Narrow Results

We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang–Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric is almost calibrated. As an application, we introduce the tangent Lagrangian phase flow (TLPF) on the space of almost calibrated $(1,1)$-forms that fits into the GIT framework for dHYM metrics recently discovered by Collins–Yau. The TLPF has some special properties that are not seen for the line bundle mean curvature flow (i.e. the mirror of the Lagrangian mean curvature flow for graphs). We show that the TLPF starting from any initial data exists for all positive time. Moreover, we show that the TLPF converges smoothly to a dHYM metric assuming the existence of a $C$-subsolution, which gives a new proof for the existence of dHYM metrics in the highest branch.

Recently an averaging principle for diffusion processes with a null-recurrent fast component without a drift term was obtained in [4]. In this note this result is widened to allow a drift in the fast component. As a corollary a new result on the homogenization for parabolic PDE's is obtained.

Mesh qualities affect both the efficiency and accuracy for solving partial differential equations (PDEs). In this paper, we present a multiobjective mesh optimization algorithm, which improves the accuracy for solving PDEs. Our algorithm is designed to simultaneously improve more than two aspects of the mesh, while being able to successfully decrease errors for solving various PDEs. Numerical experiments show that our algorithm is able to significantly decrease errors compared with existing single objective mesh optimization algorithms.

In earlier works (Davis and Lleo, 2011b; Davis and Lleo, 2012), we showed that jump-diffusion risk-sensitive asset management problem without benchmark admit a unique classical (C^{1, 2}) solution. In this article we extend these solution techniques to a benchmarked asset management problem with jumps. Benchmarked asset management problems are highly relevant to the financial industry: most investment funds have a benchmark, such as a financial index or a customized portfolio, against which their performance is assessed. We show here under two different sets of assumptions that the stochastic control problem associated with the benchmarked aset management problem admits a unique C^{1, 2} solution and that the optimal investment strategy exists and is unique.