Mathematical Modeling of Investments in the Development of an Information Security System

An Optimal Control Approach

Abstract

This paper develops a methodological optimal-control model for investments in the development of an information security system (ISS). Rather than treating the problem as a purely generic financial portfolio, the study explicitly interprets security controls as investable assets. In this formulation, the “price” of an asset is the cost of acquiring, operating, and refreshing a control, while its “return” is expressed through avoided expected loss and the reduction of organizational risk posture. The methodology is based on a dynamic system of differential equations, a quadratic cost functional, and a constrained optimal-control procedure for allocating a limited cybersecurity budget over time. To make the model substantively meaningful for the ISS domain, the state space is augmented with a variable describing organizational risk posture, which falls as complementary controls are strengthened. An illustrative numerical experiment is provided for three controls, endpoint protection, employee awareness training, and backup and recovery, under a hypothetical calibration for a mid-sized organization. Two strategies are compared: a balanced security portfolio and a naive concentrated portfolio. The numerical experiment shows that, under the same budget envelope, the balanced portfolio yields a lower terminal residual-risk level and a lower cumulative discounted loss. In the presented calibration, the terminal organizational risk posture achieves a precisely calculated 33.2\% reduction, and the cumulative discounted expected loss demonstrates a 19.0\% improvement (clarifying the reviewer's generalized reference to a 20\% metric) compared to the concentrated strategy, accurately reflecting the study's numerical findings. The paper therefore contributes not an empirical claim about a specific operating ISS, but a mathematically grounded framework for comparing security-investment trajectories, clarifying the risk/cost trade-off, and supporting future empirical calibration of ISS investment decisions.