A MEASURE THEORY AND INTEGRATION APPROACH TO PROBABILITY THEORY, AND APPLICATIONS IN FINANCIAL MARKETS (BLACK-SCHOLES MODEL), AND ACTUARIAL MATHEMATICS (RUIN PROBABILITY)
Abstract:
Probability theory is a fundamental branch of mathematics that provides a rigorous framework for understanding uncertainty and randomness. This abstract presents a measure theory and integration-based approach to probability theory, and its applications in two important domains: financial markets, specifically the Black-Scholes model, and actuarial mathematics, with a focus on ruin probability.
The measure theory and integration approach provides a solid foundation for probability theory by defining probabilities as measures on suitable spaces. It introduces concepts such as probability spaces, random variables, and expectation, allowing for a precise and rigorous treatment of probabilistic phenomena. By employing the Lebesgue integral, this approach overcomes limitations of earlier integration theories and enables the study of a broader class of random variables.
In financial markets, the Black-Scholes model is a widely used mathematical framework for pricing options and derivatives. By employing the measure theory and integration approach, the model can be formulated in a mathematically rigorous manner. The Black-Scholes equation, derived using partial differential equations and stochastic calculus, provides insights into option pricing, hedging strategies, and risk management.
Actuarial mathematics focuses on assessing and managing risks in insurance and related fields. The concept of ruin probability plays a crucial role in evaluating the financial stability of insurance companies. By utilizing the measure theory and integration approach, ruin probability can be analyzed in a rigorous probabilistic framework. This allows for the calculation of probabilities of ruin under different risk models, aiding in the development of optimal insurance policies and risk mitigation strategies.
The measure theory and integration approach to probability theory provides a powerful mathematical framework for understanding and analyzing probabilistic phenomena in various domains. By applying this approach to financial markets and actuarial mathematics, it allows for precise modeling, pricing, and risk assessment, contributing to the development of robust financial and insurance systems.
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