Welcome to the Home Page of
 Professor Tomasz Zastawniak

M. Capinski and T. Zastawniak, Mathematics for Finance. An Introduction to Financial Engineering, Springer Undergraduate Mathematics Series (SUMS), Springer-Verlag, London, 2003. Corrected 3rd printing, September 2004. See more details

Contents

How to contact me About myself Personal Information Education Employment Grants & Awards Published Books Basic Stochastic Processes Foundations of Finslerian Diffusion with Applications Probability Through Problems Mathematics for Finance: An Introduction to Financial Engineering Corrected 3rd printing, Sept.2004

Research Interests and Experience Stochastic Analysis Path Integrals Quantum Mechanics Financial Mathematics Finsler Geometry Mathematical Biology Research Papers Editorships Teaching  MSc in Mathematical Finance Other Interests How long was the coast of Atlantis? Voting power theory in the EU Council

How to contact me

About myself

Employment

Dates	Institution	Division	Position
1984-1988	Jagiellonian University, Krakow, Poland	Institute of Mathematics	Assistant Professor
1987-1988	University of Warwick, Coventry, UK	Mathematics Institute	Visiting Academic
1988-1994	Jagiellonian University, Krakow, Poland	Institute of Mathematics	Adjunct Professor
1985-1989	Polish Academy of Sciences, Krakow, Poland	Institute of Mathematics	Assistant Professor
1989-1991	University of Wales, Swansea, UK	Department of Mathematics	Research Fellow
1991-1993	University of Alberta, Edmonton, Canada	Department Mathematical Sciences	NSERC International Fellow
1993-1994	Northwestern University, Chicago, USA	Department of Mathematics	Assistant Professor
1994-1997	University of Glamorgan, Wales, UK	School of Accounting  and Mathematics	Welsh Office Funded Research Fellow
1997-2001	University of Hull, Kingston upon Hull, UK	Department of Mathematics	Lecturer
2001-2003	University of Hull, Kingston upon Hull, UK	Department of Mathematics	Senior Lecturer
Mar.2003   -Sep.2003	University of Hull, Kingston upon Hull, UK	Department of Mathematics	Reader
Oct.2003- 2005	University of Hull, Kingston upon Hull, UK	Department of Mathematics	Professor
2005- continuing	University of York York, UK	Department of Mathematics	Chair of Financial Mathematics

Grants & Awards

Dates	Grant/Award	Awarding Body
1984	J. Marcinkiewicz Prize  for Mathematics Students	Polish Mathematical Society
1984	F. Leja Prize  for Young Mathematicians	Jagiellonian University, Krakow, Poland
1986-1987	Visiting Studentship at St. John's College, Oxford	The Soros Foundation
1991-1993	NSERC International Fellowship at University of Alberta, Edmonton, Canada	Natural Sciences and Engineering  Research Council, Canada
1996	EPSRC Visiting Fellowship Research Grant (Co-investigator)	Engineering and Physical Sciences Research Council, UK
1997	EPSRC Visiting Fellowship Research Grant (Co-investigator)	Engineering and Physical Sciences Research Council, UK
1998	Royal Society Conference Grant	The Royal Society, UK
1998-2001	EPSRC Postgraduate Studentship in Financial Mathematics (Supervisor)	Engineering and Physical Sciences Research Council, UK
1999-2002	Open Postgraduate Studentship (Supervisor)	University of Hull
1999-2002	German Academic Service Exchange Scheme Grant (Team Member)	The British Council
1999-2000	EPSRC Fast Stream Research Grant (Principal Investigator)	Engineering and Physical Sciences Research Council, UK
2000-2002	Polish Academic Exchange Grant (British Team Leader)	The British Council
2002	HIMSA Seeding Grant	Hull Institute of Mathematical Sciences and Applications

Research Interests and Experience

	Stochastic Analysis diffusion on manifolds, see Refs. [AntZas93], [AntZas94a], [AntZas94b], [AntZas95b], [AntZas98], [AntZas99] , [AntZas03] stochastic mechanics, see Refs. [Zas90], [Zas92], [GliZas:95], [GliZas96], [Zas97a], [TruZas98], [GliZas98] stochastic heat equation and stochastic Schroedinger equation, see Refs. [Zas97b], [TruZas99], [TruZas00] miscellaneous, see Refs. [Zas91a], [BrzZas99], [SloZas04]
	Financial Mathematics see Ref. [CapZas01], [SloZas04], [TokZas04], [Zas04], [TokZas05],  [RouZas05], [TokRouZas06], [RouZas06]
	Quantum Mechanics, Thermodynamics, Mathematical Physics see Refs. [Zas87b], [PliZas94], [Zas97a], [TruZas99], [TruZas00], [SloZas04]
	Path Integrals see Refs. [Zas86], [Zas87a], [Zas88], [Zas89a], [Zas89b], [Zas91b], [Zas93], [Zas97b], [TruZas99], [TruZas00]
	Finsler Geometry see Refs. [AntMatZas96], [AntZas98], [AntZas99], [AntZas03]
	Mathematical Biology see Refs. [AntZas94a], [AntZas94c], [AntZas95a], [AntZas96], [AntZas97a], [AntZas97b], [AntZas99]

Published Books

	[BrzZas99] Z. Brzezniak and T. Zastawniak, Basic Stochastic Processes. A Course Through Exercises, Springer Undergraduate Mathematics Series (SUMS), Springer-Verlag, London 1999. ISBN 3-540-76175-6. This book is a final year undergraduate text on stochastic analysis, a theory used widely by statisticians and experts working, for example, in mathematical finance. A detailed treatment is given of conditional probability and expectation, a topic which is essential as a tool for stochastic processes. Exercises, complete with informal hints and fully worked solutions, are chosen as the main means of explanation, hence the course has a strong self-study element. The authors have concentrated on major topics within stochastic analysis: martingales in discrete time and their convergence, Markov chains, stochastic processes in continuous time, with emphasis on the Poisson process and Brownian motion, as well as Ito stochastic calculus including stochastic differential equations. see "Basic Stochastic Processes" web page see Springer-Verlag catalogue entry
	[AntZas99] P. Antonelli and T. Zastawniak, Fundamentals of Finslerian Diffusion with Applications, Fundamental Theories of Physics Series, Vol. 101, Kluwer Academic Publishers, Dordrecht 1999. ISBN 0-7923-5511-3 This is the first text to be published on stochastic Finslerian geometry. The theory is rigorously presented and several applications in ecology, evolution and epidemiology are described. Amongst the various topics covered are the role of curvature in Finslerian diffusions, Nelson's stochastic mechanics, nonlinear (Finslerian) filtering and entropy production. Two appendices deal with, respectively, the stochastic Hodge theory of Finslerian harmonic forms, and the theory of 2-dimensional Finsler spaces. The latter plays an important role in the applications described in the text. This volume will be of interest to probabilists, applied mathematicians, mathematical biologists and geometers. It can also be recommended as a supplementary graduate text. see publisher's catalogue entry
	[CapZas00] M. Capinski and T. Zastawniak, Probability Through Problems, Springer-Verlag, New York, 2001. This book of problems has been designed to accompany an undergraduate course in probability. The only prerequisite is basic algebra and calculus. Each chapter is divided into three parts: Problems, Hints, and Solutions. To make the book self-contained, all problem sections include expository material. Definitions and statements of important results are interlaced with relevant problems. The problems have been selected to motivate abstract definitions by concrete examples and to lead in manageable steps towards general results, as well as to provide exercises based on the issues and techniques introduced in each chapter. The book is intended as a challenge to involve students as active participants in the course. See "Probability Through Problems" web page See Springer-Verlag catalogue entry
	[CapZas01] M. Capinski and T. Zastawniak, Mathematics for Finance. An Introduction to Financial Engineering, Springer Undergraduate Mathematics Series (SUMS), Springer-Verlag, London, 2003. True to its title, this book is itself an excellent financial investment. For the price of one volume it teaches two Nobel Prize winning theories, with plenty more included for good measure. Mathematics for Finance is designed to form the basis of an undergraduate course. It builds on mathematical models of bond and stock prices and covers three major areas of mathematical finance that all have an enormous impact on the way modern financial markets operate, namely: Black-Scholes' arbitrage pricing of options and other derivative securities; Markowitz portfolio optimisation and the Capital Asset Pricing Model; Interest rates and their term structure. Assuming only a basic knowledge of probability and calculus, the book combines financial motivation with mathematical style. It covers the material in a mathematically rigorous and complete way at a level accessible to second or third year undergraduate students. The text is interspersed with a multitude of worked examples and exercises, which provides ample material for tutorials, and makes the book ideal for self-study. It is suitable not only for students of mathematics, but also students of business management, finance and economics, and anyone with an interest in finance who needs to understand the underlying theory. Corrected 3rd printing, September 2004. See "Mathematics for Finance" web page See Springer-Verlag catalogue entry

Research Papers

[Zas87a] T. Zastawniak, Path integrals and probabilistic representations for the Dirac equation in two and four space-time dimensions, PhD thesis, Jagiellonian University, Krakow 1987.

[Zas87b] T. Zastawniak, The analysis of the SVZ method applied to the Schroedinger equation with the potential \lambda\ctg^{2\pi x}, Acta Physica Polonica B18 (1987) 37-45 (MSc thesis in physics, Jagiellonian University, Krakow 1985).

[Zas88] T. Zastawniak, Path integrals for the telegrapher's and Dirac equations; the analytic family of measures and the underlying Poisson process, Bull. Polish Acad. Sci. Math. 36 (1988) 341-356.

[Zas89a] T. Zastawniak, The nonexistence of the path-space measure for the Dirac equation in four space-time dimensions, J. Math. Phys. 30 (1989) 1354-1358.

[Zas89b] T. Zastawniak, Path integrals for the Dirac equation - some recent developments in mathematical theory, in: K.D.Elworthy and J.-C.Zambrini (eds.), Stochastic Analysis, Path Integration and Dynamics (Warwick 1987), Pitman Res. Notes Math. Ser., Vol. 200, Longman Sci. Tech., Harlow 1989, pp.243-263.

[Zas90] T. Zastawniak, A relativistic version of Nelson's stochastic mechanics, Europhysics Letters 13 (1990) 13-17.

[Zas91a] T. Zastawniak, Evaluation of certain Wiener integrals, Univ. Iagel. Acta Math. 28 (1991) 177-185.

[Zas91b] T. Zastawniak, The equivalence of two approaches to the Feynman integral for the anharmonic oscillator, Univ. Iagel. Acta Math. 28 (1991 187-199.

[Zas92] T. Zastawniak, Markov diffusions in relativistic stochastic mechanics, A.Truman and I.M.Davies (eds.), Stochastics and Quantum Mechanics (Swansea 1990), World Sci. Publishing, Singapore-River Edge, NJ-London 1992, pp.280-297.

[AntZas93] P.L. Antonelli and T. J. Zastawniak, Diffusions on Finsler manifolds, Rep. Math. Phys. 33 (1993) 303-315.

[Zas93] T. Zastawniak, Evaluation of the distribution of the random variable S(t)=\int_0^t (-1)^{N(u)}du, in: H.A.Cerdeira, et.al., Lectures on Path Integration (Trieste 1991), World Scientific Publishing, Singapore-New Jersey-London-Hong Kong 1993, pp.446-448 (Appendix in S.K.Foong, Path integral solution for telegrapher equation).

[PliZas94] A. Plis and T. Zastawniak, Relativistic quantum mechanics for two interacting particles in one-time representation, Rep. Math. Phys.31 (1992) 317-327.

[AntZas94a] P.L. Antonelli and T. J. Zastawniak, Stochastic calculus on Finsler manifolds and an application in biology, Nonlinear World 1 (1994) 149-171.

[AntZas94b] P.L. Antonelli and T. J. Zastawniak, Introduction to diffusion on Finsler manifolds, Math. Comput. Modelling 20 (1994) 109-116.

[AntZas94c] P.L. Antonelli and T. J. Zastawniak, Density-dependent host/parasite systems of Rothschild type and Finslerian diffusion, Math. Comput. Modelling 20 (1994) 117-129.

[AntZas95a] P. Antonelli and T. Zastawniak, Curvature and production stability in Volterra-Hamilton systems of Finsler type, Open Systems and Information Dynamics 3 (1995) 319-324.
see publisher's catalogue entry

[AntZas95b] P.L. Antonelli and T.J. Zastawniak, Diffusion on the tangent and indicatrix bundles of a Finsler manifold, Tensor (N.S.) 56 (1995) 233-247.

[GliZas95] Yu.E. Gliklikh and T.J. Zastawniak, Solution of the Cauchy problem for the Nelson-Newton equation, in: Novoe Global. Anal., Vol.15, Izdat. Voronezh. Univ., Voronezh 1995, pp.41-50.

[AntZas96] P. Antonelli and T. Zastawniak, Stochastic Finsler geometry with biological applications, in: D.Bainov (ed.), Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, August 1995), Science Culture Technology Publishing, Sofia 1996, pp.15-20.

[WilZas96] R. Wiltshire and T.Zastawniak, Approximate Lie symmetries for diffusion equations with lon-linear perturbations, in: D.Bainov (ed.), Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, 18-23 August 1995), VSP Publishers, The Netherlands 1996, pp.361-367. ISBN 90-6764-203-7.

[AntMatZas96] P.L. Antonelli, M. Matsumoto and T.J.Zastawniak, On y-Berwald spaces of dimension two and associated heterochronic systems with Finslerian noise, Contemp. Math. 196 (1996) 203-212.

[GliZas96] Y.E. Gliklikh and T.J. Zastawniak, Solution of the Nelson-Newton equation with random initial data, in: Y.E.Gliklikh, Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics, Mathematics and its Applications, Vol.374, Kluwer Academic Publishers, Dordrecht 1996, pp.166-174.
see publisher's catalogue entry

[AntZas97a] P. Antonelli and T. Zastawniak, Noise induced transitions in a stochastic Volterra-Hamilton open system, Open Systems and Information Dynamics 4 (1997) 89-100.
see publisher's catalogue entry

[Zas97a] T. Zastawniak, Geometrical stochastic control and quantization, J. Math. Phys. 38 (1997) 173-181.

[AntZas97b] P.L. Antonelli and T.J. Zastawniak, Stochastic Finsler geometry in the theory of evolution by symbiosis, Dynamics of Continuous, Discrete and Impulsive Systems 3 (1997) 1-18.

[Zas97b] T. Zastawniak, Fresnel type path integrals for the stochastic Schroedinger equation, Lett. Math. Phys. 41 (1997) 93-99.

[GliZas98] Yu.E. Gliklikh and T. Zastawniak, Existence theorems in Nelson's stochastic mechanics on Riemannian manifolds and space-times of general relativity, in: Th.M.Rassias (ed.), Nonlinear Mathematical Analysis and Applications, Hadronic Press, Palm Harbor, Florida, 1998, pp.39-58.

[AntZas98] P.L. Antonelli and T. Zastawniak, Diffusion, Laplacian and Hodge decomposition on Finsler spaces, in: P.L.Antonelli and B.Lackey (eds.), The Theory of Finslerian Laplacians and Applications, Mathematics and Its Applications, Vol.459, Kluwer Academic Publishers, Dordrecht-Boston-London, 1998, pp.141-149.
see publisher's catalogue entry

[TruZas99] A. Truman and T. Zastawniak, Stochastic PDE's of Schroedinger type and stochastic Mehler kernels. A path integral approach, in: R.Dalang, M.Dozzi, and F.Russo (eds.), Seminar on Stochastic Analysis, Random Fields and Applications, Centro Stefano Franscini, Ascona, Italy, September 1996, Birkhauser Prog. Prob., Vol.45, Birkhauser Verlag, Basel, Switzerland, 1999, pp.275-283

[TruZas00] A. Truman and T. Zastawniak, Stochastic Mehler kernels via oscillatory path integrals, J. Korean Math. Soc., 38 (2001) 469-483.

[AntZas03] P.L. Antoneli and T.J. Zastawniak, Fundamentals of Finslerian diffusion, in: P.L. Antonelli (ed.), Handbook of Finsler Geometry, Vol.1, Part 3, Kluwer Academic Publishers, Dordrecht-Boston-London, 2003, pp.177-355.

[SloZas04] W. Slomczynski and T. Zastawniak, Utility maximising entropy and the second law of thermodynamics, The Annals of Probability 32 (2004) 2261-2285.
download paper
go to official journal site

[TokZas04] K. Tokarz and T. Zastawniak, Dynamic programming algorithms for the ask and bid prices of American options under small proportional transaction costs, preprint, 2004.
download paper from this site or from http://ssrn.com/abstract=581543

[Zas04] T. Zastawniak, American contingent claims with physical delivery under small proportional
transaction costs, preprint, 2004.
download paper from this site or from http://ssrn.com/abstract=595941

[TokZas05] K. Tokarz and T. Zastawniak, American contingent under small proportional
transaction costs, Journal of Mathematical Economics 43 (2006) 65-85.
http://dx.doi.org/10.1016/j.jmateco.2006.09.003
download preprint from this site

[RouZas05] A. Roux and T. Zastawniak, A counter-example to an option pricing formula under transaction costs, Finance and Stochastics 10 (2006) 575-578.
http://dx.doi.org/10.1007/s00780-006-0016-2
download preprint from this site or an earlier version from http://ssrn.com/abstract=665283

[TokRouZas06] A. Roux, K. Tokarz and T. Zastawniak, European options under proportional transaction costs: An algorithmic approach to pricing and hedging, preprint, 2006.
download paper from this site or from http://ssrn.com/abstract=897965

[RouZas06] A. Roux and T. Zastawniak, American options under proportional transaction costs: Seller's price algorithm, hedging strategy and optimal stopping, preprint, 2006.
download paper from this site or from http://ssrn.com/abstract=906337

Teaching

A selection of courses taught/designed in recent years:

Stochastic Analysis

Applied Probability

Decision Making for Business and Management

Portfolio Theory and Risk Management (MSc level)

Mathematical Methods of Finance (MSc level)

Mathematical Finance: Investments

Editorships

	Deputy Editor of the English Edition, Izvestiya Mathematics, London Mathematical Society, Turpion Ltd, and the Russian Academy of Sciences
	Member of the Editorial Board, Open Systems and Information Dynamics, Springer Netherlands
	Invited Editor, Mathematical and Computer Modelling, Vol. 20, Elsevier Science

Other interests

How long was the coast of Atlantis?

Did a pre-Homeric Greek artist try to represent a fractal? The blue shape in this 3500 years old wall painting from Akrotiri, Thera known as "Shipwreck" bears a striking resemblance to the fractal known as von Koch coastline, bringing to mind Mandelbrot's celebrated paper "How long is the coast of Great Britain?" The painting was preserved under a layer of ashes, following a volcanic eruption some 3500 years ago, which contributed to the destruction the Minoan civilisation. The island of Thera is believed by many archaeologists to have been no less than the mythical Atlantis...

To find out more about fractal shapes in Minoan art see:
W. Slomczynski and T. Zastawniak, How long was the coast of Atlantis?, Mathematical Intelligencer 21 (1999) 38-41.

Voting power theory in the EU Council

W. Słomczyński, T. Zastawniak and K. Życzkowski, The root of the matter: voting in the EU Council, to appear in Physics World, March 2006.
download paper from this site