A model‐free approach to continuous‐time finance
提出一种不依赖概率概念的连续时间金融路径方法，基于因果泛函微积分定义自融资组合，证明在通用市场情景下无套利，并利用微分博弈得到超对冲成本的路径动态规划原理，对亚式期权给出显式解。
Abstract We present a pathwise approach to continuous‐time finance based on causal functional calculus. Our framework does not rely on any probabilistic concept. We introduce a definition of continuous‐time self‐financing portfolios, which does not rely on any integration concept and show that the value of a self‐financing portfolio belongs to a class of nonanticipative functionals, which are pathwise analogs of martingales. We show that if the set of market scenarios is generic in the sense of being stable under certain operations, such self‐financing strategies do not give rise to arbitrage. We then consider the problem of hedging a path‐dependent payoff across a generic set of scenarios. Applying the transition principle of Rufus Isaacs in differential games, we obtain a pathwise dynamic programming principle for the superhedging cost. We show that the superhedging cost is characterized as the solution of a path‐dependent equation. For the Asian option, we obtain an explicit solution.