Proceedings of the Singapore National Academy of Science

It is well known that the derivative of the primitive of 1-dimensional Henstock integral exists almost everywhere. Point-interval pairs used in the derivative are Henstock point-interval pairs, which are consistent with point-interval pairs used in the Henstock integral. Note that “almost everywhere” is a set of points, more precisely, the derivative does not exist on a set of points with measure zero. We can transform a set of Henstock point-interval pairs to a set of points with measure zero because of Vitali’s covering theorem. For 1-dimensional McShane integrals, $n$-dimensional McShane and Henstock integrals, covering theorems of Vitali’s type cannot be applied. In this paper, we shall discuss differentiation of $n$-dimensional McShane and Henstock integrals.

In T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016), the topology of ${ℝ}^{\infty }$ was replaced with a new topology and denoted by ${ℝ}_I^{\infty }$. This space was then used to construct Lebesgue measure on ${ℝ}_I^{\infty }$ in a manner that is no more difficult than the same construction on ${ℝ}^n$. More important for us, a new class of separable Banach spaces ${\mathrm{\text{KS}}}^p[{ℝ}^n]$, $1\le p\le \infty$, for the HK-integrable functions, was introduced. These spaces also contain the $L^p$ spaces and the Schwartz space as continuous dense embeddings. This paper extends the work in T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016) from ${\mathrm{\text{KS}}}^p[{ℝ}^n]$ to ${\mathrm{\text{KS}}}^p[{ℝ}_I^{\infty }]$.

In V. Boonpogkrong, On measurability of $C[0,1]$, Thai J. Math. (accepted) we have proved that $C[0,1]$ is measurable in a setting of a dyadic Henstock integral, where dimension sets are subsets of the dyadic rational. In this paper, we replace “dyadic rational” by “countable dense set” and use a setting of Henstock integral, where dimension sets are subsets of this fixed countable dense set. The collection of measurable sets in this new setting is smaller than that in the old setting. We still can prove that $C[0,1]$ is measurable.

In this paper, we formulate a version of convergence theorem using double Lusin condition and a version of Vitali convergence theorem for the Itô–Henstock integral of an operator-valued stochastic process with respect to a $Q$-Wiener process.

We prove that the controlled continuity, induced by controlled convergence, and continuity in norm are equivalent for linear operators defined on the space of Henstock integrable vector-valued functions.

The classical pricing theory requires that the simple sets of outcomes are extended, using the Kolmogorov Extension Theorem, to a sigma-algebra of measurable sets in an infinite-dimensional sample space whose elements are continuous paths; the process involved are represented by appropriate stochastic differential equations (using Itô calculus); a suitable measure for the sample space can be found by means of the Girsanov and Radon–Nikodym Theorems; the derivative asset valuation is determined by means of an expression using Lebesgue integration. It is known that if we replace Lebesgue’s by the generalized Riemann integration to obtain the expectation, the same result can be achieved by elementary methods. In this paper, we consider the Black–Scholes PDE subject to impulse action. We replace the process which follows a geometric Brownian motion by a process which has additional impulsive displacements at random times. Instead of constants, the volatility and the risk-free interest rate are considered as continuous functions which can vary in time. Using the Feynman–Ka$ă$ formulation based on generalized Riemann integration, we obtain a pricing formula for a European call option which copes with many discontinuities. This paper seeks to develop techniques of mathematical analysis in derivative pricing theory which are less constrained by the standard assumption of lognormality of prices. Accordingly, the paper is aimed primarily at analysis rather than finance. An example is given to illustrate the main results.

The Schrödinger equation is fundamental in quantum mechanics as it makes it possible to determine the wave function from energies and to use this function in the mean calculation of variables, for example, as the most likely position of a group of one or more massive particles. In this paper, we present a survey on some theories involving the Schrödinger equation and the Feynman path integral. We also consider a Feynman–Kac-type formula, as introduced by Patrick Muldowney, with the Henstock integral in the description of the expectation of random walks of a particle. It is well known that the non-absolute integral defined by R. Henstock “fixes” the defects of the Feynman integral. Possible applications where the potential in the Schrödinger equation can be highly oscillating, discontinuous or delayed are mentioned in the end of the paper.