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Dynamic Risk Measures for Anticipated Backward Doubly Stochastic Volterra Integral Equations
Liangliang Miao1, Zhang Liu2, Yijun Hu1
1School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China.
This study introduces anticipated backward doubly stochastic Volterra integral equations (ABDSVIEs) for financial market risk quantification. The research provides the theory for these equations and discusses their application in dynamic convex risk measures.
Area of Science:
- Financial Mathematics
- Stochastic Analysis
- Risk Management
Background:
- Financial markets generate complex data influencing risk.
- Existing risk quantification methods require dynamic adaptation.
- Stochastic Volterra integral equations offer a framework for modeling such dynamics.
Purpose of the Study:
- Introduce a novel class of equations for financial risk.
- Develop the theoretical underpinnings for these new equations.
- Explore their application in dynamic risk measure construction.
Main Methods:
- Definition and theoretical analysis of anticipated backward doubly stochastic Volterra integral equations (ABDSVIEs).
- Establishing existence, uniqueness, and comparison theorems for ABDSVIEs.
- Application of ABDSVIEs to derive dynamic convex risk measures.
Main Results:
- A new class of anticipated backward doubly stochastic Volterra integral equations (ABDSVIEs) is formally introduced.
- The existence, uniqueness, and a comparison theorem for ABDSVIEs are proven.
- Dynamic convex risk measures are successfully derived using ABDSVIEs.
Conclusions:
- ABDSVIEs provide a robust mathematical framework for financial risk quantification.
- The established theory supports the practical application of ABDSVIEs.
- This work advances the field of dynamic risk management in financial markets.
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