Continuous Viscosity Solution Research Articles

Backward reachability approach to state-constrained stochastic optimal control problem for jump-diffusion models

AbstractIn this paper, we consider the stochastic optimal control problem for jump-diffusion models with state constraints. In general, the value function of such problems is the discontinuous viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation since the regularity cannot be guaranteed at the boundary of the state constraint. By adapting the stochastic target theory, we obtain an equivalent representation of the original value function by means of the backward reachable set. We then show that this backward reachable can be characterized by the zero-level set of the auxiliary value function for the unconstrained stochastic control problem, which includes two additional unbounded controls as a consequence of the martingale representation theorem. We prove that the auxiliary value function is the unique continuous viscosity solution of the associated HJB equation, which is the second-order nonlinear partial integro-differential equation (PIDE). Our paper provides an explicit way to characterize the original (possibly discontinuous) value function as a zero-level set of the continuous solution of the auxiliary HJB equation. The proof of the existence and uniqueness requires a new technique due to the unbounded control sets and the presence of the singularity of the corresponding Lévy measure in the nonlocal operator of the HJB equation.

A note on the density of the partial regularity result in the class of viscosity solutions

We establish the density of the partial regularity result in the class of continuous viscosity solutions. Given a fully nonlinear equation, we prove the existence of a sequence entitled to the partial regularity result, approximating its solutions. Distinct conditions on the operator driving the equation lead to density in different topologies. Our findings include applications to inhomogeneous problems, with variable-coefficients models.

Optimal sizing of the sediment replenishment capacity based on robust ergodic control of subordinator-driven dynamics

A countermeasure against sediment depletion in rivers is sand replenishment. The cost-efficient sizing of its capacity is critical for sustainable environmental management. However, only temporally discrete replenishment is possible, and sediment transport is intermittent and uncertain. In this paper, we approach this new design problem using a robust ergodic control. First, a stochastic differential equation driven by a (tempered) stable subordinator describes the sediment storage dynamics. Sand replenishment is considered an impulse control, wherein observation/replenishment chances arrive at Poisson jump times. The subordinator is ambiguous since the decision-maker does not precisely know its Lévy measure. Using a dynamic risk measure that balances replenishment cost, disutility triggered by sediment depletion, and penalization of ambiguity, we derived the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation to determine the optimal storage capacity. We obtained that the HJBI equation has an analytical solution when a stable subordinator is used. Furthermore, we prove it admits a unique continuous viscosity solution and that it is optimal and applied the proposed model to real data.

Stochastic optimal control in infinite dimensions with state constraints

We consider the state-constrained stochastic optimal control problem in infinite-dimensional separable Hilbert spaces, where the state process is driven by the Q-Wiener process and the (possibly unbounded) linear operator. By applying the stochastic target theory and the backward reachability approach, we show that the original (possibly discontinuous) value function can be represented by the zero-level set of the auxiliary (continuous) value function. The auxiliary value function is obtained from the penalized unconstrained stochastic control problem (in infinite dimensions) that includes an additional control variable as a consequence of the (infinite-dimensional) martingale representation theorem. We then prove that the auxiliary value function is a unique (continuous) viscosity solution to the associated Hamilton–Jacobi–Bellman (HJB) equation in infinite dimensions. Note that the viscosity analysis developed in our paper generalizes that presented in the existing literature, since the corresponding infinite-dimensional HJB equation includes an additional operator-valued control variable in the Hamiltonian maximization and depends on an additional initial state variable.

Viscosity Solution of System of Integro-Partial Differential Equations with Interconnected Obstacles of Non-local Type Without Monotonicity Conditions

In this paper, we study a system of second order integro-partial differential equations with interconnected obstacles with non-local terms, related to an optimal switching problem with the jump-diffusion model. Getting rid of the monotonicity condition on the generators with respect to the jump component, we construct a continuous viscosity solution which is unique in the class of functions with polynomial growth. In our study, the main tool is the associated of reflected backward stochastic differential equations with jumps with interconnected obstacles for which we show the existence of a unique Markovian solution.

Differential Games with Incomplete Information and with Signal Revealing: The Symmetric Case

In this paper, we investigate the existence of value for a two-person zero-sum differential game with symmetric incomplete information and with signal revealing. Before the game begins, the initial state of the dynamic is chosen randomly among a finite number of points in $$\mathbb {R}^n$$ , while both players have only a probabilistic knowledge of the chosen initial state. During the game, if the system reaches a fixed closed target set K, the current state of the system at the hitting time is revealed to both players. We prove in this paper that this game has a value and its value function is the unique bounded continuous viscosity solution of a suitable Hamilton–Jacobi–Isaacs equation.

Continuous solution for a non-linear eikonal system

<p style='text-indent:20px;'>In this work, we are dealing with a non-linear eikonal system in one dimensional space that describes the evolution of interfaces moving with non-signed strongly coupled velocities. We prove a global existence result in the framework of continuous viscosity solution. The approach is made by adding a viscosity term and passing to the limit for vanishing viscosity, relying on a new gradient entropy and <inline-formula><tex-math id="M1">\begin{document}$ BV $\end{document}</tex-math></inline-formula> estimates. A uniqueness result is also proved through a comparison principle property.</p>

Hamilton–Jacobi–Bellman Equation for Control Systems With Friction

This article proposes a new framework for modeling control systems, in which a dynamic friction occurs. The model consists of a controlled differential inclusion with a discontinuous right-hand side, which still preserves existence and uniqueness of the solution for each given input function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$u(t)$</tex-math></inline-formula> . Under general hypotheses, we are able to derive the Hamilton–Jacobi–Bellman equation for the related free-time-optimal control problem and to characterize the value function as the unique locally Lipschitz continuous viscosity solution.

Variational approach to the asymptotic mean-value property for the [formula omitted]-Laplacian on Carnot groups

Let 1≤p≤∞. We provide an asymptotic characterization of continuous viscosity solutions u of the normalized p-Laplacian ΔpGNu=0 in any Carnot group G.

Continuous Viscosity Solutions to Linear-Quadratic Stochastic Control Problems with Singular Terminal State Constraint

This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a linear-quadratic stochastic control problem with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous viscosity sub- and supersolutions for PDEs with singular terminal value. Continuity of the viscosity solution is enough to carry out the verification argument.

Optimal Market Making under Partial Information with General Intensities

ABSTRACT Starting from the Avellaneda–Stoikov framework, we consider a market maker who wants to optimally set bid/ask quotes over a finite time horizon, to maximize her expected utility. The intensities of the orders she receives depend not only on the spreads she quotes but also on unobservable factors modelled by a hidden Markov chain. We tackle this stochastic control problem under partial information with a model that unifies and generalizes many existing ones under full information, combining several risk metrics and constraints, and using general decreasing intensity functionals. We use stochastic filtering, control and piecewise-deterministic Markov processes theory, to reduce the dimensionality of the problem and characterize the reduced value function as the unique continuous viscosity solution of its dynamic programming equation. We then solve the analogous full information problem and compare the results numerically through a concrete example. We show that the optimal full information spreads are biased when the exact market regime is unknown, and the market maker needs to adjust for additional regime uncertainty in terms of P&L sensitivity and observed order flow volatility. This effect becomes higher, the longer the waiting time in between orders.

Hölder behavior of viscosity solutions of some fully nonlinear equations in the Heisenberg group

In this paper we prove the $C^{0,\alpha}$ regularity of bounded and uniformly continuous viscosity solutions of some degenerate fully nonlinear equations in the first Heisenberg group.

Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation

In this paper, we investigate positive viscosity solutions of a third degree homogeneous parabolic equation \begin{document}$ u^{2}u_{t} = \Delta_{\infty}u $\end{document} . We prove a comparison principle, existence and uniqueness of continuous positive viscosity solutions.

Optimal liquidation under partial information with price impact

We study the optimal liquidation problem in a market model where the bid price follows a geometric pure jump process whose local characteristics are driven by an unobservable finite-state Markov chain and by the liquidation rate. This model is consistent with stylized facts of high frequency data such as the discrete nature of tick data and the clustering in the order flow. We include both temporary and permanent effects into our analysis. We use stochastic filtering to reduce the optimal liquidation problem to an equivalent optimization problem under complete information. This leads to a stochastic control problem for piecewise deterministic Markov processes (PDMPs). We carry out a detailed mathematical analysis of this problem. In particular, we derive the optimality equation for the value function, we characterize the value function as continuous viscosity solution of the associated dynamic programming equation, and we prove a novel comparison result. The paper concludes with numerical results illustrating the impact of partial information and price impact on the value function and on the optimal liquidation rate.

Viscosity solutions of systems of PDEs with interconnected obstacles and switching problem without monotonicity condition

We show the existence and uniqueness of a continuous viscosity solution of a system of partial differential equations (PDEs for short) without assuming the usual monotonicity conditions on the driver function as in Hamad\`ene and Morlais's article \cite{hamadene2013viscosity}. Our method strongly relies on the link between PDEs and reflected backward stochastic differential equations with interconnected obstacles for which we already know that the solution exists and is unique for general drivers.

Reflected Backward Stochastic Differential Equation with Jumps and Viscosity Solution of Second Order Integro-Differential Equation Without Monotonicity Condition: Case with the Measure of Lévy Infinite

We consider the problem of viscosity solution of integro-partial differential equation(IPDE in short) with one obstacle via the solution of reflected backward stochastic differential equations(RBSDE in short) with jumps. We show the existence and uniqueness of a continuous viscosity solution of equation with non local terms, if the generator is not monotonous and Levy’s measure is infinite.

Uniqueness of Viscosity Mean Curvature Flow Solution in Two Sub-Riemannian Structures

We provide a uniqueness result for a class of viscosity solutions to sub-Riemannian mean curvature flows. In a sub-Riemannian setting, uniqueness cannot be deduced by the comparison principle, which is known only for graphs and for radially symmetry surfaces. Here we use a definition of continuous viscosity solutions of sub-Riemannian mean curvature flows motivated from a regularized Riemannian approximation of the flow. With this definition, we prove that any continuous viscosity solution of the equation is a limit of a sequence of solutions of Riemannian flow and obtain as a consequence uniqueness and the comparison principle. The results are provided in the settings of both 3-dimensional rototranslation group ${SE}(2)$ and Carnot groups of step 2, which are particularly important due to their relation to the surface completion problem of a model of the visual cortex.

Infinite Horizon Differential Games with Asymmetric Information

This article is concerned with the existence of a value for a zero-sum differential game with an asymmetric information on initial state with an infinite horizon running cost. Before the game begins, an initial state is chosen randomly from a finite set and each player receives a private signal generated by the chosen initial state. The main result is that the game has a value with random non-anticipative strategies with delay and that its value function can be characterized as the unique bounded continuous viscosity solution of a suitable Hamilton–Jacobi–Isaacs equation.

Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations

We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for operators of the form $\nabla^2 \psi + L(x,\psi,\nabla \psi)$ which are non-decreasing in $\psi$.

Existence of Value for a Differential Game with Incomplete Information and Revealing

This article is devoted to the existence of value for a zero-sum differential game with incomplete information on initial state with partial revealing of information and with a running cost. Before the game begins, an initial state is chosen from among several states, and both players do not know this initial state until the dynamic system reaches a fixed target during the game. The main result is the existence of value for the game defined with nonanticipative strategies with delay. Moreover, the value is the unique bounded continuous viscosity solution of a suitable Hamilton--Jacobi--Isaacs equation.