Non-anticipative functional calculus and applications to stochastic processes

Abstract : This thesis focuses on various mathematical questions arising in the non-anticipative functional calculus, which is based on a notion of pathwise directional derivatives for functionals. We extend the scope and results of this calculus to functionals which may not admit such derivatives, either through approximations (Part I) or by defining a notion of weak vertical derivative (Part II). In the first part, we consider the representation of conditional expectations as non-anticipative functionals. We show that it is possible under very general conditions to approximate such functionals by a sequence of smooth functionals in an appropriate sense. This approach provides a systematic method for computing explicit approximations to martingale representations for a large class of Brownian functionals. We also derive explicit convergence rates of the approximations. These results are then applied to the problem of sensitivity analysis and dynamic hedging of (path-dependent) contingent claims. In the second part, we propose a concept of weak vertical derivative for non-anticipative functionals which may fail to possess directional derivatives. The definition of the weak vertical derivative is based on the notion of pathwise quadratic variation and makes use of the duality associated to the associated bilinear form. We show that the notion of weak vertical derivative leads to a functional characterization of local martingales with respect to a reference process, and allows to define a concept of pathwise weak solution for path-dependent partial differential equations.
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Yi Lu. Non-anticipative functional calculus and applications to stochastic processes. General Mathematics [math.GM]. Université Pierre et Marie Curie - Paris VI, 2017. English. ⟨NNT : 2017PA066418⟩. ⟨tel-01759395⟩

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