Searcharxiv⌕ Search

arXiv subjects

Kiyuob Jung

Publications and source records attributed to Kiyuob Jung.

8 recordsLinked to original sources
arXiv2026

Specular differentiation in normed vector spaces

This paper develops specular differentiation for mappings between real normed vector spaces, extending the classical notions of Gâteaux and Fréchet differentiability. These notions apply to a broad class of nonsmooth functions, including functions for which certain other generalized derivatives do not exist. We investigate their fundamental theoretical properties and establish a Quasi-Mean Value Theorem for real-valued functions. Finally, we prove that the specular Gâteaux derivative of a semi-convex functional, whenever it exists at an interior point of its effective domain, determines a global quadratic lower bound and belongs to its Fréchet subdifferential.

math.CA↗
arXiv2026

American condor contracts with a cash component: A variational inequality approach

We study a jointly exercised American condor contract whose payoff is a call condor payoff plus a positive cash amount. Its plateau and positive tails produce two inner exercise boundaries and two outer cash-exercise boundaries. We construct the unique bounded solution of the associated variational inequality, strong on each side of the plateau, and identify it with the optimal stopping value. We prove monotonicity and continuity of the four boundaries and determine their limits at expiry. Following Broadie and Detemple, we express the inner boundaries as cutoffs of the boundaries of uncapped contracts with the same cash components. We also prove call-side convexity and volatility monotonicity under a sufficient condition, solve the stationary problems explicitly and present numerical results.

math.AP↗
arXiv2026

Regular specular differentiation in Euclidean spaces

We study the regular specular derivative, a generalized derivative defined at every point where both one-sided derivatives exist and are finite. Geometrically, it is the slope of the mirror that reflects the left tangent ray into the right one. In one variable we derive computational formulas, prove inverse function and rotation rules, establish Quasi-Rolle's Theorem and the Quasi-Mean Value Theorem, and obtain a derivative-limit theorem, which shows that twice regularly specularly differentiable functions are continuously differentiable. We also prove both parts of the Fundamental Theorem of Calculus. In several variables we introduce specular gradients, directional derivatives, tangent hyperplanes, and normal vectors, show that a continuous specular gradient forces classical differentiability, and characterize when the specular tangent hyperplane is unique.

math.CA↗
arXiv2026

The specular ellipse method for scalar ordinary differential equations: exactness and accuracy up to fourth order

This paper introduces a family of one-step implicit methods for solving scalar ordinary differential equations. At each step, the update uses a scaled angular mean of two vector-field evaluations, and the positive scale may vary from step to step. The leading terms of the local truncation error can be expressed in terms of the derivative of the signed curvature of the scaled solution graph. We prove that the proposed method reproduces the exact solution at the mesh points when the solution graph has constant signed curvature under a fixed positive scaling and each implicit update is unique. When this special geometric condition is not satisfied, we establish second-order consistency and convergence for positive scale sequences satisfying suitable uniform conditions. Furthermore, third- and fourth-order consistency and convergence can be achieved by choosing the scale to cancel the relevant curvature terms in the local truncation error. Using only the given problem data, we classify when these improvements are possible and determine the corresponding scale choices. An example shows that the proposed fourth-order method can yield smaller errors than the classical fourth-order Runge--Kutta method at the same step size.

math.NA↗
arXiv2026

Specular differentiation in one dimension: a quasi-mean value theorem, regularity, and discontinuities

We develop a theory of specular differentiation on real intervals. The specular derivative is defined by averaging the angles associated with the forward and backward difference quotients and extends classical differentiation. For specularly differentiable and continuous functions, we establish a quasi-mean value theorem and use it to show that continuity of the specular derivative implies $C^1$-regularity. Without assuming continuity, we prove that the discontinuities at which the specular derivative is nonzero form an at most countable set; a nowhere-continuous example with identically zero specular derivative shows that this restriction is sharp. In the second-order theory of specular differentiation, we impose additional conditions on twice specularly differentiable functions to obtain a class that lies strictly between the classes $C^1$ and $C^2$.

math.NA↗
arXiv2026

Specular gradient methods for nonsmooth convex optimization in Euclidean spaces: a subgradient selection strategy

This paper deals with nonsmooth convex optimization problems in Euclidean spaces. We identify special elements of the subdifferential of a convex function, called specular gradients. Based on this observation, we propose three numerical methods that use specular gradients in subgradient methods. We prove the convergence of the proposed methods under suitable step sizes. Numerical experiments demonstrate that the proposed methods are capable of minimizing non-differentiable functions that classical methods fail to minimize.

math.OC↗
arXiv2022

The wave equation with specular derivatives

In this paper, we construct the transport equation and the wave equation with specular derivatives and solve these equations in one-dimension. To solve these equations, we introduce new function spaces, which we term specular spaces, consisting of certain specularly differentiable functions.

math.AP↗