Guyon, Julien (2006) Probabilistic modeling in finance and biology - Limit theorems and applications. PhD thesis, ENPC.

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Abstract

In this PhD dissertation, I strove to propose a both precise and handy probabilistic modeling in some areas of finance and biology. The first chapter details my goals and objectives. It sums up the main results of my research compare them to existing works and suggest possible extensions. In Chapter 2, after the articles of Talay and Tubaro (1990) and Bally and Talay (1996), I mesure the error one commits when one approaches the law of the solution of a stochastic differential equation by that of its Euler scheme. Under an ellipticity hypothesis, a joint use of probabilistic and analytic techniques leads me to a functional expansion of the "transition kernel" of the Euler scheme, in powers of the time step, in spaces of smooth gaussian-like functions. This result naturally applies to financial mathematics. It gives the rate of convergence of the prices, deltas and gammas of European options for an extremely wide range of payoffs. In Chapter 3, I analyse a stochastic volatility model proposed by Fouque, Papanicolaou and Sircar (2000). The results in Chapter 2 allow me to build a pricing and hedging algorithm for European options which adaptively guarantees the balance between the statistical error, due to Monte Carlo sampling, and the time discretization error. The last chapter deals with cellular aging and results from a cooperation with biologists from the Faculté de Médecine Necker in Paris. The experimental data consists of a binary tree of growth rates from which my colleagues wish to detect two subpopulations. To explain this data, I propose a bifurcating autoregressive model generalizing that of Cowan and Staudte (1986) and then build and implement statistical procedures to estimate parameters and test biological hypothesis. To this end, I introduce the concept of "bifurcating Markov chains" and I prove that such stochastic processes satisfy original limit theorems which I apply to the model and the data, confirming the intuition and the preliminary computations of the biologists.

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