Path-dependent equations and viscosity solutions in infinite dimension

Path-dependent PDEs (PPDEs) are natural objects to study when one deals with non Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus (see [15]), in the case of finite-dimensional underlying space various papers have been devoted to studying the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [15, 9]) and viscosity solutions (see e.g. [16]). In this paper, motivated by the study of models driven by path-dependent stochastic PDEs, we give a first well-posedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finite-dimensional case, our well-posedness result, even in the Markovian case, applies to equations which cannot be treated, up to now, with the known theory of viscosity solutions.Comment: To appear in the Annals of Probabilit

Topics in stochastic calculus in infinite dimension for financial applications

This thesis is devoted to study delay/path-dependent stochastic differential equations and their connection with partial differential equations in infinite dimensional spaces, possibly path-dependent. We address mathematical problems arising in hedging a derivative product for which the volatility of the underlying assets as well as the claim may depend on the past history of the assets themselves. The starting point is to provide a robust framework for working with mild solutions to path-dependent SDEs: well-posedness, continuity with respect to the data, regularity with respect to the initial condition. This is done in Chapter 1. In Chapter 2, under Lipschitz conditions on the data, we prove the directional regularity needed in order to write the hedging strategy. In Chapter 3 we introduce a new notion of viscosity solution to semilinear path-dependent PDEs in Hilbert spaces (PPDEs), we prove well-posedness and show that the solution is given by the Fyenman-Kac formula. In Chapter 4 we extend to Hilbert spaces the functional It\uafo calculus and, under smooth assumptions on the data, we prove a path-dependent It\uafo\u2019s formula, show existence of classical solutions to PPDEs, and obtain a Clark-Ocone type formula. In Chapter 5 we introduce a new notion of C0-semigroup suitable to be applied to Markov transition semigroups, hence to mild solutions to Kolmogorov PDEs, and we prove all the basic results analogous to those available for C0-semigroups in Banach spaces. Additional theoretical results for stochastic analysis in Hilbert spaces, regarding stochastic convolutions, are given in Appendix A. Our methodology varies among different chapters. Path-dependent models can be studied in their original path-dependent form or by representing them as non-pathdependent models in infinite dimension. We exploit both approaches. We treat pathdependent Kolmogorov equations in infinite dimension with two notions of solution: classical and viscosity solutions. Each approach leads to original results in each chapter

Path-dependent Hamilton-Jacobi-Bellman equation: Uniqueness of Crandall-Lions viscosity solutions

We prove existence and uniqueness of Crandall-Lions viscosity solutions of Hamilton-Jacobi-Bellman equations in the space of continuous paths, associated to the optimal control of path-dependent SDEs. This seems the first uniqueness result in such a context. More precisely, similarly to the seminal paper of P.L. Lions, the proof of our core result, that is the comparison theorem, is based on the fact that the value function is bigger than any viscosity subsolution and smaller than any viscosity supersolution. Such a result, coupled with the proof that the value function is a viscosity solution (based on the dynamic programming principle, which we prove), implies that the value function is the unique viscosity solution to the Hamilton-Jacobi-Bellman equation. The proof of the comparison theorem in P.L. Lions' paper, relies on regularity results which are missing in the present infinite-dimensional context, as well as on the local compactness of the finite-dimensional underlying space. We overcome such non-trivial technical difficulties introducing a suitable approximating procedure and a smooth gauge-type function, which allows to generate maxima and minima through an appropriate version of the Borwein-Preiss generalization of Ekeland's variational principle on the space of continuous paths

Optimal control of path-dependent McKean-Vlasov SDEs in infinite dimension

We study the optimal control of path-dependent McKean-Vlasov equations valued in Hilbert spaces motivated by non Markovian mean-field models driven by stochastic PDEs. We first establish the well-posedness of the state equation, and then we prove the dynamic programming principle (DPP) in such a general framework. The crucial law invariance property of the value function V is rigorously obtained, which means that V can be viewed as a function on the Wasserstein space of probability measures on the set of continuous functions valued in Hilbert space. We then define a notion of pathwise measure derivative, which extends the Wasserstein derivative due to Lions [41], and prove a related functional It{\^o} formula in the spirit of Dupire [24] and Wu and Zhang [51]. The Master Bellman equation is derived from the DPP by means of a suitable notion of viscosity solution. We provide different formulations and simplifications of such a Bellman equation notably in the special case when there is no dependence on the law of the control.Comment: 54 page

Viscosity solutions of path-dependent PDEs with randomized time

We introduce a new definition of viscosity solution to path-dependent partial differential equations, which is a slight modification of the definition introduced in [8]. With the new definition, we prove the two important results till now missing in the literature, namely, a general stability result and a comparison result for semicontinuous sub-/super-solutions. As an application, we prove the existence of viscosity solutions using the Perron method. Moreover, we connect viscosity solutions of path-dependent PDEs with viscosity solutions of partial differential equations on Hilbert spaces

C0-sequentially equicontinuous semigroups

We present and apply a theory of one-parameter C0-semigroups of linear operators in locally convex spaces. Replacing the notion of equicontinuity considered by the literature with the weaker notion of sequential equicontinuity, we prove the basic results of the classical theory of C0-equicontinuous semigroups: we show that the semigroup is uniquely identified by its generator and we provide a generation theorem in the spirit of the celebrated Hille–Yosida theorem. Then we particularize the theory in some functional spaces and identify two locally convex topologies that allow us to gather—under a unified framework—various notions of C0-semigroups introduced by some authors to deal with Markov transition semigroups. Finally, we apply the results to transition semigroups associated to stochastic differential equations (SDEs)

Robustness for path-dependent volatility models

In this paper, we consider a generalisation of the Hobson–Rogers model proposed by Foschi and Pascucci (Decis Eocon Finance 31(1):1–20, 2008) for financial markets where the evolution of the prices of the assets depends not only on the current value but also on past values. Using differentiability of stochastic processes with respect to the initial condition, we analyse the robustness of such a model with respect to the so-called offset function, which generally depends on the entire past of the risky asset and is thus not fully observable. In doing this, we extend previous results of Blaka Hallulli and Vargiolu (2007) to contingent claims, which are globally Lipschitz with respect to the price of the underlying asset, and we improve the dependence of the necessary observation window on the maturity of the contingent claim, which now becomes of linear type, while in Blaka Hallulli and Vargiolu (2007), it was quadratic. Finally, in this framework, we give a characterisation of the stationarity assumption used in Blaka Hallulli and Vargiolu (2007), and prove that this model is stationary if and only if it is reduced to the original Hobson–Rogers model. We conclude by calibrating the model to the prices of two indexes using two different volatility shapes