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...

We consider minimization problems of the form ${\min }_{u\in \varphi +W_0^{1,1}\left(\Omega \right)}{\int }_{\Omega }[f\left(Du\left(x\right)\right)-u\left(x\right)]\mathrm{d}x$ where $\Omega \subseteq {ℝ}^N$ is a bounded convex open set, and the Borel function $f:{ℝ}^N\to [0,+\infty ]$ is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of $\Omega$ and the zero level set of $f$, we prove that the viscosity solution of a related Hamilton–Jacobi equation provides a minimizer for the integral functional.

We consider minimization problems of the form ${\min }_{u\in \varphi +W_0^{1,1}\left(\Omega \right)}{\int }_{\Omega }[f\left(Du\left(x\right)\right)-u\left(x\right)]\mathrm{d}x$ where $\Omega \subseteq {ℝ}^N$ is a bounded convex open set, and the Borel function $f:{ℝ}^N\to [0,+\infty ]$ is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of Ω and the zero level set of f, we prove that the viscosity solution of a related Hamilton–Jacobi equation provides a minimizer for the integral functional.

Let $\Lambda$ be a Lagrangian submanifold of $T^{*}X$ for some closed manifold X. Let $S(x,\xi )$ be a generating function for $\Lambda$ which is quadratic at infinity, and let W(x) be the corresponding graph selector for $\Lambda ,$ in the sense of Chaperon-Sikorav-Viterbo, so that there exists a subset $X_0\subset X$ of measure zero such that W is Lipschitz continuous on X, smooth on $X\setminus X_0$ and $(x,\partial W/\partial x(x\left)\right)\in \Lambda$ for $X\setminus X_0.$ Let H(x,p)=0 for $(x,p)\in \Lambda$. Then W is a classical solution to $H(x,\partial W/\partial x(x\left)\right)=0$ on $X\setminus X_0$ and extends to a Lipschitz function on the whole of X. Viterbo refers to W as a variational...