Sigma-martingale

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In the mathematical theory of probability, a sigma-martingale is a semimartingale with an integral representation. Sigma-martingales were introduced by C.S. Chou and M. Emery in 1977 and 1978.[1] In financial mathematics, sigma-martingales appear in the fundamental theorem of asset pricing as an equivalent condition to no free lunch with vanishing risk (a no-arbitrage condition).[2]

Mathematical definition[edit]

An \mathbb{R}^d-valued stochastic process X = (X_t)_{t = 0}^T is a sigma-martingale if it is a semimartingale and there exists an \mathbb{R}^d-valued martingale M and an M-integrable predictable process \phi with values in \mathbb{R}_+ such that

X = \phi \cdot M. \,[1]

References[edit]

  1. ^ a b F. Delbaen; W. Schachermayer (1998). "The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes" (pdf). Mathematische Annalen 312: 215–250. doi:10.1007/s002080050220. Retrieved October 14, 2011. 
  2. ^ Delbaen, Freddy; Schachermayer, Walter. "What is... a Free Lunch?" (pdf). Notices of the AMS 51 (5): 526–528. Retrieved October 14, 2011.