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We study the Dirichlet boundary value problem for eikonal type equations of ray
light propagation in an inhomogeneous medium with discontinuous
refraction index. We prove a comparison principle
that allows us to obtain existence and uniqueness of a continuous
viscosity solution when the Lie algebra generated by the coefficients satisfies a Hörmander
type condition. We require the refraction index to be piecewise continuous across Lipschitz hypersurfaces. The results characterize the value...
We study here the impulse control minimax problem. We allow the cost functionals and dynamics to be unbounded and hence the value functions can possibly be unbounded. We prove that the value function of the problem is continuous. Moreover, the value function is characterized as the unique viscosity solution of an Isaacs quasi-variational inequality. This problem is in relation with an application in mathematical finance.
We study a quasiconvex conjugation that transforms the level sum of functions into the pointwise sum of their conjugates and derive
new duality results for the minimization of the max of two quasiconvex functions. Following Barron and al., we show that the level
sum provides quasiconvex viscosity solutions for Hamilton-Jacobi equations in which the initial condition is a general continuous
quasiconvex function which is not necessarily Lipschitz or bounded.
We consider an illiquid financial market with different regimes modeled by a continuous time finite-state Markov chain. The investor can trade a stock only at the discrete arrival times of a Cox process with intensity depending on the market regime. Moreover, the risky asset price is subject to liquidity shocks, which change its rate of return and volatility, and induce jumps on its dynamics. In this setting, we study the problem of an economic agent optimizing her expected utility from consumption...
In this paper, we study one kind of stochastic recursive optimal control problem for the systems described by stochastic differential equations with delay (SDDE). In our framework, not only the dynamics of the systems but also the recursive utility depend on the past path segment of the state process in a general form. We give the dynamic programming principle for this kind of optimal control problems and show that the value function is the viscosity solution of the corresponding infinite dimensional...
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