对偶控制问题渐近展开的Malliavin演算法
Malliavin calculus method for asymptotic expansion of dual control problems
摘要：
我们开发了一种技术的基础上Malliavin-Bismut演算的想法，渐近展开的双重控制问题所产生的索赔与指数无差异估值，并与最小化的相对熵，在不完全市场。这些问题涉及优化功能的布朗路径上的维纳空间，与路径扰动漂移涉及的控制。此外，还有一个惩罚项，其中控制功能二次。利用吉尔萨诺夫定理将漂移扰动解释为一种测量变化，从而得到一种分部积分公式，其中计算了维纳空间上的方向导数。这使得渐近分析的控制问题。应用到不完全伊藤过程市场，其中无差异价格近似在低风险厌恶极限。我们也给出了一个应用，以确定最小熵鞅测度作为一个扰动的最小鞅测度的随机波动率模型。
英文摘要：
We develop a technique based on Malliavin-Bismut calculus ideas, for asymptotic expansion of dual control problems arising in connection with exponential indifference valuation of claims, and with minimisation of relative entropy, in incomplete markets. The problems involve optimisation of a functional of Brownian paths on Wiener space, with the paths perturbed by a drift involving the control. In addition there is a penalty term in which the control features quadratically. The drift perturbation is interpreted as a measure change using the Girsanov theorem, leading to a form of the integration by parts formula in which a directional derivative on Wiener space is computed. This allows for asymptotic analysis of the control problem. Applications to incomplete Itô process markets are given, in which indifference prices are approximated in the low risk aversion limit. We also give an application to identifying the minimal entropy martingale measure as a perturbation to the minimal martingale measure in stochastic volatility models.
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