Initial boundary value problem of an equation from mathematical finance

Abstract

Consider the initial boundary value problem of the strong degenerate parabolic equation $\begin{array}{*{20}c} {\partial _{xx} u + u\partial _y u - \partial _t u = f(x,y,t,u),} & {(x,y,t) \in Q_T = \Omega \times (0,T)} \\ \end{array}$ with a homogeneous boundary condition. By introducing a new kind of entropy solution, according to Oleinik rules, the partial boundary condition is given to assure the well-posedness of the problem. By the parabolic regularization method, the uniform estimate of the gradient is obtained, and by using Kolmogoroff’s theorem, the solvability of the equation is obtained in BV (Q _T) sense. The stability of the solutions is obtained by Kruzkov’s double variables method.

Keywords

Mathematical finance / Oleinik rules / Partial boundary condition / Entropy solution / Kruzkov’s double variables method

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