A martingale approach for fractional Brownian motions and related path dependent PDEs
A martingale approach for fractional Brownian motions and related path dependent PDEs
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DOI：
10.1214/19-aap1486
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发表时间：
2017-12
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影响因子：
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通讯作者：
F. Viens;Jianfeng Zhang
中科院分区：
文献类型：
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作者：
F. Viens;Jianfeng Zhang
本文以（前向）状态过程满足沃尔泰拉型方程为框架，以分数布朗运动为典型，以条件期望的计算为主要目标，研究动态后向问题.这些过程既不是马尔可夫过程，也不是半鞅，最值得注意的是，它们具有一定的时间不一致性，这使得任何直接应用马尔可夫思想，如流性质，不可能不通过路径依赖框架。我们的主要结果是一个功能的It\^{o}公式，扩展了Dupire \cite{Dupire}的开创性工作，我们更一般的框架。特别是，与只需要考虑停止路径的\cite{Dupire}不同，这里我们需要将当前时间的观察路径与从未来路径分布中导出的某个平滑的可观察曲线连接起来。这一新的特征是由于本文所涉及的时间不一致性。然后，我们推导出路径依赖的偏微分方程的向后问题。最后给出了在具有粗糙波动率的金融市场中期权定价的一个应用。
In this paper we study dynamic backward problems, with the computation of conditional expectations as a main objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional It\^{o} formula, extending the seminal work of Dupire \cite{Dupire} to our more general framework. In particular, unlike in \cite{Dupire} where one needs only to consider the stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths. This new feature is due to the time inconsistency involved in this paper. We then derive the path dependent PDEs for the backward problems. Finally, an application to option pricing in a financial market with rough volatility is presented.