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Existence of solutions to initial value problem for a parabolic Monge-Ampère equation and application. (English) Zbl 1100.35047
The author deals with the following initial value problem for a parabolic Monge-Ampère equation \[ \begin{cases} V_t\cdot V_{xx}+r\cdot x \cdot V_x\cdot V_{xx}-\theta V_x^2=0, &(x,t)\in\mathbb{R}^1\times[0,T)\\ V(x,T)=g(x), &x\in\mathbb{R}^1\\ V_{xx}<0,\quad g'(x)>0,\end{cases}\tag{1} \] where \(V=V(x,t)\) is the unknown function, \(\theta>0\) and \(r\) are given constants. In a typical case, \(g(x)=1-e^{-\lambda x}\) with \(\lambda>0\) is a positive constant. In order that the solution of (1) can be used to the corresponding problem of optimal investment, the author assumes that \(V(x,t)\) is smooth and \(\frac{V_x(x,t)}{V_{xx} (x,t)}\) is Lipschitz continuous with respect to \(x\). The author proves existence of a solution for (1).

MSC:
35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
91B28 Finance etc. (MSC2000)
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[1] Bénilan, P.; Crandall, M.G.; Pierre, M., Solutions of the porous medium equation in \(R^n\) under optimal conditions on initial values, Indiana univ. math. J., 30, 162-177, (1981)
[2] Bernard, G., Existence theorems for fast diffusion equations, Nonlinear analysis, 43, 575-590, (2001) · Zbl 0963.35090
[3] Ivochkina, N.M.; Ladyzhenskaja, O.A., On the parabolic equations generated by symmetric functions of the principal curvatures of the evolution surface, or of the eigenvalues of the Hessian, part imonge – ampère equations, Saint |St. Petersburg math. J., 6, 375-394, (1995)
[4] Krylov, N.V., Boundedly inhomogeneous elliptic and parabolic equations, Izv. akad. nauk SSSR ser. mat., 46, 485-523, (1982), (English transl. in Math. USSR Izv. 20 (1983) · Zbl 0511.35002
[5] Krylov, N.V., Nonlinear elliptic and parabolic equations, (1987), D. Reidel Dordrecht, Holland · Zbl 0697.35043
[6] Lieberman, G.M., Second order parabolic differential equations, (1996), World Scientific Singapore, New Jersey, London, Hong Kong · Zbl 0884.35001
[7] Vázquez, J.L., Nonexistence of solutions for nonlinear heat equations of fast diffusion type, J. math. pure appl., 71, 503-526, (1992) · Zbl 0694.35088
[8] Wang, G., The first boundary value problem for parabolic monge – ampère equation, Northeast. math. J., 3, 463-478, (1987) · Zbl 0694.35100
[9] Yong, J., Introduction to mathematical finance, (), 19-137
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