Abstract: The portfolio decision and contribution plan are the paramount important problems of the defined benefit pension funds. So the paper creates a Heston stochastic volatility model for the defined pension funds management, and transfers the primal problem to the dual problem by applying optimal control theory and the Legendre transform. It provides an analytic solution to the primal optimal problem by studying the dual problem. At last, an optimal asset allocation strategy (between a risky asset and a riskless asset) and the least contribution policy are obtained.

Key words:  Heston model, stochastic volatility, Legendre transform, duality, defined benefit pension funds