Miwaka Yamashita, CFA   BlackRock Japan Co., Ltd.  miwaka.yamashita@blackrock.com

1-8-3 MTTM Marunouchi Chiyoda-ku, Tokyo, Japan Phone: 81-3-6703-4533, FAX: 81-3-6703-4160

Miwaka Yamashita is Director of BlackRock Japan Co., Ltd. and responsible client advisory. BS and MA holder of Geophysics from Tokyo University and MBA from School of Business and Economy, Michigan Technological University. In The Institute of Actuaries of Japan, acting Head of AFIR study session.

Abstract

This paper surveys dynamic asset allocation models which uses stochastic optimal control problem methods. Merton model and its valiational models, use of Hamilton Jacobi Bellman Equation (HJB), Backward Stochastic Differential Equation (BSDE), and Malliavin calculus, arrangements for the model, numerical simulation are discussed.

Contents

1. Introduction          

2. Problem Setting

3. Viscosity Solution

3.1 Viscosity Solution in general

3.2 Viscosity properties inside domain and strong comparison principle

3.3 Viscosity Solution for kinked utility

4. Backward Stochastic Differential Equation (BSDE)

4.1 BSDE in general

4.2 BSDE and utility maximization

4.3 CRRA utility and BSDEs

4.4 Link to PDE and Viscosity Solutions

4.5 Stopping time problem, Reflected BSDEs and Kinked Utility

5. Relation with Malliavin calculus

5.1 BSDE and  Malliavin

5.2 Malliavin calculus in general

6. Summary and Future Challenging

References