COSSO Andrea

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.

We study the Bellman equation in the Wasserstein space arising in the study of mean field control problems, namely stochastic optimal control problems for McKean-Vlasov diffusion processes. Using the standard notion of viscosity solution à la Crandall-Lions extended to our Wasserstein setting, we prove a comparison result under general conditions, which coupled with the dynamic programming principle, implies that the value function is the unique viscosity solution of the Master Bellman equation. This is the first uniqueness result in such a second-order context. The classical arguments used in the standard cases of equations in finite-dimensional spaces or in infinite-dimensional separable Hilbert spaces do not extend to the present framework, due to the awkward nature of the underlying Wasserstein space. The adopted strategy is based on finite-dimensional approximations of the value function obtained in terms of the related cooperative n-player game, and on the construction of a smooth gauge-type function, built starting from a regularization of a sharpe estimate of the Wasserstein metric. such a gauge-type function is used to generate maxima/minima through a suitable extension of the Borwein-Preiss generalization of Ekeland's variational principle on the Wasserstein space.

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 Itô 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.

We formulate an equilibrium model of intraday trading in electricity markets. Agents face balancing constraints between their customers consumption plus intraday sales and their production plus intraday purchases. They have continuously updated forecast of their customers consumption at maturity with decreasing volatility error. Forecasts are prone to idiosyncratic noise as well as common noise (weather). Agents production capacities are subject to independent random outages, which are each modelled by a Markov chain. The equilibrium price is defined as the price that minimises trading cost plus imbalance cost of each agent and satisfies the usual market clearing condition. Existence and uniqueness of the equilibrium are proved, and we show that the equilibrium price and the optimal trading strategies are martingales. The main economic insights are the following. (i) When there is no uncertainty on generation, it is shown that the market price is a convex combination of forecasted marginal cost of each agent, with deterministic weights. Furthermore, the equilibrium market price follows Almgren and Chriss's model and we identify the fundamental part as well as the permanent market impact. It turns out that heterogeneity across agents is a necessary condition for the Samuelson's effect to hold. (ii) When there is production uncertainty, the price volatility becomes stochastic but converges to the case without production uncertainty when the number of agents increases to infinity. Further, on a two-agent case, we show that the potential outages of a low marginal cost producer reduces her sales position.

We address our interest to the development of a theory of viscosity solutions à la Crandall-Lions for path-dependent partial differential equations (PDEs), namely PDEs in the space of continuous paths C([0, T ]. R^d). Path-dependent PDEs can play a central role in the study of certain classes of optimal control problems, as for instance optimal control problems with delay. Typically, they do not admit a smooth solution satisfying the corresponding HJB equation in a classical sense, it is therefore natural to search for a weaker notion of solution. While other notions of generalized solution have been proposed in the literature, the extension of the Crandall-Lions framework to the path-dependent setting is still an open problem. The question of uniqueness of the solutions, which is the more delicate issue, will be based on early ideas from the theory of viscosity solutions and a suitable variant of Ekeland's variational principle. This latter is based on the construction of a smooth gauge-type function, where smooth is meant in the horizontal/vertical (rather than Fréchet) sense. In order to make the presentation more readable, we address the path-dependent heat equation, which in particular simplifies the smoothing of its natural "candidate" solution. Finally, concerning the existence part, we provide a new proof of the functional Itô formula under general assumptions, extending earlier results in the literature.

The aim of the present work is the introduction of a viscosity type solution, called strong-viscosity solution to distinguish it from the classical one, with the following peculiarities: it is a purely analytic object. it can be easily adapted to more general equations than classical partial differential equations. First, we introduce the notion of strong-viscosity solution for semilinear parabolic partial differential equations, defining it, in a few words, as the pointwise limit of classical solutions to perturbed semilinear parabolic partial differential equations. we compare it with the standard definition of viscosity solution. Afterwards, we extend the concept of strong-viscosity solution to the case of semilinear parabolic path-dependent partial differential equations, providing an existence and uniqueness result.

We analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities. These equations are related to robust switching control problems introduced in [3]. We prove that, as time horizon goes to infinity (resp. discount factor goes to zero) the long run average solution to the parabolic system (resp. the limiting discounted solution to the elliptic system) is characterized by a solution of a nonlinear system of ergodic variational inequalities. Our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term. Our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities.

We provide a stochastic representation for a general class of viscous Hamilton-Jacobi (HJ) equations, which has convexity and superlinear nonlinearity in its gradient term, via a type of backward stochastic differential equation (BSDE) with constraint in the martingale part. We compare our result with the classical representation in terms of (super)quadratic BSDE, and show in particular that existence of a solution to the viscous HJ equation can be obtained under more general growth assumptions on the coefficients, including both unbounded diffusion coefficient and terminal data.

We consider a robust switching control problem. The controller only observes the evolution of the state process, and thus uses feedback (closed-loop) switching strategies, a non standard class of switching controls introduced in this paper. The adverse player (nature) chooses open-loop controls that represent the so-called Knightian uncertainty, i.e., misspecifications of the model. The (half) game switcher versus nature is then formulated as a two-step (robust) optimization problem. We develop the stochastic Perron method in this framework, and prove that it produces a viscosity sub and supersolution to a system of Hamilton-Jacobi-Bellman (HJB) variational inequalities, which envelope the value function. Together with a comparison principle, this characterizes the value function of the game as the unique viscosity solution to the HJB equation, and shows as a byproduct the dynamic programming principle for robust feedback switching control problem.

We analyze a stochastic optimal control problem, where the state process follows a McKean-Vlasov dynamics and the diffusion coefficient can be degenerate. We prove that its value function V admits a nonlinear Feynman-Kac representation in terms of a class of forward-backward stochastic differential equations, with an autonomous forward process. We exploit this probabilistic representation to rigorously prove the dynamic programming principle (DPP) for V. The Feynman-Kac representation we obtain has an important role beyond its intermediary role in obtaining our main result: in fact it would be useful in developing probabilistic numerical schemes for V. The DPP is important in obtaining a characterization of the value function as a solution of a non-linear partial differential equation (the so-called Hamilton-Jacobi-Belman equation), in this case on the Wasserstein space of measures. We should note that the usual way of solving these equations is through the Pontryagin maximum principle, which requires some convexity assumptions. There were attempts in using the dynamic programming approach before, but these works assumed a priori that the controls were of Markovian feedback type, which helps write the problem only in terms of the distribution of the state process (and the control problem becomes a deterministic problem). In this paper, we will consider open-loop controls and derive the dynamic programming principle in this most general case. In order to obtain the Feynman-Kac representation and the randomized dynamic programming principle, we implement the so-called randomization method, which consists in formulating a new McKean-Vlasov control problem, expressed in weak form taking the supremum over a family of equivalent probability measures. One of the main results of the paper is the proof that this latter control problem has the same value function V of the original control problem.

Path Dependent PDE's (PPDE's) are natural objects to study when one deals with non Markovian models. Recently, after the introduction (see [12]) of the so-called pathwise (or functional or Dupire) calculus, various papers have been devoted to study the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [18]) and viscosity solutions (see e.g. [13]), in the case of finite dimensional underlying space. In this paper, motivated by the study of models driven by path dependent stochastic PDE's, we give a first well-posedness result for viscosity solutions of PPDE's when the underlying space is an infinite dimensional Hilbert space. The proof requires a substantial modification of the approach followed in the finite dimensional case. We also observe that, differently from the finite dimensional case, our well-posedness result, even in the Markovian case, apply to equations which cannot be treated, up to now, with the known theory of viscosity solutions.

We study a class of reflected backward stochastic differential equations with nonpositive jumps and upper barrier. Existence and uniqueness of a minimal solution is proved by a double pena\-lization approach under regularity assumptions on the obstacle. In a suitable regime switching diffusion framework, we show the connection between our class of BSDEs and fully nonlinear variational inequalities. Our BSDE representation provides in particular a Feynman-Kac type formula for PDEs associated to general zero-sum stochastic differential controller-and-stopper games, where control affect both drift and diffusion term, and the diffusion coefficient can be degenerate. Moreover, we state a dual game formula of this BSDE minimal solution involving equivalent change of probability measures, and discount processes. This gives in particular a new representation for zero-sum stochastic differential controller-and-stopper games.

We are interested in stochastic control problems coming from mathematical finance and, in particular, related to model uncertainty, where the uncertainty affects both volatility and intensity. This kind of stochastic control problems is associated to a fully nonlinear integro-partial differential equation, which has the peculiarity that the measure $(\lambda(a,\cdot))_a$ characterizing the jump part is not fixed but depends on a parameter $a$ which lives in a compact set $A$ of some Euclidean space $\R^q$. We do not assume that the family $(\lambda(a,\cdot))_a$ is dominated. Moreover, the diffusive part can be degenerate. Our aim is to give a BSDE representation, known as nonlinear Feynman-Kac formula, for the value function associated to these control problems. For this reason, we introduce a class of backward stochastic differential equations with jumps and partially constrained diffusive part. We look for the minimal solution to this family of BSDEs, for which we prove uniqueness and existence by means of a penalization argument. We then show that the minimal solution to our BSDE provides the unique viscosity solution to our fully nonlinear integro-partial differential equation.