Path dependent partial differential equation: theory and applications

Abstract : In the previous works, Ekren, Keller, Touzi & Zhang [35] and Ekren, Touzi & Zhang [37, 38], the new notion of viscosity solutions to path dependent PDEs is introduced, and a wellposedness theory is proved by a ‘path-frozen’ argument. This new notion generalizes that of viscosity solutions to PDEs developed intensively in the years of 80’s and 90’s, and can be used to characterize the value function of non-Markovian stochastic control problem. In this thesis, we report the recent development of the new theory. We improve the argument for the comparison result, and provide a PDE-style Perron’s method for proving the existence of viscosity solutions to semi- linear path dependent PDEs. As in the seminar work of Barles and Souganidis [4] in the context of PDEs, we show that a family of numerical schemes satisfying the so- called monotonicity condition provides numerical solutions converging to viscosity solutions of fully nonlinear path dependent PDEs. Further, we develop a notion of elliptic path dependent PDEs, and provide a wellposedness theory by following the lines of arguments in [38]. This thesis also includes some interesting applications of path dependent PDEs. One of them is on the large deviations of non-Markovian dif- fusion. As Fleming used the stability of viscosity solutions of PDEs to establish the large deviation principle in Markovian case (see [51]), we use the theory of backward stochastic differential equations and that of path dependent PDEs to generalize his result for non-Markovian diffusions. Moreover, the large deviation result is applied to investigate the short maturity asymptotics of the implied volatility surface in financial mathematics. Finally, a study of dual algorithm for stochastic control pro- blems is presented. As Monte-Carlo simulations for the stochastic control problems provide low-biased estimate, a dual algorithm offer upper bounds of the true values. The idea of ‘path-frozen’ is exploited to give a dual representation of non-Markovian stochastic control problems.
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Zhenjie Ren. Path dependent partial differential equation: theory and applications. Analysis of PDEs [math.AP]. Ecole Doctorale Polytechnique, 2015. English. ⟨tel-01265462⟩

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