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 stochas-

tic 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 stochas-

tic 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

Dissertations and Thesis:

(a)        AYOOLA, E.O. (1987). Solut

IonADissertation and thesis

(a)        AYOOLA,E.O.(1978).Solution of an Elliptic Boundary Value Problem on Polygonal Domain by Fredholm Theory.

M.Sc. Dissertation, Department of Mathematics, University of Ibadan.

(b) AYOOLA, E.O. (1999). On Numerical Procedures for Solving Lipschitzian Quantum Stochastic Differential Equations.

                    Ph.D. Thesis, Department of Mathematics, University of Ibadan.

Contribution to Scholarship

My contribution to scholarship can broadly be classified into two groups A and B.

Group A

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 value problems.

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 rocedure 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 and the Lebesgue integrals to quantum stochastic integrals under some suitable 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 pro-

cesses 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 topologicalproperties 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.

Group 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 re-

cently 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.

IX.   PUBLICATIONS

(i) Books or chapters in books already published:

[1] AYOOLA, E. O. (2008): MAT 351: Numerical Analysis, Theory and Computations,

Ibadan Distance Learning Centre Series in Mathematics,

Publisher:Distance Learning Centre, University of Ibadan, 140 pages.

ISBN 928-021-355-4.

[2] AYOOLA, E. O. (2008): Ordinary Differential Equations and Applications,

Publisher: GOLD Printing and Publishing, Ibadan, 155 pages.

ISBN 978 - 245 - 832 - 5.

(ii) Patents:  Nil

(iii) Articles that have already appeared in Learned Journals:

[3] AYOOLA, E.O. (1998). Solutions of Lipschitzian quantum stochastic differential equations in a locally convex space.     Journal of Science Research,Faculty of Science, University of Ibadan, Vol. 4, No. 1, 15 - 23.

[4] AYOOLA, E.O. (1999). Discrete approximations of weak solutions of a classof quantum stochastic evolution equations.     Journal of Science Research,Faculty of Science, University of Ibadan, Vol. 5, No.1, 17 - 24.

[5] AYOOLA, E.O. (1999). Error estimations in the Gauss and Newton Cotes

quadrature schemes for weak solutions of quantum stochastic differential equations.    Journal of the Nigerian Association of Mathematical Physics,

Vol. 3 No. 1, 12-35. Publisher:NAMP, Benin City, Nigeria.

[6] AYOOLA, E.O. (2000).Converging multistep schemes for weak solutions of

quantum stochastic differential equations.

Stochastic Analysis and Applications, Vol. 18, No. 4, 525 - 554.

Publishers:  Marcel Dekker Inc, New York, USA.

Mathematical Review: MR 2001e:81065, Zentralblatt Math: Zbl 0964.60070.

ISI Web of Science/Journal Citation Reports- Times Cited : 8.

[7] AYOOLA, E.O. (2000). Convergence of general multistep schemes for weak solutions of quantum stochastic differential equations. Proceedings of

                     the National Mathematical Centre - Ordinary Differential Equations

Vol. 1, No.1, 43 - 55. Publishers: NMC, Abuja, Nigeria. MR: 2004e:81072.

[8] AYOOLA, E.O. (2001). On Convergence of one-step schemes for weak solutions of quantum stochastic differential equations.

Acta Applicandae Mathematicae, Vol. 67, No. 1, 19 - 58.

Publishers:  Kluwer Academic Publishers, The Netherlands.

Mathematical Review: MR 2002f: 65017, Zentralblatt Math: Zbl 0998.60056.ISI

Web of Science/Journal Citation Reports - Times Cited: 3.

[9] AYOOLA, E.O. (2001). Lipschitzian quantum stochastic differential equations and the associated Kurzweil equations.

Stochastic Analysis and Applications, Vol. 19, No. 4, 581-603.

Publishers:  Marcel Dekker Inc., New York, USA.

10

Mathematical Review: MR: 2002g:81078, Zentralblatt Math: Zbl 0987.60072.

ISI Web of Science/Journal Citation Reports - Time Cited : 1.

[10] AYOOLA, E.O. (2001). Construction of approximate attainability sets for Lip-

schitzian quantum stochastic differential inclusions.

Stochastic Analysis and Applications, Vol. 19, No. 3, 461 - 471.

Publishers:  Marcel Dekker Inc, New York, USA.

Mathematical Reviews: MR 2002f:65018, Zentralblatt Math: Zbl 0985.60064.

ISI Web of Science/Journal Citation Reports - Times Cited: 3.

[11] AYOOLA, E.O. (2002). Lagrangian quadrature schemes for computing weak solutions of quantum stochastic differential equations.

      SIAM Journal on Numerical Analysis, Vol. 39, No. 6, 1835 - 1864.

Publishers:  SIAM Publications, Philadelphia, USA.

Mathematical Review: MR: 2003e: 60121, Zentralblatt Math:Zbl 1008.60077.ISI

Web of Science/Journal Citation Reports - Times Cited: 3.

[12] AYOOLA, E. O. (2002). On Computational procedures for weak solutions of quantum stochastic differential equations.

Stochastic Analysis and Applications, Vol. 20, No. 1, 1 - 20.

Publishers:  Marcel Dekker Inc, New York, USA.

Mathematical Review: MR: 2002m: 81124, Zentralblatt Math: Zbl 1009.81031.

ISI Web of Science/Journal Citation Reports:- Time cited: 1

[13] AYOOLA, E. O. (2002). Existence and stability results for strong solutions of quantum stochastic differential equations.

Stochastic Analysis and Applications, Vol. 20, No. 2, 263 - 281.

Publishers:  Marcel Dekker Inc, New York, USA.

Mathematical Review: MR: 2003b: 60081, Zentralblatt Math: Zbl 0997.60063.

ISI Web of Science/Journal Citation reports- Time cited: 1.

[14] AYOOLA, E. O. (2003). Exponential formula for the reachable sets of quantum stochastic differential inclusions.

Stochastic Analysis and Applications, Vol. 21, No 3, 515 - 543.

Publishers:  Marcel Dekker Inc, New York, USA.

Mathematical Review: MR 2004e: 81073, Zentralblatt Math: Zbl 1047.81048.

ISI Web of Science/Journal Citation Reports: Times Cited: 2.

[15] AYOOLA, E . O (2003). Error Estimates for discretized quantum stochastic

differential inclusions.

Stochastic Analysis and Applications, Vol. 21, No. 6, 1215 - 1230.

Publishers:  Marcel Dekker Inc, New York, USA.

Mathematical Review: MR: 2005a: 60109, Zentralblatt Math: Zbl 1033.60068.

ISI Web of Science/Journal Citation Reports- Time Cited: 1.

[16] AYOOLA, E. O. (2004) On the properties of weak solutions of Lipschitzian

quantum stochastic differential equations.

Journal of the International Centre for Mathematical

and Computer Sciences, (ICMCS, Lagos, Nigeria.), Vol. 1. 361- 369.

Publishers:  ICMCS Publications, Lagos, Nigeria.

Mathematical Review:   MR: 2005j:00013.

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[17] AYOOLA, E. O. (2004) Continuous selections of solution sets of Lipschitzian quantum stochastic differential inclusions.

International Journal of Theoretical Physics , Vol. 43, No. 10 , 2041 - 2059.

Publishers:  Springer Science, The Netherlands.

Mathematical Review: MR: 2005i : 81076, Zentralblatt Math:                                                                         Zbl 1074.81041.

ISI Web of Science/Journal Citation Reports- Time Cited: 1.

[18] AYOOLA, E. O. and GBOLAGADE, A. W (2005) Further results on the ex-

istence, uniqueness and stability of strong solutions of quantum stochastic differential equations.

Applied Mathematics Letters. Volume 18, 219 - 227.

Percentage Contribution : 90 percent .

Publishers: Elseviers Science, USA. Mathematical Review: MR :2005m:81179.

Zentralblatt fur Math:   Zbl 1071.81071.

ISI Web of Science/Journal Citation Reports- Time Cited: 0.

[19] AYOOLA, E. O; ADEYEYE, JOHN: (2007) Continuous interpolation of solu-

tion sets of Lipschitzian quantum stochastic differential inclusions.

Journal of Applied Mathematics and Stochastic Analysis,

Volume 2007 (2007), Article ID 80750, 12 pages, doi: 10.1155/2007/80750.

Percentage Contribution:80 percent.

Publishers:  Hindawi Publishers, New York, USA.

AMS Mathematical Review in process.

[20] ATONUJE, A. O; AYOOLA, E. O. (2007): On noise contribution to the oscil-

latory behavior of solutions of stochastic delay differential equations.

Journal of the Institute of Mathematics and Computer Sciences

(Computer Science Series) Volume 18, No.2, 51 - 59.

Percentage Contribution:50 percent.

Publishers:  Indian Mathematics institute, India.

AMS Mathematical Review in process.

[21] AYOOLA, E. O: (2007) : On the properties of Continuous selections of solution and reachable sets of quantum stochastic differential inclusions.

Journal of the Nigerian Association of Mathematical Physics,

Vol. 11, 71 - 82.

Publishers:  Nigerian Association of Mathematical Physics.

AMS Mathematical Review in process.

[22] AYOOLA, E. O (2008): Further results on the existence of continuous selections of solution sets of quantum stochastic differential inclusions.

Dynamic Systems and Applications, Volume 17, 609 - 624.

Publishers:Dynamic Publishers, Atlanta, Georgia, USA.

AMS Mathematical Review in process.

[23] AYOOLA, E. O. (2008) : Quantum stochastic differential inclusions satisfying a general Lipschitz condition.

Dynamic Systems and Applications, volume 17, 487 - 502.

Publishers:  Dynamic Publishers, Atlanta, Georgia, USA).

AMS Mathematical Review in process

[24] AYOOLA, E. O.(2008): Topological properties of solution sets of Lipschitzian quantum stochastic differential inclusions.

Acta Applicandae Mathematicae, Volume 100, Number 1, 15 - 37.

Publishers:  Springer Science + Business Media.

AMS Mathematical Review in process.

[25] NJOSEH, IGNATIUS N.; AYOOLA, EZEKIEL O. (2008): Finite element

method for a strongly damped stochastic wave equation driven by space -

time noise.

Journal of Mathematical Sciences, Volume 19, Number 1, 61 - 72.

Percentage Contribution:50 percent.

Publishers:  International Center for Advance Studies, Dattapukur,

India .

AMS Mathematical Review in process.

[26] ATONUJE, A.O.; AYOOLA, E. O.(2008) : On the complementary roles of

noise and delay in the oscillatory behavior of stochastic delay differential equations.

Journal of Mathematical Sciences, Volume 19, Number 1, 11 -20.

Percentage Contribution:       50 percent.

Publishers:  International Center for Advance Studies, Dattapukur,

India.

AMS Mathematical Review in process.

[27] ATONUJE, A.O.; AYOOLA, E. O.(2008) : Oscillation in solutions of stochas-

tic delay differential equations with real coefficients and several constant time lags.

Journal of the Nigerian Association of Mathematical Physics,

Volume 13, November 2008, 87 -94.

Percentage Contribution:       50 percent.

Publisher:   Nigerian Association of Mathematical Physics, Nigeria.

[28] OGUNDIRAN, M.; AYOOLA, E. O.(2010) : Mayer Problems for Optimal

Quantum Stochastic Control.

Journal of Mathematical Physics, Vol. 51, 023521, doi: 10.1063/1.3300332.

Publisher: American Institute of Physics,(AIP), Jan 2010

(iv) Books, Chapters in Books and Articles already accepted for Publications: Nil.

(v) Technical Report and/or Monographs:

[29] AYOOLA, E.O. (2001): Strongly stable multistep schemes for a class of quantum stochastic differential equations.

ICTP Preprint No. IC/2001/53.

Published at the URL: http://www.ictp.trieste.it/∼pub off

[30] AYOOLA, E. O. (2002): Approximate solutions of quantum stochastic differential equations and applications to stochastic models for T cell dynamics.

Proceeding of the RAMAD International Conference on

Bio - Mathematics, Bamako, Mali, July 01 - 13, 2002.

Abstract published at the URL: http://www.chez.com/ramad/ayoola2002.htm

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[31] AYOOLA, E. O: (2006): Implicit Multistep schemes for solving a system of

quantum stochastic differential equations.

A peer-reviewed paper in the Proceedings of the International Conference

on New Trends in Mathematical and Computer Sciences and Applications

to Real World Problems, C. K Ayo, C. R. Nwozo, etc (eds),

June 19-23, 2006, Covenant University, Ota, Nigeria, 595 -620.

[32] AYOOLA, E. O: (2006): Noncommutative quantum formulation of classical

stochastic differential equations, quadrature solutions and applications to financial mathematics.

An invited seminar paper presented as a Visiting Professor at the

Department of Mathematics and Applied Mathematics, University of

Pretoria, South Africa, Monday, December 4, 2006, 18 pages

Ten Groups of Publications Which Best Reflect My Contribution

To Scholarship

(i) Following the establishment of the existence of solutions and some qualitative aspects of Lipschitzian quantum stochastic differential inclusions by Ekhaguere in 1992, utilizing the framework of Hudson and Parthasarathy formulation of quantum stochastic calculus, I proposed, developed and studied in paper [4], multistep schemes for solving numerically Lipschitzian quantum stochastic

differential equation (LQSDE) of the form:

dX(t) = E(t, X(t))dΛπ⁽t) + F (t, X(t))dA⁺f⁽t)+G(t, X (t))dA_g(t) + H(t, X (t))dt        (1) in an interval [t₀, T ] with initial condition X(t₀) = X₀. The driving processes

Λπ^{, A}+_f, A_gare the stochastic integrators in the Boson Fock space quantum

stochastic calculus. Convergence of the discrete schemes to the exact solutions and error estimates were obtained for explicit scheme of class A in the locally convex space of solutions. Results published in [4] contain the Euler Maruyama schemes for Ito stochastic differential equations as a special case.

Numerical examples were given. Explicit and exact solutions of LQSDE (1)

are rarely available making the search for approximate solutions a necessary and worthwhile endeavour. Prior to the publication of paper [4], very little, if any at all, was known about the features of numerical solutions of LQSDE

(1). As LQSDE (1) is a noncommutative generalization of the classical Ito

stochastic differential equation (Ito SDE), driven by Brownian motion, the implementation of the multistep schemes and other discrete schemes developed in my subsequent works completely eliminated the need for the computation of random increments by random number generators as obtained in the implementation of stochastic Taylor schemes for simulation of sample paths and functionals of solutions of classical Ito SDE. This paper has opened further research directions concerning the refinement of the schemes in several ways as well as study of numerical stability associated with the multistep schemes .

See the AMS Mathematical Review of paper [4] by K. R. Parthasarathy in

MATHSCINET online with review number :MR 2001e:81065.

(ii) Paper [6] is concerned with the development, analysis and applications of several one-step schemes for computing weak solutions of LQSDE (1). The workwas accomplished in the framework of Hudson and Parthasarathy formulation of quantum stochastic calculus and subject to the matrix elements of solution being sufficiently differentiable. The results here concern non commutative generalization of the usual Euler scheme, Runge - Kutta sch an integral scheme for computing solutions of LQSDE (1).The paper contains

results for the Ito SDE as a special case with Ito processes as multiplication valued operators in a simple Fock space. The schemes here exhibit important implementation benefits as in paper [4]. Paper [6] is 40 page long and contains the main existence results of paper [1] as appendix as well as some numerical experiments to illustrate the main features of the different schemes

and their error estimates. The one step schemes here also generalize discrete schemes reported in papers [2] and [3]. Extension of the results here to the caseof continuous time Euler approximation scheme and a computational scheme under Caratheodory conditions was undertaken in [10]. This paper has created further research questions involving extensions to LQSDE (1) of various improvements already established for classical discrete schemes in the finite dimensional setting. See the mathematical review of paper [6] by Rolando Rebolledo Berroe in Zentralblatt Mathematics Database with review number :        Zbl 0998.60056   and the Abstract in the AMS Mathematical Review

with number MR 2002f:65017.

(iii) In paper [7], I introduced and studied Kurzweil equations associated with

LQSDE (1). Non commutative quantum extensions of classical Kurzweil in-

tegrals and some technical results were established. In addition, I proved the interesting equivalence between LQSDE (1) in integral form and the Kurzweil equation of the form :

d

< η, X (τ)ξ >= DΦ(X(τ ), t)(η, ξ)

dτ

on [t₀, T ] and for t       ∈      [t₀, T ], for a suitable map Φ and η, ξ    belonging to anappropriate class. Investigation of approximate solutions of LQSDE (1) bymutilizing established results on Kurzweil integrals and equations was afforded by the equivalence results. It was shown in the paper that the associated Kurzweil equation may be used to obtain reasonably high accurate solutions of LQSDE (1). This paper extends established relationship between Lebesgue and Kurzweil integrals to quantum stochastic integrals. The work here generalize some numerical results in paper [6] since the results in [7] hold under pure Caratheodory conditions where the matrix elements of solutions need not be differentiable more than once. The result here also generalize several analogous results for classical initial value problems to the non commutative quantum setting involving unbounded linear operators on a Hilbert space. Further research problems have been opened by this paper concerning the issue of variational stability of LQSDE (1). See the AMS Review of paper [7] by Debashish Goswami in MATHSCINET online with review number MR 2002g:81078 .

(iv) In paper [8], I presented a numerical method for constructing with a specified accuracy, attainability set R⁽T )₍x₀)(η, ξ) defined by

R⁽T )₍x₀)(η, ξ) = {< η, Φ(T )ξ >: Φ(·) ∈ S⁽T )₍x₀)}

(2)

for the Lipschitzian quantum stochastic differential inclusion (LQSDI) in integral form:

∫ t

X(t) ∈ x₀+

t₀(E(s, X(s))dλπ⁽s) + F (s, X(s))dA_f(s)

+ G(s, X(s))dA⁺g⁽s) + H(s, X(s)      ds, t ∈ [t₀, T ].

(3)    where S⁽T )₍x₀) is the set of solutions to LQSDI (3). An algorithm is described

for numerically approximating the attainability set within any prescribed accuracy. Results here generalize an analogous classical result of Komarov and Pevchikh to noncommutative quantum stochastic differential inclusion (3). Attainability sets are important for several characterization of the set of trajectories of LQSDI (3). In Paper [21], I established the existence of solutions of QSDI (3) satisfying a general Lipschitz condition. The Lipschitz condition of paper [8] is a special case. Extension of the numerical algorithm of paper[8] to general case is still open. See the AMS review of paper [8] by Volker Wihstutz in MATHSCINET online with review number MR 2002f:65018).

(v) Paper [9] is devoted to the analysis of the Lagrangian quadrature schemes for

computing weak solutions of LQSDE (1) with matrix elements that are sufficiently smooth. Results concerning the convergence of Lagrangian schemes to exact solutions were obtained. Precise estimates for an error term were given in the case when the nodes of approximations are chosen to be roots of the Chebyshev polynomials. Some important features of the quadrature schemes are the conversion of LQSDE (1) to solvable algebraic equations in term of the nodal values and that the nodes need not be equally spaced. This paper established the possibility of applying numerous results in linear and computer algebra

for investigating numerical solutions to LQSDE (1). Numerical experiments

were performed by solving associated linear systems taking into consideration,

computational complexity of the algorithm and round off errors. See the AMS

review of this paper by Vassili N. Kolokoltsov in MATSCINET online with

review number MR 2003e:60121.

(vi) In Paper [22], I established a continuous mapping of the space of the matrixelements of an arbitrary nonempty set of quasi solutions of Lipschitzian QSDI

(3) into the space of the matrix elements of its solutions. As a corollary, I

furnished a generalization of my previous selection result in paper [15]. In

particular, when the coefficients of the inclusion are integrally bounded, it was

shown that the space of the matrix elements of solutions is an absolute re-

tract, contractible, locally and integrally connected in an arbitrary dimension.

As usual, we employ the Hudson and Parthasarathy formulation of quantum

stochastic calculus. See the mathematical review of paper [22] by Vassili N.

Kolokoltsov  in the AMS Mathematics Review.

(vii) In paper [11], the existence, uniqueness and stability of strong solutions of LQSDE (1) were established.

The locally convex topology on the space of quantum stochastic processes in this case is generated by a family of seminorms induced by the norm of the Fock space. The second fundamental formula of Hudson and Parthasarathy concerning the estimate of the square of the norm of the values of stochastic processes on exponential vectors facilitates the existence results by method of successive approximations. Results

here generalize analogous results concerning classical SDE driven by Brownian motion. Convergence in the sense of this paper generalize the root mean square convergence of successive approximation in the case of classical Ito process considered as quantum stochastic process in a simple Fock space. The work in [16] is a continuation of [11] concerning the existence and stability of solutions of QSDE satisfying a general Lipschitz condition in the strong topology. Paper [16] established a class of Lipschitzian QSDE where the coefficients

are merely continuous on the locally convex space of stochastic processes. See the AMS Mathematical Review of paper [11] by Vassili N. Kolokoltsov in

MATHSCINET online with review number MR 2003b: 60081.

(viii) Paper [12] is my second major work on quantum stochastic differential in-

clusions (QSDI). The paper is a continuation of my previous work in paper

[8] concerning QSDI (3) where the coefficients are assumed to have suitable regularity properties. The basic setup of the paper is that of multivalued functions with appropriately defined multivalued stochastic integrals. By endowing the family of closed subsets of the locally convex space of quantum stochastic processes with a Hausdorff topology, the paper established the following exponential formula:

R⁽T )₍x₀) = lim

N→∞(I +TNPN(x₀)