Ray methods for free boundary problems
Authors:
J. A. Addison, S. D. Howison and J. R. King
Journal:
Quart. Appl. Math. 64 (2006), 41-59
MSC (2000):
Primary 35K60, 35K65, 80M35, 41A60, 41A63
DOI:
https://doi.org/10.1090/S0033-569X-06-00993-4
Published electronically:
January 24, 2006
MathSciNet review:
2211377
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We discuss the use of the WKB ansatz in a variety of parabolic problems involving a small parameter. We analyse the Stefan problem for small latent heat, the Black–Scholes problem for an American put option, and some nonlinear diffusion equations, in each case constructing an asymptotic solution by the use of ray methods.
- D. G. Aronson, O. Gil, and J. L. Vázquez, Limit behaviour of focusing solutions to nonlinear diffusions, Comm. Partial Differential Equations 23 (1998), no. 1-2, 307–332. MR 1608532, DOI https://doi.org/10.1080/03605309808821347
- G. I. Barenblatt, On self-similar motions of a compressible fluid in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh. 16 (1952), 679–698 (Russian). MR 0052948
chadam Chen, X., and Chadam, J. (2004). A mathematical analysis for the optimal exercise boundary of an American put option. Preprint.
chevalier Chevalier, E. (2004). Free boundary near the maturity of an American option on several assets. Working paper.
- John Crank, Free and moving boundary problems, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1984. MR 776227
- J. N. Dewynne, S. D. Howison, J. R. Ockendon, and Wei Qing Xie, Asymptotic behavior of solutions to the Stefan problem with a kinetic condition at the free boundary, J. Austral. Math. Soc. Ser. B 31 (1989), no. 1, 81–96. MR 1002093, DOI https://doi.org/10.1017/S0334270000006494
- C. M. Elliott, M. A. Herrero, J. R. King, and J. R. Ockendon, The mesa problem: diffusion patterns for $u_t=\nabla \cdot (u^m\nabla u)$ as $m\to +\infty $, IMA J. Appl. Math. 37 (1986), no. 2, 147–154. MR 983523, DOI https://doi.org/10.1093/imamat/37.2.147
- J. D. Evans, R. Kuske, and Joseph B. Keller, American options of assets with dividends near expiry, Math. Finance 12 (2002), no. 3, 219–237. MR 1910594, DOI https://doi.org/10.1111/1467-9965.02008
green Green, G. (1837). On the motion of waves in a variable canal of small depth and width. Trans. Camb. Phil. Soc. 6.
grinberg Grinberg, G.A., and Chekmareva, O.M. (1971). Motion of phase interface in Stefan problems. Sov. Phys. Tech. Phys. 15 1579–1583.
- Morton E. Gurtin, Thermodynamics and the supercritical Stefan equations with nucleations, Quart. Appl. Math. 52 (1994), no. 1, 133–155. MR 1262324, DOI https://doi.org/10.1090/qam/1262324
- William L. Kath and Donald S. Cohen, Waiting-time behavior in a nonlinear diffusion equation, Stud. Appl. Math. 67 (1982), no. 2, 79–105. MR 670736, DOI https://doi.org/10.1002/sapm198267279
- J. R. King, Multidimensional singular diffusion, J. Engrg. Math. 27 (1993), no. 4, 357–387. MR 1244218, DOI https://doi.org/10.1007/BF00128761
- J. R. King, Exact multidimensional solutions to some nonlinear diffusion equations, Quart. J. Mech. Appl. Math. 46 (1993), no. 3, 419–436. MR 1233988, DOI https://doi.org/10.1093/qjmam/46.3.419
kingnew King, J.R., Koerber, A.J., Croft, J.M., Ward, J.P., Williams, P., and Sockett, R.E. (2003). Modelling host tissue degradation by extracellular bacterial pathogens. Math. Med. Biol. 20, 227–260.
- J. R. King, D. S. Riley, and A. M. Wallman, Two-dimensional solidification in a corner, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1989, 3449–3470. MR 1807692, DOI https://doi.org/10.1098/rspa.1999.0460
- Charles Knessl, Asymptotic analysis of the American call option with dividends, European J. Appl. Math. 13 (2002), no. 6, 587–616. MR 1949725, DOI https://doi.org/10.1017/S0956792502004898
keller Kuske, R., and Keller, J.B. (1998). Optimal exercise boundary for an American put option. Applied Mathematical Finance 5 107–116.
- A. A. Lacey and J. R. Ockendon, Ill-posed free boundary problems, Control Cybernet. 14 (1985), no. 1-3, 275–296 (1986) (English, with Russian and Polish summaries). MR 839524
lame Lamé, G., and Clapeyron, B.P. (1831). Mémoir sur la solidification par refroidissement d’un globe liquide. Ann. Chem. Phys. 47 250–256.
liouville Liouville, J. (1837). Sur le développement des fonctions ou parties de fonctions en séries. J. Math. Pures Appl. 2 (1837) 16–35.
- B. Sherman, Limiting behavior in some Stefan problems as the latent heat goes to zero, SIAM J. Appl. Math. 20 (1971), 319–327. MR 293259, DOI https://doi.org/10.1137/0120034
- A. M. Soward, A unified approach to Stefan’s problem for spheres and cylinders, Proc. Roy. Soc. London Ser. A 373 (1980), no. 1752, 131–147. MR 592750, DOI https://doi.org/10.1098/rspa.1980.0140
stewartson Stewartson, K., and Waechter, T.T. (1976). On Stefan’s problem for spheres. Proc. R. Soc. Lond. A348 415–426.
- Paul Wilmott, Sam Howison, and Jeff Dewynne, The mathematics of financial derivatives, Cambridge University Press, Cambridge, 1995. A student introduction. MR 1357666
aronson Aronson, D.G., Gil, O., and Vazquez, J.–L. (1998). Limit behavior of focusing solutions to nonlinear diffusions. Comm. Partial Differential Equations 23 307–332.
barenblatt Barenblatt, G.I. (1952). On self-similar motions of a compressible fluid in a porous medium. (Russian) Akad. Nauk SSSR. Prikl. Mat. Meh. 16 419–436.
chadam Chen, X., and Chadam, J. (2004). A mathematical analysis for the optimal exercise boundary of an American put option. Preprint.
chevalier Chevalier, E. (2004). Free boundary near the maturity of an American option on several assets. Working paper.
crank Crank, J. (1984). Free and Moving Boundary Problems. Oxford University Press.
dewynne Dewynne, J.N., Howison, S.D., Ockendon, J.R., and Xie, W. (1989). Asymptotic behaviour of solutions to the stefan problem with a kinetic condition at the free boundary. J. Austral. Math. Soc. Ser. B 31 81–96.
elliott Elliott, C.M., Herrero, M.A., King, J.R., and Ockendon, J.R. (1986). The mesa problem: diffusion patterns for $u_t=\boldsymbol {\nabla } \cdot (u^m\boldsymbol {\nabla }u)$ as $m\rightarrow +\infty$. IMA J. Appl. Math. 37 147–154.
evans Evans, J.D., Keller, J.B, and Kuske, R. (2002). American options with dividends near expiry. Mathematical Finance 12 219–237.
green Green, G. (1837). On the motion of waves in a variable canal of small depth and width. Trans. Camb. Phil. Soc. 6.
grinberg Grinberg, G.A., and Chekmareva, O.M. (1971). Motion of phase interface in Stefan problems. Sov. Phys. Tech. Phys. 15 1579–1583.
gurtin Gurtin, M.E. (1994). Thermodynamics and the supercritical Stefan equations with nucleation. Quart. Appl. Math. 52, 133–155.
kath Kath, W.L., and Cohen, D.S. (1982). Waiting-time behaviour in a nonlinear diffusion equation. Stud. Appl. Math. 67 79–105.
king1 King, J.R. (1993). Multidimensional singular diffusion. J. Eng. Math. 27 357–387.
king2 King, J.R. (1993). Exact multidimensional solutions to some nonlinear diffusion equations. Quart. J. Mech. Appl. Math. 46 419–436.
kingnew King, J.R., Koerber, A.J., Croft, J.M., Ward, J.P., Williams, P., and Sockett, R.E. (2003). Modelling host tissue degradation by extracellular bacterial pathogens. Math. Med. Biol. 20, 227–260.
kingetal King, J.R., Riley, D.S., and Wallman, A.M. (1999). Two-dimensional solidification in a corner. Proc. R. Soc. Lond. A455 3449–3470.
knessl Knessl, C. (2002). Asymptotic analysis of the American call option with dividends. Europ. J. Appl. Math. 13 587–616.
keller Kuske, R., and Keller, J.B. (1998). Optimal exercise boundary for an American put option. Applied Mathematical Finance 5 107–116.
lacey Lacey, A.A., and Ockendon, J.R. (1985). Ill-posed free boundary problems. Control and Cybernetics 14 275–296.
lame Lamé, G., and Clapeyron, B.P. (1831). Mémoir sur la solidification par refroidissement d’un globe liquide. Ann. Chem. Phys. 47 250–256.
liouville Liouville, J. (1837). Sur le développement des fonctions ou parties de fonctions en séries. J. Math. Pures Appl. 2 (1837) 16–35.
sherman Sherman, B. (1971). Limiting behavior in some Stefan problems as the latent heat goes to zero. SIAM J. Appl. Math. 20 319–327.
soward Soward, A.M. (1980). A unified approach to Stefan’s problem for spheres and cylinders. Proc. R. Soc. Lond. A373 131–147.
stewartson Stewartson, K., and Waechter, T.T. (1976). On Stefan’s problem for spheres. Proc. R. Soc. Lond. A348 415–426.
wilmott Wilmott, P., Howison, S.D., and Dewynne, J.N. (1995). The Mathematics of Financial Derivatives. Cambridge University Press.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2000):
35K60,
35K65,
80M35,
41A60,
41A63
Retrieve articles in all journals
with MSC (2000):
35K60,
35K65,
80M35,
41A60,
41A63
Additional Information
J. A. Addison
Affiliation:
Mathematical Institute, 24–29 St. Giles’, Oxford, OX1 3LB, U.K.
S. D. Howison
Affiliation:
Mathematical Institute, 24–29 St. Giles’, Oxford, OX1 3LB, U.K.
J. R. King
Affiliation:
Theoretical Mechanics Section, University Park, Nottingham, NG7 2RD, U.K.
Received by editor(s):
January 1, 2005
Published electronically:
January 24, 2006
Additional Notes:
J. A. Addison and J. R. King gratefully acknowledge financial support from the Engineering and Physical Sciences Research Council (EPSRC)
Article copyright:
© Copyright 2006
Brown University
