禹王宫
Dewen Xiong: Jump Process in Finance
Mo14-16
gouwu>送元二使安西的诗意
Abstract
We will first discuss the general mimartingale theory and stochastic integrals with respect to general mimartingales. We introduce a random measure to describe the jumps and finally introduce the characteristics of mimartingales. Then we discuss Itô’s formula and Girsanov’s theorem for general mimartingales.
After the technical preparations, we apply the above theory to mathematical finance problems. We a
ssume that the price process of a risky ast or a stock is a general mimartingale with jumps described by a random measure. We will add some conditions to make sure that the in general incomplete market is arbitrage free. In the general jump framework we will mainly discuss the following topics which are extremely relevant for up-to-date applications of mathematical finance :
i) the mean-variance hedging of a contingent claim: we introduce the variance optimal martingale measure and give an explicit form of the optimal strategy by a backward mimartingale equation (BSE);
ii) the $p$-optimal martingale measures and their convergence: we will introduce a backward mimartingales equations to describe the p-optimal martingale measures and prove that the p-optimal martingale measures will converge to the minimal entropy martingale measure as p goes to 1.
This lecture requires the knowledge of Stochastics II. It is strongly recommended to attend this lecture parallel to or after attending Stochastics III.
Literature :
全国扶贫日J. Jacod and A. N. Shiryaev, Limit theorems for stochastic process
Grundlehren der Mathematischen Wisnschaften, vol. 288, Springer-Verlag, Berlin, Heidelberg, New York, 1987, xvii + 600 pp.汽车分级
电脑病毒代码Phillip E. Protter: Stochastic Integration and Differential equations (Second edition) (2004) Springer-Verlag, (xiv)+415 pp.李鸿章去美国
摘要
我们将首先讨论一般半鞅和关于一般半鞅的随机积分。我们将引入“整数值随机测度”来刻画一般半鞅的跳及其特征，然后我们将讨论关于一般半鞅的Itô公式与Girsanov定理。然后我们将讨论一般半鞅的理论在金融中的应用。我们假定风险资产（或股票）的价格过程为一个具有“一般跳”的半鞅的数学模型，其跳由一个“整数值测度”描述，在一些基本的假定下，市场为不完全市场。我们将主要讨论在数学金融学中及其重要的两个问题：
i) 均值方差套期保值问题：我们引入方差最优鞅测度，利用一个倒向半鞅方程的解给出了最优套期保值策略的一种自反馈形式；
ii) $p$-最优鞅测度及其收敛性：我们将引入一个倒向半鞅方程来刻画$p$-最优鞅测度，我们将证明当$p$收敛到1时，$p$-最优鞅测度将收敛到最小熵测度。
本课程要求一些Stochastics II知识，强烈推荐修本课程的同时修Stochastics III或在修Stochastics III之后再修。杭州到宁波有多少公里
