This paper reports the finding of a new financial chaotic system. A new control law for completely synchronizing the new financial chaotic system with itself has been established using adaptive integral sliding mode control. We also find that the new financial chaotic system has fascinating traits including symmetry, equilibrium points, multistability, Lyapunov exponents and bifurcation diagrams. We illustrate all the main results of this research work using MATLAB phase plots. The Lyapunov characteristic exponents and analysis using bifurcation diagrams have resulted in a new financial chaos system showing chaos phenomena in the intervals of parameters 0 < a < 15, and parameters 0 < b < 0.25. The results of this study can be used to predict if there is chaos in financial risk. Chaotic systems have many applications in engineering like cryptosystems and secure communication systems.

Countries prone to earthquakes face increasing seismic activity, often resulting in losses that exceed national budgets. To mitigate these losses, earthquake bonds present a promising alternative funding source; however, pricing them is complex, requiring simultaneous accounting for financial and seismic risks. Therefore, this study aimed to model earthquake bond pricing. The model incorporates earthquake intensity to account for rising seismic activity. It also includes depth and maximum magnitude as correlated dual trigger indices, making the bonds more attractive to investors, as claims are generated if both events occur. These three factors were modeled together as a compound stochastic process. The bond price was then formulated using a risk-neutral pricing measure with a stochastic interest rate under the Cox-Ingersoll-Ross model. Since the model lacks a closed-form solution, we employed an algorithm based on the Monte Carlo method for estimation. Through this algorithm, we showed that bond prices for terms of one to six years follow a normal distribution. The use of stochastic interest rates becomes significant as the bond term increases. We also found that earthquake intensity and bond terms negatively correlate with bond prices, while annual coupons positively correlate. Additionally, including dual triggers lowers claim probability and increases the bond demand, but is compensated by higher prices. This study can assist issuers in pricing earthquake bonds based on earthquake severity-maximum magnitude, depth, and intensity-and aid geological institutions in estimating earthquake risk in observed areas.

This study proposes a nonstationary Poisson-Lindley Hidden Markov Model (PL-HMM) as a novel framework for modeling the frequency of community-based disaster insurance claims. The model accounts for both serial dependence and overdispersion in claim counts through hidden risk states, while nonstationary transition probabilities are introduced via a sliding-window mechanism. Parameters are estimated using the Generalized Expectation-Maximization (GEM) algorithm, supported by a theoretical foundation to ensure a monotonic improvement of the complete log-likelihood. The model was simulated using monthly claim frequency data from West Java Province, Indonesia. A comparative analysis against nonstationary Poisson HMMs with varying numbers of hidden states showed that the two-state nonstationary PL-HMM achieved the lowest Bayesian information criterion ( $BIC$), thus indicating the best fit. A sensitivity analysis of sliding-window horizons (12, 24, and 36 months) demonstrated that persistence patterns of claim risk-states remained robust, with horizon changes reflecting alternative risk measurement periods. The results highlight that the proposed model effectively captures time-varying claim risks, particularly the alternation between low- and high-claim periods, while realistically reflecting the empirical dominance of high-claim regimes. Beyond the simulation data, a nonstationary PL-HMM is flexible and applicable to other regions that exhibit overdispersed claim data, making it a valuable framework for adaptive premium design and disaster risk financing in community-based insurance schemes.

There are many opinions about multiplication and division formulas in interval numbers, but there is a common weakness in these formulas: That the division between two equal interval numbers does not produce the identity. Similar to the interval number sequence, many concepts about the convergence of the interval sequence are offered by various authors but they cannot prove that the basic properties of the convergence of the real number sequence can be generalized to the properties of convergence of the interval number sequence. Here, we used the algebra for interval numbers from the author, as contained in Mashadi et al. (2023), that is, the algebra for interval numbers using midpoints, which guarantees the existence of the inverse of any interval number. In this article, we showed that some basic properties of the sequence of real numbers can be generalized to the sequence of interval numbers. In addition to the convergence properties of the interval number sequence, the convergence of the interval number sequence with positive, negative, and fractional powers were also shown. Based on the definition of convergence of interval sequences given along with various basic theorems for convergence given in this paper, it was expected that all theorems related to the convergence of real number sequences can be generalized to interval number sequences; for example, the properties of tail sequences, the Monotone Convergence Theorem, the Existence of Monotone Subsequences, Subsequences and the Bolzano-Weierstrass Theorem, and the Cauchy criterion for the convergence of interval number sequences.