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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).

35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
91B28 Finance etc. (MSC2000)
Full Text: DOI
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