Beliefs and Uncertainty in Stochastic Modeling

Abstract

Belief specification, as well as the identification of sources and statistical properties of uncertainty, is a crucial stage in stochastic model development. In much of the classical literature, one would begin by pinpointing a future event whose outcome would effectively determine the conclusion of some scenario. Next, one would hypothesize a particular distribution for the event's outcome. Answering theoretical and practical questions would then be a matter of careful argument and computation.

In this thesis, we are concerned with the following two questions, especially in cases when they can be motivated by financial applications: What if one is unable to select a single distribution which most appropriately characterizes the likelihoods of future outcomes? What if one has made a choice, even the best conceivably available choice, but it is simply wrong?

The financial mathematics community has investigated our first question since the seminal works of Avellaneda et al. and Lyons. Volatility is a key parameter when pricing certain instruments, and these papers examine what unfolds, if the volatility is not precisely known. In support of this line of research, others have investigated the transference of statistical properties from classical objects to their counterparts under this new scheme. Chapters 2 - 3 fall into the latter category. Intuitively, they primarily focus on features of large aggregations of future events where the corresponding distributions are uncertain.

To the best of our knowledge, our second question has attracted less scholarly attention in the contexts of index tracking during a reconstitution, parimutuel wagering, and mini-flash crashes. Briefly, index funds aim to replicate a chosen market benchmark. Parimutuel wagering is a popular betting system used in finance, sports, lotteries, and prediction markets. Mini-flash crashes are violent, rapid spikes or crashes in the prices of securities. Chapters 4 - 6 can be viewed as suggesting that it is natural to make mistakes in these situations, whether by picking a seemingly reasonable (but imperfect) objective or relying upon a sophisticated (but faulty) model. These innocuous errors can have surprising and occasionally disastrous consequences.