路径依赖导数的凸阶：一种动态规划方法
Convex order for path-dependent derivatives: a dynamic programming approach
摘要：
我们研究了各种连续鞅过程的（泛函）凸序，无论是关于Lévy驱动的随机微分方程的扩散系数还是关于随机积分的被积函数。主要结果是由反例接壤。在局部波动率模型中，路径欧式期权价格的上界和下界是不同的。考虑到数值应用，我们采用了一种系统的（对称的）方法：（a）通过反向归纳（或线性动力学原理）传播{ \em simulatable}支配/支配离散时间模型的凸性;（b）将泛函弱收敛结果应用于满足（a）的连续时间鞅的数值格式/时间离散，以转移凸序性质.各种界限推导出欧洲期权写凸路径依赖的支付。我们检索和扩展以前的结果，自1985年的开创性论文Hajek的几个作者。在第二部分中，我们将这种方法推广到最优停止问题，使用的Snell包络满足（a '）向后动态规划原理传播凸离散时间;（b'）满足抽象收敛结果下的非退化假设的过滤。应用于凸路径支付过程美式期权价格的比较，得到了一个纯概率的论点。
英文摘要：
We investigate the (functional) convex order of for various continuous martingale processes, either with respect to their diffusions coefficients for Lévy-driven SDEs or their integrands for stochastic integrals. Main results are bordered by counterexamples. Various upper and lower bounds can be derived for path wise European option prices in local volatility models. In view of numerical applications, we adopt a systematic (and symmetric) methodology: (a) propagate the convexity in a {\em simulatable} dominating/dominated discrete time model through a backward induction (or linear dynamical principle); (b) Apply functional weak convergence results to numerical schemes/time discretizations of the continuous time martingale satisfying (a) in order to transfer the convex order properties. Various bounds are derived for European options written on convex pathwise dependent payoffs. We retrieve and extend former results obtains by several authors since the seminal 1985 paper by Hajek . In a second part, we extend this approach to Optimal Stopping problems using a that the Snell envelope satisfies (a') a Backward Dynamical Programming Principle to propagate convexity in discrete time; (b') satisfies abstract convergence results under non-degeneracy assumption on filtrations. Applications to the comparison of American option prices on convex pathwise payoff processes are given obtained by a purely probabilistic arguments.
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