Web renewal counting processes and their applications in insurance

Area of Science:

• Probability Theory
• Stochastic Processes
• Actuarial Science

Background:

• Renewal processes are fundamental in modeling random event occurrences.
• Dependent inter-arrival times introduce complexities not captured by standard models.
• Understanding aggregate claim amounts is critical for insurance risk management.

Purpose of the Study:

• To introduce and analyze a nonstandard renewal counting process with dependent inter-arrival times, termed the web renewal process.
• To derive essential limit properties of this novel process.
• To apply these findings to actuarial science for precise deviation formulas in insurance.

Main Methods:

• Investigation of a nonstandard renewal counting process.
• Derivation of limit properties, focusing on the tail of the exponential moment.
• Application of derived properties to insurance risk models.

Main Results:

• Established limit properties for the web renewal process.
• Identified the significance of the tail of the exponential moment condition.
• Developed precise large deviations and moderate deviation formulas for aggregate insurance claims.

Conclusions:

• The web renewal process offers a more realistic model for systems with dependent event times.
• The derived limit properties are crucial for theoretical advancements and practical applications.
• The study provides enhanced tools for actuarial analysis of aggregate claim distributions.