[Paper Review] Drift operator in a market affected by the expansion of information flow : a case study

Shiqi Song|arXiv (Cornell University)|Jul 6, 2012

Auction Theory and Applications21 references8 citations

TL;DR

This paper studies market viability under expanded information flow using the drift operator in filtration enlargement. It establishes sufficient conditions—based on the drift operator's classical form and the martingale representation property—under which a market remains viable after information expansion, with explicit integrability and positivity constraints on the drift and jump components.

ABSTRACT

We consider a viable market model. Suppose that new information arrives at the market. We are interested in modeling the market reaction facing to the change of information. In particular we seek for the limit on the intensity of information change below which the market stays always viable. We succeed to find such a limit with the drift operator when the market possesses the martingale representation property.

Motivation & Objective

• To analyze how market viability is affected when information flow expands, particularly in the context of filtration enlargement.
• To identify the critical threshold of information expansion intensity that preserves market viability.
• To develop a framework using the drift operator as a gauge for controlling information expansion while maintaining viability.
• To address the technical challenges introduced by jumps in price processes under expanded information.
• To establish sufficient conditions for viability in the presence of jumps, leveraging the martingale representation property.

Proposed method

• Models information flow via filtrations $\mathbb{F} \subset \mathbb{G}$, with $\mathbb{G}$ representing expanded information.
• Applies the theory of enlargement of filtrations, assuming Hypothesis $(H')$ to ensure local martingales in $\mathbb{F}$ remain semimartingales in $\mathbb{G}$.
• Represents the drift operator $\Gamma(X)$ in classical form: $\Gamma(X) = {}^{\top}\!\overline{\varphi} \centerdot [N,X]^{\mathbb{F}-p}$, where $N$ is an $\mathbb{F}$-local martingale and $\overline{\varphi}$ is $\mathbb{G}$-predictable.
• Imposes technical conditions on the increasing processes ${{}^{\top}\!}\overline{\varphi}(\centerdot[N^{c},{}^{\top}\!N^{c}])\overline{\varphi}$, $\frac{1}{\mathsf{u}}\centerdot[D^{d},D^{d}]$, and $\frac{1}{\mathsf{u}}{{}^{\top}\!}\overline{\varphi}(\centerdot[N^{d},{}^{\top}\!N^{d}])\overline{\varphi}$ to ensure local integrability.
• Localizes the analysis at jump times and constructs global solutions by integrating local solutions, particularly under the martingale representation property.

Experimental results

Research questions

• RQ1 What is the maximal intensity of information expansion below which the market remains viable?
• RQ2 How does the drift operator $\Gamma(X)$ serve as a quantitative gauge for information expansion in a viable market?
• RQ3 What conditions ensure market viability when jumps are present in the price process after information expansion?
• RQ4 How can the martingale representation property be leveraged to ensure viability under jump-diffusion dynamics?
• RQ5 What role do the predictable dual projections and integrability constraints play in maintaining viability after filtration enlargement?

Key findings

• The market with expanded information flow $\mathbb{G}$ remains viable on $[0,T]$ if the drift operator $\Gamma(X)$ is in classical form and the associated increasing processes are locally integrable.
• The condition $1 + {}^{\top}\!\overline{\varphi}\Delta N \geq \mathsf{u}$ with $\mathsf{u}$ a $\mathbb{G}$-predictable process ensures positivity and avoids arbitrage in the expanded market.
• For the continuous case, viability is preserved under the classical form of $\Gamma(X)$ and the integrability of the quadratic variation terms.
• In the presence of jumps, the problem is localized at jump times, and viability is maintained if the local drift and jump components satisfy the stated integrability and positivity constraints.
• The key result, Theorem 4.4, establishes that under the martingale representation property and the classical form of $\Gamma(X)$, the market remains viable after information expansion.
• Corollary 4.5 confirms that the viability condition holds when the drift operator satisfies the specified integrability and positivity conditions on the jump components.