Robust Utility Maximization in Non-dominated Models with 2BSDEs
Robust Utility Maximization in Non-dominated Models with 2BSDEs
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发表时间：
2012-01
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通讯作者：
A. Matoussi;Dylan Possamai;Chao Zhou
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作者：
A. Matoussi;Dylan Possamai;Chao Zhou
考虑了波动率不确定的不完全市场中的鲁棒效用最大化问题，即市场的波动率仅被假定在两个给定的界之间。这里考虑的所有可能模型（概率测度）的集合是非支配的。我们建议在具有二次增长生成元的二阶倒向随机微分方程（简称2BSDEs）的框架下研究这个问题。我们表明，指数，功率和对数效用的问题的值函数可以写为一个特定的2Btons的初始值，并证明存在的最优策略。最后，提供了几个例子，阐明了这个问题及其与经典效用最大化的联系。特别地，我们证明了在某些情况下，波动率区间的上界起着核心作用，就像在[2]的不确定波动率模型的期权定价问题中一样。
The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is non-dominated. We propose studying this problem in the framework of second-order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models of [2].