# zbMATH — the first resource for mathematics

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)
##### Keywords:
unbounded initial function; optimal portfolio
Full Text:
##### References:
  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)  Bernard, G., Existence theorems for fast diffusion equations, Nonlinear analysis, 43, 575-590, (2001) · Zbl 0963.35090  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)  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  Krylov, N.V., Nonlinear elliptic and parabolic equations, (1987), D. Reidel Dordrecht, Holland · Zbl 0697.35043  Lieberman, G.M., Second order parabolic differential equations, (1996), World Scientific Singapore, New Jersey, London, Hong Kong · Zbl 0884.35001  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  Wang, G., The first boundary value problem for parabolic monge – ampère equation, Northeast. math. J., 3, 463-478, (1987) · Zbl 0694.35100  Yong, J., Introduction to mathematical finance, (), 19-137
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.