Abstract
Applied mathematics has a history going back several thousands of years at least to the time of the Babylonians. In a sense, (pure) mathematics evolved from applied mathematics. Over the centuries, applied mathematics became closely associated with mechanics and new developments were called applicable mathematics. This terminology has faded in recent decades due to the widespread use of mathematics in the solution of problems in any fields which can be treated quantitatively. One recent field is that of financial mathematics, and we illustrate a few of the problems which have been solved using the techniques of applied mathematics.
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Notes
- 1.
One would hope that this initial condition would not apply in financial matters! Unfortunately there are some instances of financial instability in which such an initial condition is far too accurate a model. Note that the paper [16] with more realistic conditions appeared earlier, but the content of [13] had already been presented at a seminar in the Department of Physics, The University of the Witwatersrand, in 1996.
- 2.
The solution symmetries, \(\varGamma _{\infty } \), can play no role in this as their action on \(u (T,x) = U \) produces a linear combination of linearly independent solutions.
References
K. Andriopoulos, S. Dimas, P.G.L. Leach, D. Tsoubelis, On the systematic approach to the classification of differential equations by group theoretical methods. J. Comput. Appl. Math. 230, 224–232 (2009). doi:10.1016/j.cam.2008.11.002
F. Black, M. Scholes, The valuation of option contracts and a test of market efficiency. J. Financ. 27, 399–417 (1972)
F. Black, M. Scholes, The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)
R. Brown, A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Philos. Mag. 4, 161–173 (1828)
K. Chan, A. Karolyi, F. Longstaff, A. Sanders, An empirical comparison of alternate models of the short-term interest rate. J. Financ. 47, 1209–1227 (1992)
J.C. Cox, J.E. Ingersoll, S.A. Ross, An intertemporal general equilibrium model of asset prices. Econometrica 53, 363–384 (1985)
S. Dimas, D. Tsoubelis, SYM: A New Symmetry-Finding Package for Mathematica, ed. by N.H. Ibragimov, C. Sophocleous, P.A. Damianou. Group Analysis of Differential Equations (University of Cyprus, Nicosia, 2005), pp. 64–70. http://www.math.upatras.gr/~spawn
S. Dimas, D. Tsoubelis, A New Mathematica-Based Program for Solving Overdetermined Systems of PDEs. 8th International Mathematica Symposium, Avignon, France, 2006
S. Dimas, Partial differential equations, algebraic computing and nonlinear systems, Thesis, University of Patras, Patras, Greece, 2008
S. Dimas, K. Andriopoulos, D. Tsoubelis, P.G.L. Leach, Complete specification of some partial differential equations that arise in financial mathematics. J. Nonlinear Math. Phys. (2009) (submitted)
U.L. Dothan, On the term structure of interest rates. J. Financ. Econ. 6, 59–69 (1978)
R.A. Fisher, The wave of advance of advantageous genes. Ann. Eugenics 7, 353–369 (1937)
R. Gasizov, N.H. Ibragimov, Lie symmetry analysis of differential equations in finance. Nonlinear Dyn. 17, 387–407 (1998)
J. Goard, New solutions to the bond-pricing equation via Lie’s classical method. Math. Comput. Model. 32, 299–313 (2000)
D. Heath, E. Platin, M. Schweizer, Numerical Comparison of Local Risk-minimisation and Mean-variance Hedging, ed. by E. Jouini, J. Cvitanić, M. Musiela. Option Pricing, Interest Rates and Risk Management (Cambridge University Press, Cambridge, 2001), pp. 509–537
N.H. Ibragimov, C. Wafo Soh, Solution of the Cauchy Problem for the Black-Scholes Equation Using Its Symmetries Modern Group Analysis, ed. by N.H. Ibragimov, K.R. Naqvi, E. Straume. International Conference at the Sophus Lie Conference Centre (MARS Publishers, Norway, 1997)
R. Jiwari, A. Verma, Analytic, Power series and numerical solutions of nonlinear diffusion equations via symmetry reductions, preprint school of mathematics and computer applications, Thapar University, Patiala—147004, India (2014)
R.L. Lemmer, P.G.L. Leach, A classical viewpoint on quantum chaos. Arab. J. Math. Sci. 5, 1–17 (1999)
R.C. Merton, On the pricing of corporate data: the risk structure of interest rates. J. Financ. 29, 449–470 (1974)
V.V. Morozov, Classification of six-dimensional nilpotent Lie algebras. Izv. Vyssh. Uchebn. Zavad. Mat. 5, 161–171 (1958)
G.M. Mubarakzyanov, On solvable Lie algebras. Izv. Vyssh. Uchebn. Zavad. Mat. 32, 114–123 (1963)
G.M. Mubarakzyanov, Classification of real structures of five-dimensional Lie algebras. Izv. Vyssh. Uchebn. Zavad. Mat. 34, 99–106 (1963)
G.M. Mubarakzyanov, Classification of solvable six-dimensional Lie algebras with one nilpotent base element. Izv. Vyssh. Uchebn. Zavad. Mat. 35, 104–116 (1963)
V. Naicker, K. Andriopoulos, P.G.L. Leach, Symmetry reductions of a Hamilton-Jacobi-Bellman equation arising in financial mathematics. J. Nonlinear Math. Phys. 12, 268–283 (2005)
C.A. Pooe, F.M. Mahomed, S.C. Wafo, Fundamental solutions for zero-coupon bond-pricing models. Nonlinear Dyn. 36, 69–76 (2004)
O.O. Vaneeva, R.O. Popovych, C. Sophocleous, Group Classification of the Fisher Equation with Time-dependent Coefficients, ed. by O.O. Vaneeva, C. Sophocleous, R.O. Popovych, P.G.L. Leach, V.M. Boyko, P.A. Damianou. Group Analysis of Differential Equations and Integrable Systems VI (University of Cyprus, Lefkosia, 2013), pp. 225–236
Acknowledgments
I thank Professor KM Tamizhmani and the Department of Mathematics for generous provision of facilities and support and furthermore thank the University of KwaZulu-Natal and the National Research Foundation of South Africa for their continued support. The opinions expressed in this paper should not be construed as being those of either institution.
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Leach, P. (2015). The Globalisation of Applied Mathematics. In: Sarkar, S., Basu, U., De, S. (eds) Applied Mathematics. Springer Proceedings in Mathematics & Statistics, vol 146. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2547-8_2
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