Abstract
In this second part of the paper, through applying semigroup theory procedures, we study initial boundary problems associated with degenerate second-order differential operators of the form Lu(x) ≔ α(x) u″(x)+β(x)u′(x)+γ(x) u(x) in the framework of weighted continuous function spaces on an arbitrary real interval, when particular boundary conditions are imposed. By using the general results stated in the first part, we show that such operators, frequently occurring in Mathematical Finance, generate positive strongly continuous semigroups, which are, in turn, the transition semigroups associated with suitable Markov processes. Finally, an application to the Black-Scholes equation is discussed, as well.
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References
F. Altomare and A. Attalienti, Degenerate evolution equations in weighted continuous function spaces, Markov processes and the Black-Schoies equation-Part I, preprint, Bari University, 2001.
F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applications, Vol.17, de Gruyter Series Studies in Mathematics, de Gruyter, Berlin-New York, 1994.
F. Altomare and I. Carbone, On some Degenerate Differential Operators on Weighted Function Spaces, J. Math. Anal. & Appl. 213 (1997), 308–333.
F. Altomare and E. M. Mangino, On a class of elliptic-parabolic equations on unbounded intervals, Positivity 5 (2001), 239–257.
A. Attalienti and M. Campiti, On the generation of C0-semigroups in L1(I), preprint, Bari University, 1998.
J. M. Bony, P. Courrège and P. Priouret, Semi-groupes de Feller sur une variété a bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum, Ann. Inst. Fourier, Grenoble 18, 2 (1968), 369–521.
M. Campiti, G. Metafune and D. Pallara, One-dimensional Feller semigroups with reflecting barriers, J. Math. Anal. & Appl. 244 (2000), 233–250.
Ph. Clément and C. A. Timmermans, On C0-semigroups generated by differential operators satisfying Ventcel’s boundary conditions, Indag. Math. 89 (1986), 379–387.
K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, Berlin, 2000.
W. Feller, The parabolic differential equations and the associated semi-groups of transformation, Ann. of Math. 55 (1952), 468–519.
Y. K. Kwok, Mathematical Models of Financial Derivatives, Springer-Verlag, Berlin, 1998.
J. Lamperti, Stochastic Processes, Applied Mathematical Sciences, n. 23, Springer-Verlag, Berlin, 1977.
E. M. Mangino, Differential operators with second order degeneracy and positive approximation processes, to appear in Constr. Approx. 2002.
E. M. Mangino, A positive approximation sequence related to Black and Scholes equation, to appear in Suppl. Rend. Circ. Mat. Palermo 2002.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
R. Sato and T. Ueno, Multi-dimensional diffusion and the Markov process on the boundary, J. Math. Kyoto University 4 (1965), 529–605.
K. Taira, Diffusion Processes and Partial Differential Equations, Academic Press, San Diego, CA, 1988.
C. A. Timmermans, On C0-semigroups in a space of bounded continuous functions in the case of entrance or natural boundary points in: Approximation and Optimization, J.A. Gómez Fernández et al. (eds), Lecture Notes in Math. 1354, Springer-Verlag, Berlin-New York, 1988, 209–216.
D. Williams, Diffusions, Markov Processes and Martingales, Vol. 1, John Wiley & Sons, New York, 1979.
P. Wilmott, S. Howison and J. Dewynne, The Mathematics of Financial Derivatives, Cambridge University Press, 1995.
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Altomare, F., Attalienti, A. Degenerate Evolution Equations in Weighted Continuous Function Spaces, Markov Processes and the Black-Scholes Equation-Part II. Results. Math. 42, 212–228 (2002). https://doi.org/10.1007/BF03322851
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DOI: https://doi.org/10.1007/BF03322851
Key words and phrases
- Positive semigroup
- Markov Transition Function
- Markov Process
- Differential Operator
- Weighted Function Space
- Pricing Theory