Modeling in Finance and Insurance with Levy-Itsigma Driven Dynamic Processes under Semi Markov-type Switching Regimes and Time Domains

Mathematical and statistical modeling have been at the forefront of many significant advances in many disciplines in both the academic and industry sectors. From behavioral sciences to hard core quantum mechanics in physics, mathematical modeling has made a compelling argument for its usefulness and its necessity in advancing the current state of knowledge in the 21rst century. In Finance and Insurance in particular, stochastic modeling has proven to be an effective approach in accomplishing a vast array of tasks: risk management, leveraging of investments, prediction, hedging, pricing, insurance, and so on. However, the magnitude of the damage incurred in recent market crisis of 1929 (the great depression), 1937 (recession triggered by lingering fears emanating from the great depression), 1990 (1 year recession following a decade of steady expansion) and 2007 (the great recession triggered by the sub-prime mortgage crisis) has suggested that there are certain aspects of financial markets not accounted for in existing modeling. Explanations have abounded as to why the market underwent such deep crisis and how to account for regime change risk. One such explanation brought forth was the existence of regimes in the financial markets. The basic idea of market regimes underscored the principle that the market was intrinsically subjected to many different states and can switch from one state to another under unknown and uncertain internal and external perturbations. Implementation of such a theory has been done in the simplifying case of Markov regimes. The mathematical simplicity of the Markovian regime model allows for semiclosed or closed form solutions in most financial applications while it also allows for economically interpretable parameters. However, there is a hefty price to be paid for such practical conveniences as many assumptions made on the market behavior are quite unreasonable and restrictive. One assumes for instance that each market regime has a constant propensity of switching to any other state irrespective of the age of the current state. One also assumes that there are no ı́ntermediateśtates as regime changes occur in a discrete manner from one of the finite states to another. There is therefore vi no telling how meaningful or reliable interpretation of parameters in Markov regime models are. In this thesis, we introduced a sound theoretical and analytic framework for Levy driven linear stochastic models under a semi Markov market regime switching process and derived Itó formula for a general linear semi Markov switching model generated by a class of Lévy Itó processes (1). Itó formula results in two important byproducts, namely semi closed form formulas for the characteristic function of log prices and a linear combination of duration times (2). Unlike Markov markets, the introduction of semi Markov markets allows a time varying propensity of regime change through the conditional intensity matrix. This is more in line with the notion that the market’s chances of recovery (respectively, of crisis) are affected by the recession’s age (respectively, recovery’s age). Such a change is consistent with the notion that for instance, the longer the market is mired into a recession, the more improbable a fast recovery as the the market is more likely to either worsens or undergo a slow recovery. Another interesting consequence of the time dependence of the conditional intensity matrix is the interpretation of semi Markov regimes as a pseudo-infinite market regimes models. Although semi Markov regime assume a finite number of states, we note that while in any give regime, the market does not stay the same but goes through an infinite number of changes through its propensity of switching to other regimes. Each of those separate ı́ntermediateśtates endows the market with a structure of pseudo-infinite regimes which is an answer to the long standing problem of modeling market regime with infinitely many regimes. We developed a version of Girsanov theorem specific to semi Markov regime switching stochastic models, and this is a crucial contribution in relating the risk neutral parameters to the historical parameters (3). Given that Levy driven markets and regime switching markets are incomplete, there are more than one risk neutral measures that one can use for pricing derivative contracts. Although much work has been done about optimal choice of the pricing measure, two of them jump out of the current literature: the minimal martingale measure and the minimum entropy martingale measure. We first presented a general version of Girsanov theorem explicitly accounting for semi Markov regime. Then we presented Siu and Yang pricing kernel. In addition, we developed the conditional and unconditional minimum entropy martingale measure which minimized the dissimilarity