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Some weak self-adjoint Hamilton-Jacobi-Bellman equations arising in financial mathematics. (English) Zbl 1238.35048
Summary: The first author [J. Phys. A, Math. Theor. 44, No. 26, Article ID 262001, 6 p. (2011; Zbl 1223.35203)] introduced the concept of weak self-adjoint equations. This definition generalizes the concept of self-adjoint and quasi self-adjoint equations that were introduced by N. H. Ibragimov [J. Math. Anal. Appl. 318, No. 2, 742–757 (2006; Zbl 1102.34002)]. In this paper we find a class of weak self-adjoint Hamilton-Jacobi-Bellman equations which are neither self-adjoint nor quasi self-adjoint. By using a general theorem on conservation laws proved in [N. H. Ibragimov, J. Math. Anal. Appl. 333, 329–346 (2007; Zbl 1117.83127)] and the new concept of weak self-adjointness we find conservation laws for some of these partial differential equations.
35K59Quasilinear parabolic equations
35Q91PDEs in connection with game theory, economics, social and behavioral sciences
91G80Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
35A30Geometric theory for PDE, characteristics, transformations
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