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Keywords:
Black–Scholes equation; volatility; controllability; observability; Carleman estimates
Black–Scholes equation; volatility; controllability; observability; Carleman estimates
Summary:
In this paper we discuss the exact null controllability of linear as well as nonlinear Black–Scholes equation when both the stock volatility and risk-free interest rate influence the stock price but they are not known with certainty while the control is distributed over a subdomain. The proof of the linear problem relies on a Carleman estimate and observability inequality for its own dual problem and that of the nonlinear one relies on the infinite dimensional Kakutani fixed point theorem with $L^2$ topology.
References:
[1] Adams R. A., Fournier J. F.: Sobolev Spaces. Second edition. Academic Press, New York 2003 MR 2424078 | Zbl 1098.46001
[2] Khodja F. Ammar, Benabdallah, A., Dupaix C.: Null controllability of some reaction diffusion systems with one control force. J. Math. Anal. Appl. 320 (2006), 928–943 MR 2226005
[3] Amster P., Averbuj C. G., Mariani M. C.: Solutions to a stationary nonlinear Black–Scholes type equation. J. Math. Anal. Appl. 276 (2002), 231–238 MR 1944348 | Zbl 1027.60077
[4] Amster P., Averbuj C. G., Mariani M. C., Rial D.: A Black–Scholes option pricing model with transaction costs. J. Math. Anal. Appl. 303 (2005), 688–695 MR 2122570 | Zbl 1114.91044
[5] Anita S., Barbu V.: Null controllability of nonlinear convective heat equation. ESAIM: Control, Optimization and Calculus of Variations 5 (2000), 157–173 MR 1744610
[6] Barbu V.: Controllability of parabolic and Navier–Stokes equations. Scientiae Mathematicae Japonicae 56 (2002), 143–211 MR 1911840 | Zbl 1010.93054
[7] Black F., Scholes M.: The pricing of options and corporate liabilities. J. Political Economics 81 (1973), 637–659 Zbl 1092.91524
[8] Bouchouev I., Isakov V.: The inverse problem of option pricing. Inverse Problems 13 (1997), L11–L17 MR 1474358 | Zbl 0894.90014
[9] Bouchouev I., Isakov V.: Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets. Inverse problems 15 (1999), R95–R116 MR 1696930 | Zbl 0938.35190
[10] Chae D., Imanivilov, O. Yu., Kim M. S.: Exact controllability for semilinear parabolic equations with Neumann boundary conditions. J. Dynamical Control Systems 2 (1996), 449–483 MR 1420354
[11] Doubova A., Fernandez-Cara E., Gonzalez-Burgos, M., Zuazua E.: On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41 (2002), 789–819 MR 1939871 | Zbl 1038.93041
[12] Egger H., Engl H. W.: Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates. Inverse Problems 21 (2005) 1027–1045 MR 2146819 | Zbl 1205.65194
[13] Fursikov A. V., Imanuvilov O. Yu.: Controllability of Evolution Equations. Lecture Notes Series 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul 1996 MR 1406566 | Zbl 0862.49004
[14] Hörmander L.: Linear Partial Differential Operators I – IV. Springer–Verlag, Berlin 1985
[15] Imanuvilov O. Yu.: Boundary controllability of parabolic equations. Sbornik Mathematics 187 (1995), 879–900 MR 1349016
[16] Imanuvilov O. Yu., Yamamoto M.: Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Publ. Res. Inst. Math. Sci. 39 (2003), 227–274 MR 1987865 | Zbl 1065.35079
[17] Ingber L., Wilson J. K.: Statistical mechanics of financial markets: Exponential modifications to Black–Scholes. Mathematical and Computer Modelling 31 (2000) 167–192 MR 1761486 | Zbl 1042.91524
[18] Jódar L., Sevilla-Peris P., Cortes J. C., Sala R.: A new direct method for solving the Black–Scholes equations. Appl. Math. Lett. 18 (2005), 29–32 MR 2121550
[19] Kangro R., Nicolaides R.: Far field boundary conditions for Black–Scholes equations. SIAM J. Numer. Anal. 38 (2000), 1357–1368 MR 1790037 | Zbl 0990.35013
[20] Sakthivel K., Balachandran, K., Sritharan S. S.: Controllability and observability theory of certain parabolic integro-differential equations. Comput. Math. Appl. 52 (2006), 1299–1316 MR 2307079
[21] Sakthivel K., Balachandran, K., Lavanya R.: Exact controllability of partial integrodifferential equations with mixed boundary conditions. J. Math. Anal. Appl. 325 (2007), 1257–1279 MR 2270082 | Zbl 1191.93013
[22] Sowrirajan R., Balachandran K.: Determination of a source term in a partial differential equation arising in finance. Appl. Anal. (to appear) MR 2536788 | Zbl 1170.35328
[23] Widdicks M., Duck P. W., Andricopoulos A. D., Newton D. P.: The Black–Scholes equation revisited: Asymptotic expansions and singular perturbations. Mathematical Finance 15 (2005), 373–371 MR 2132196 | Zbl 1124.91342
[24] Wilmott I., Howison, S., Dewynne J.: The Mathematics of Financial Derivatives. Cambridge University Press, Cambridge 1995 MR 1357666 | Zbl 0842.90008
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