This paper considers a class of continuous functions constructed as a series of iterates of the “tent map” multiplied by variable signs. This class includes Takagi’s nowhere-differentiable function, and contains the functions studied by Hata and Yamaguti [Japan J. Appl. Math., 1 (1984), 183-199] and Kono [Acta Math. Hungar. 49 (1987), 315324] as a proper subclass. A complete description is give...

Motivated by a problematic coming from mathematical finance, the paper deals with existing and additional results on the continuity and the differentiability of the Itô map associated to rough differential equations. These regularity results together with the Malliavin calculus are applied to the sensitivities analysis of stochastic differential equations driven by multidimensional Gaussian pro...

This paper describes the use of a modified Differential Evolution strategy that identifies multiple solutions to the numerical optimization of multidimensional objective functions. Traditional approaches to this class of problems, such as Newton’s Method, are restricted for use on continuous, differentiable functions; in addition, the solution identified by these approaches i often dependent up...

Motivated by extremal problems of weighted Dirichlet or Neumann eigenvalues, we will establish two fundamental results on the dependence of weighted eigenvalues of the one-dimensional p-Laplacian on indefinite integrable weights. One is the continuous differentiability of eigenvalues in weights in the Lebesgue spaces L with the usual norms. Another is the continuity of eigenvalues in weights wi...

This paper studies a regularization approach for an inverse problem of estimating a spatially-and-temporally dependent Robin coefficient arising in the analysis of convective heat transfer. The parameter-to-state map is analyzed, especially a differentiability result is established. A regularization approach is proposed, and the properties, e.g., existence and optimality system, of the function...

In game theory, players have continuous expected payoff functions and can use fixed point theorems to locate equilibria. This optimization method requires that players adopt a particular type of probability measure space. Here, we introduce alternate probability measure spaces altering the dimensionality, continuity, and differentiability properties of what are now the game’s expected payoff fu...

The convex envelope of a given function was recently characterized as the solution of a fully nonlinear partial differential equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main result is an optimal regularity result. Differentiability (C1,α regularity) of the boundary data implies the corresponding result for the solution in the in...

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz co...

We aim to construct a general framework for portfolio management in continuous time, encompassing both stocks and bonds. In these lecture notes we give an overview of the state of the art of optimal bond portfolios and we re-visit main results and mathematical constructions introduced in our previous publications (Ann. Appl. Probab. 15, 1260–1305 (2005) and Fin. Stoch. 9, 429–452 (2005)). A sol...

We consider superlinearly convergent analogues of Newton methods for nondifferentiable operator equations in function spaces. The superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally Lipschitzian operator in Rn is based on Rademacher’s theorem which does not hold in function spaces. We introduce a concept of slant differentiability and us...