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Dynamic Risk Measures for Processes via Backward Stochastic Differential Equations Associated with Lévy Processes.
Liangliang Miao1, Zhang Liu2, Yijun Hu1
1School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China.
Entropy (Basel, Switzerland)
|July 2, 2021
Summary
This study introduces dynamic risk measures for processes using backward stochastic differential equations with Teugel's martingales and Lévy processes. It characterizes their time consistency, coherency, and convexity, supported by numerical examples.
Area of Science:
- Stochastic Analysis
- Financial Mathematics
- Risk Management
Background:
- Dynamic risk measures are crucial for financial risk assessment.
- Backward stochastic differential equations (BSDEs) provide a framework for modeling these risks.
- Teugel's martingales and Lévy processes offer advanced tools for stochastic modeling.
Purpose of the Study:
- To study dynamic risk measures for processes driven by BSDEs involving Teugel's martingales and Lévy processes (BSDELs).
- To provide a representation theorem for generators of BSDELs.
- To characterize the time consistency, coherency, and convexity of these dynamic risk measures.
Main Methods:
- Utilizing backward stochastic differential equations (BSDELs).
- Developing a representation theorem for BSDEL generators.
- Analyzing the properties of dynamic risk measures through their generators.
Main Results:
- A representation theorem for generators of BSDELs is established.
- Time consistency, coherency, and convexity of dynamic risk measures are characterized.
- The findings are illustrated with two numerical examples.
Conclusions:
- The study provides a theoretical framework for dynamic risk measures in complex stochastic environments.
- The characterization of risk measure properties offers practical insights for financial modeling.
- The proposed methods and examples enhance the understanding and application of dynamic risk measures.
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