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Register a new userPath-dependent Hamilton-Jacobi-Bellman equation: Uniqueness of Crandall-Lions viscosity solutions
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Authors
Andrea Cosso
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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 Ekelands variational principle on the space of continuous paths.
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We address our interest to the development of a theory of viscosity solutions {`a} 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 Ekelands 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 Fr{e}chet) 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 It{^o} formula under general assumptions, extending earlier results in the literature.
In quantitative genetics, viscosity solutions of Hamilton-Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function I(t) that arises as the counterpart of a non-negativity constraint on the solution at each time. Although the uniqueness of viscosity solutions is known for many variants of Hamilton-Jacobi equations, the uniqueness for this particular type of constrained problem was not resolved, except in a few particular cases. Here, we provide a general answer to the uniqueness problem, based on three main assumptions: convexity of the Hamiltonian function H(I, x, p) with respect to p, monotonicity of H with respect to I, and BV regularity of I(t).
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 {`a} 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 Ekelands variational principle on the Wasserstein space.
The non-exponential Schilder-type theorem in Backhoff-Veraguas, Lacker and Tangpi [Ann. Appl. Probab., 30 (2020), pp. 1321-1367] is expressed as a convergence result for path-dependent partial differential equations with appropriate notions of generalized solutions. This entails a non-Markovian counterpart to the vanishing viscosity method. We show uniqueness of maximal subsolutions for path-dependent viscous Hamilton-Jacobi equations related to convex super-quadratic backward stochastic differential equations. We establish well-posedness for the Hamilton-Jacobi-Bellman equation associated to a Bolza problem of the calculus of variations with path-dependent terminal cost. In particular, uniqueness among lower semi-continuous solutions holds and state constraints are admitted.
For any compact connected manifold $M$, we consider the generalized contact Hamiltonian $H(x,p,u)$ defined on $T^*Mtimesmathbb R$ which is conex in $p$ and monotonically increasing in $u$. Let $u_epsilon^-:Mrightarrowmathbb R$ be the viscosity solution of the parametrized contact Hamilton-Jacobi equation [ H(x,partial_x u_epsilon^-(x),epsilon u_epsilon^-(x))=c(H) ] with $c(H)$ being the Ma~ne Critical Value. We prove that $u_epsilon^-$ converges uniformly, as $epsilonrightarrow 0_+$, to a specfic viscosity solution $u_0^-$ of the critical equation [ H(x,partial_x u_0^-(x),0)=c(H) ] which can be characterized as a minimal combination of associated Peierls barrier functions.
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