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| Financial models were
generally formulated in terms of stochastic differential equations.
However, it was soon found that under certain restrictions these
models could be written as linear evolutionary partial differemtial
equation with variable coefficients. The Black and Scholes equation
is an evolution equation dealing with random phenomena in the
specific context of finance.
The class of
Hamilton-Jacobi-Bellman equations arises in the sphere of stochastic
control theory.
Conservation laws are useful in both general theory and the analysis
of concrete systems. They are usually relevant for their physical
interpretation and they are also useful to determine potential
symmetries. S. Anco and G. Bluman gave a general treatment of a
direct conservation law method for partial differential equations
expressed in a standard Cauchy-Kovaleskaya form.
One of the present authors has introduced the concept of weak self-adjoint equations. This definition
generalizes4 the concept of self-adjoint and quasi self-adjoint equations that were introduced by
N.H.
Ibragimov. We have found 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 by N.H. Ibragimov, the Lie symmetries of
the Hamilton-Jacobi-Bellman equations derived by V. Naicker, K. Andriopulos and P.G.L. Leach and the concept
of weak self-adjointness we find conservation laws for some of these
partial differential equations. |
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