Existence and uniqueness of solutions to a quasilinear parabolic equation with quadratic gradients in financial markets

Nonlinear Analysis, Theory, Methods and Applications

Volumn 62, Issue 3, 2005, Pages 519-544

  Düring, Bertram ^a   Jüngel, Ansgar ^a  

^a JOHANNES GUTENBERG UNIVERSITY   (Germany)

Author keywords

Existence and uniqueness of solutions; Incomplete financial markets; Quadratic gradient; Quasilinear equation

Indexed keywords

COMPUTATIONAL COMPLEXITY; FINANCE; FUNCTIONS; MATHEMATICAL MODELS; OPTIMIZATION; THEOREM PROVING;

EXISTENCE AND UNIQUENESS OF SOLUTIONS; INCOMPLETE FINANCIAL MARKETS; QUADRATIC GRADIENT; QUASILINEAR EQUATION;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 19844368333     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2005.03.068     Document Type: Article

References (33)

1. H.W. Alt, and S. Luckhaus Quasilinear elliptic-parabolic differential equations Math. Z. 183 1983 311 341
2. M. Avellaneda, A. Levy, and A. Parás Pricing and hedging derivative securities in markets with uncertain volatilities Appl. Math. Finance 2 1995 73 88
3. G. Barles, and F. Murat Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions Arch. Ration. Mech. Anal. 133 1 1995 77 101
4. G. Barles, and H. Soner Option pricing with transaction costs and a nonlinear Black-Scholes equation Finance Stoch. 2 1998 369 397
5. S. Benninga, and J. Mayshar Heterogeneity and option pricing Review Derivatives Res. 4 2000 7 27
6. A. Bensoussan, L. Boccardo, and F. Murat On a nonlinear partial differential equation having natural growth terms and unbounded solution Ann. Inst. H. Poincaré, Anal. non linéaire 5 4 1988 347 364
7. F. Black, and M. Scholes The pricing of options and corporate liabilities J. Political Economy 81 1973 637 654
8. L. Boccardo, T. Gallouët, F. Murat, A unified presentation of two existence results for problems with natural growth, in: M. Chipot (Ed.), Progress in Partial Differential Equations: The Metz Surveys 2, Pitman Res. Notes Math. Ser. 296 (1993) 127-137.
9. L. Boccardo, F. Murat, and J.-P. Puel Existence de solutions non bornées pour certaines équations quasi-linéaires Port. Math. 41 1982 507 534
10. L. Boccardo, F. Murat, J.-P. Puel, Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique, Nonlinear Partial Differential Equations and Their Applications, vol. 4, Res. Notes Math. 84 (1983) 19-73.
11. L. Boccardo, F. Murat, and J.-P. Puel Résultats d'existence pour certains problèmes elliptiques quasilinéaires Ann. Sc. Norm. Super. Pisa, IV Ser. 11 1984 213 235
12. L. Boccardo, F. Murat, and J.-P. Puel Existence of bounded solutions for non linear elliptic unilateral problems Ann. Mat. Pura Appl. IV Ser. 152 1988 183 196
13. L. Boccardo, F. Murat, and J.-P. Puel L ∞ estimate for some nonlinear elliptic partial differential equations and application to an existence result SIAM J. Math. Anal. 23 1992 326 333
14. Q. Chen A nonlinear parabolic system arising from the eddy currents problem Nonlinear Anal. 42 2000 759 770
15. L.C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics, Providence, RI, 1998.
16. V. Ferone, and M.R. Posteraro On a class of quasilinear elliptic equations with quadratic growth in the gradient Nonlinear Anal. 20 6 1993 703 711
17. V. Ferone, M.R. Posteraro, and J.M. Rakotoson L ∞ -estimates for nonlinear elliptic problems with p-growth in the gradient J. Inequal. Appl. 3 2 1999 109 125
18. J. Frehse, Existence and perturbation theorems for nonlinear elliptic systems, Nonlinear Partial Differential Equations and Their Applications, vol. 4, Res. Notes Math. 84 (1983) 87-110.
19. R. Frey Perfect option hedging for a large trader Finance Stoch. 2 1998 115 141
20. N. Grenon-Isselkou, and J. Mossino Existence de solutions bornées pour certaines équations elliptiques quasilinéaires C. R. Acad. Sci., Paris, Sér. I 321 1 1995 51 56
21. I. Karatzas, and S.E. Shreve Methods of Mathematical Finance 1999 Springer New York
22. M. Kobylanski Backward stochastic differential equations and partial differential equations with quadratic growth Ann. Probab. 28 2 2000 558 602
23. D. Kramkov, and W. Schachermayer The asymptotic elasticity of utility functions and optimal investment in incomplete markets Ann. Appl. Probab. 9 3 1999 904 950
24. J. Leitner Continuous time CAPM, price for risk and utility maximization M. Kohlmann Mathematical Finance. Workshop of the Mathematical Finance Research Project, Konstanz, Germany 2001 Birkhäuser Basel
25. C. Maderna, C.D. Pagani, and S. Salsa Quasilinear elliptic equations with quadratic growth in the gradient J. Diff. Equations 97 1 1992 54 70
26. R.C. Merton Lifetime portfolio selection under uncertainty: the continuous-time case Rev. Econom. Statist. 51 1969 247 257
27. R.C. Merton Optimum consumption and portfolio rules in a time-continuous world J. Econ. Theory 3 1971 373 413
28. M. Papi A generalized Osgood condition for viscosity solutions to fully nonlinear parabolic degenerate equations Adv. Differential Equations 7 9 2002 1125 1151
29. J.-M. Rakotoson Réarrangement relatif dans les équations elliptiques quasi-linéaires avec un second membre distribution: Application à un théorème d'existence et de régularité J. Diff. Equations 66 1987 391 419
30. J. Simon Compact sets in the space L p (0, T ; B) Ann. Mat. Pura Appl., IV. Ser. 146 1987 65 96
31. K. Sircar, and G. Papanicolaou General Black-Scholes models accounting for increased market volatility from hedging strategies Appl. Math. Finance 5 1998 45 82
32. X. Xu Existence and uniqueness for the nonstationary problem of the electrical heating of a conductor due to the Joule-Thomson Effect Int. J. Math. Math. Sci. 16 1 1993 125 138
33. E. Zeidler Nonlinear Functional Analysis and Its Applications II/A 1990 Springer New York