Unique Viscosity Solution Research Articles

Equivalent Extensions of Hamilton–Jacobi–Bellman Equations on Hypersurfaces

We present a new formulation for the computation of solutions of a class of Hamilton Jacobi Bellman (HJB) equations on closed smooth surfaces of co-dimension one. For the class of equations considered in this paper, the viscosity solution of the HJB equation is equivalent to the value function of a corresponding optimal control problem. In this work, we extend the optimal control problem given on the surface to an equivalent one defined in a sufficiently thin narrow band of the co-dimensional one surface. The extension is done appropriately so that the corresponding HJB equation, in the narrow band, has a unique viscosity solution which is identical to the constant normal extension of the value function of the original optimal control problem. With this framework, one can easily use existing (high order) numerical methods developed on Cartesian grids to solve HJB equations on surfaces, with a computational cost that scales with the dimension of the surfaces. This framework also provides a systematic way for solving HJB equations on the unstructured point clouds that are sampled from the surface.

A Viscosity Solution Method for Optimal Stopping Problems with Regime Switching

We employ the viscosity solution technique to analyze optimal stopping problems with regime switching. To be specific, we show the viscosity property of value functions with the help of dynamic programming, and more importantly, provide a mild and verifiable condition and an available bound that both can guarantee the uniqueness of viscosity solutions.

Entire Solutions of Cauchy Problem for Parabolic Monge–Ampère Equations

Abstract In this paper, we study the Cauchy problem of the parabolic Monge–Ampère equation - u t ⁢ det ⁡ D 2 ⁢ u = f ⁢ ( x , t ) -u_{t}\det D^{2}u=f(x,t) and obtain the existence and uniqueness of viscosity solutions with asymptotic behavior by using the Perron method.

Reflected Backward Stochastic Differential Equation with Rank-Based Data

In this paper, we study reflected backward stochastic differential equation (reflected BSDE) with rank-based data in a Markovian framework; that is, the solution to the reflected BSDE is above a prescribed boundary process in a minimal fashion and the generator and terminal value of the reflected BSDE depend on the solution of another stochastic differential equation (SDE) with rank-based drift and diffusion coefficients. We derive regularity properties of the solution to such reflected BSDE and show that the solution at the initial starting time t and position x, which is a deterministic function, is the unique viscosity solution to some obstacle problem (or variational inequality) for the corresponding parabolic partial differential equation.

Singular elliptic equations with directional diffusion

We investigate conditions for the existence and uniqueness of viscosity solutions of the Dirichlet problem for a degenerate elliptic equation describing a stationary diffusion, which may take place in a partial number of spatial directions, with a possibly singular reaction term.

Optimal dynamic reinsurance strategies in multidimensional portfolio

The present paper addresses the issue of choosing an optimal dynamic reinsurance policy, which is state-dependent, for an insurance company that operates under multiple insurance business lines. For each line, the Cramer–Landberg model is adopted for the risk process and one of the contracts such as Proportional reinsurance, excess-of-loss reinsurance (XL) and limited XL reinsurance (LXL) is intended for transferring a portion of the risk to the reinsurance. In the optimization method used in this paper, the survival function is maximized with respect to to the dynamic reinsurance strategies. The optimal survival function is characterized as the unique nondecreasing viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation with limit one at infinity. The finite difference method (FDM) has been utilized for the numerical solution of the optimal survival function and optimal dynamic reinsurance strategies and the proof for the convergence of the numerical solution to the survival probability function is provided. The findings of this paper provide insights for the insurance companies as such that based upon the lines in which they are operating, they can choose a vector of the optimal dynamic reinsurance strategies and consequently transfer some part of their risks to several reinsurers. With the numerical examples, the significance of the elicited results in reducing the probability of ruin is demonstrated in comparison with the previous findings.

The Dynkin game with regime switching and applications to pricing game options

This paper is concerned with the Dynkin game (a zero-sum optimal stopping game). The dynamic of the system is modeled by a regime switching diffusion, in which the regime switching mechanism provides the structural changes of the random environment. The goal is to find a saddle point for the payoff functional up to one of the players exiting the game. Taking advantage of the method of penalization and the dynamic programming principle, the value function of the game problem is shown to be the unique viscosity solution to the associated variational inequalities. We also consider a financial example of pricing game option under a regime switching market. Both optimal stopping rules for the buyer and the seller and fair price of the option are numerically demonstrated in this example. All the results are markedly different from the traditional cases without regime switching.

A Bank Salvage Model by Impulse Stochastic Controls

The present paper is devoted to the study of a bank salvage model with a finite time horizon that is subjected to stochastic impulse controls. In our model, the bank’s default time is a completely inaccessible random quantity generating its own filtration, then reflecting the unpredictability of the event itself. In this framework the main goal is to minimize the total cost of the central controller, which can inject capitals to save the bank from default. We address the latter task, showing that the corresponding quasi-variational inequality (QVI) admits a unique viscosity solution—Lipschitz continuous in space and Hölder continuous in time. Furthermore, under mild assumptions on the dynamics the smooth-fit W l o c ( 1 , 2 ) , p property is achieved for any 1 < p < + ∞ .

Optimal Control of Infinite-Dimensional Piecewise Deterministic Markov Processes: A BSDE Approach. Application to the Control of an Excitable Cell Membrane

In this paper we consider the optimal control of Hilbert space-valued infinite-dimensional Piecewise Deterministic Markov Processes (PDMP) and we prove that the corresponding value function can be represented via a Feynman–Kac type formula through the solution of a constrained Backward Stochastic Differential Equation. A fundamental step consists in showing that the corresponding integro-differential Hamilton–Jacobi–Bellman equation has a unique viscosity solution, by proving a suitable comparison theorem. We apply our results to the control of a PDMP Hodgkin-Huxley model with spatial component.

Nonlocal Averages in Space and Time Given by Medians and the Mean Curvature Flow

We deal with existence and uniqueness for continuous solutions to the equation u(x,t)={\textrm{med}}_{(y,s)\in J(x,t)} \, {u}(y,s) for (x,t) \in \mathbb R^n\times (0,\infty) with a prescribed datum u (x,t)=f(x) for (x,t) \in \mathbb R^n \times (-\infty, 0] . Here {\textrm{med}}_{(y,s)\in J(x,t)} \, {u}(y,s) stands for the median value of u in the set J(x,t)=B(x,R) \times [t -(\delta + \gamma),t -\delta] . In addition we show that when we consider the family of sets J_r(x,t) := B(x,rR) \times [t - r^2(\delta +\gamma),t - r^2\delta] then the corresponding solutions u_r converge as r\to 0 to the unique viscosity solution to the local degenerate parabolic PDE, u_t(x,t)=C \triangle_{V}u(x,t) , where V is the orthogonal subspace to \nabla u(x,t) and C is a positive constant. This PDE turns out to be the equation that describes the mean curvature flow in its level sets formulation.

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 reinsurance under the mean–variance premium principle to minimize the probability of ruin

We consider the problem of minimizing the probability of ruin by purchasing reinsurance whose premium is computed according to the mean–variance premium principle, a combination of the expected-value and variance premium principles. We derive closed-form expressions of the optimal reinsurance strategy and the corresponding minimum probability of ruin under the diffusion approximation of the classical Cramér–Lundberg risk process perturbed by a diffusion. We find an explicit expression for the reinsurance strategy that maximizes the adjustment coefficient for the classical risk process perturbed by a diffusion. Also, for this risk process, we use stochastic Perron’s method to prove that the minimum probability of ruin is the unique viscosity solution of its Hamilton–Jacobi–Bellman equation with appropriate boundary conditions. Finally, we prove that, under an appropriate scaling of the classical risk process, the minimum probability of ruin converges to the minimum probability of ruin under the diffusion approximation.

Optimal dividend policy in an insurance company with contagious arrivals of claims

In this paper we consider the optimal dividend problem for an insurance company whose surplus follows a classical Cramer-Lundberg process with a feature of self-exciting. A Hawkes process is applied so that the occurrence of a jump in the claims triggers more sequent jumps. We show that the optimal value function is a unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation with a given boundary condition and declare its concavity. We introduce a barrier curve strategy and verify its optimality. Finally, some numerical results are exhibited.

Hamilton\u2013Jacobi equations for optimal control on multidimensional junctions with entry costs

We consider an infinite horizon control problem for dynamics constrained to remain on a multidimensional junction with entry costs. We derive the associated system of Hamilton–Jacobi equations (HJ), prove the comparison principle and that the value function of the optimal control problem is the unique viscosity solution of the HJ system. This is done under the usual strong controllability assumption and also under a weaker condition, coined ‘moderate controllability assumption’.

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.

Optimal control of multiagent systems in the Wasserstein space

This paper concerns a class of optimal control problems, where a central planner aims to control a multi-agent system in $${\mathbb {R}}^d$$ in order to minimize a certain cost of Bolza type. At every time and for each agent, the set of admissible velocities, describing his/her underlying microscopic dynamics, depends both on his/her position, and on the configuration of all the other agents at the same time. So the problem is naturally stated in the space of probability measures on $${\mathbb {R}}^d$$ equipped with the Wasserstein distance. The main result of the paper gives a new characterization of the value function as the unique viscosity solution of a first order partial differential equation. We introduce and discuss several equivalent formulations of the concept of viscosity solutions in the Wasserstein spaces suitable for obtaining a comparison principle of the Hamilton Jacobi Bellman equation associated with the above control problem.

Two-player zero-sum stochastic differential games with regime switching

This paper is concerned with the two-player zero-sum stochastic differential game in a regime switching model with an infinite horizon. The state of the system is characterized by a number of diffusions coupled by a continuous-time finite-state Markov chain. Based on the dynamic programming principle (DPP), the lower and upper value functions are shown to be the unique viscosity solutions of the associated lower and upper Hamilton–Jacobi–Bellman–Isaacs (HJBI) equations, respectively. Moreover, the lower and upper value functions coincide under the Isaacs’ condition, which implies that the game admits a value. All the proofs in this paper are markedly different from those for the case when there is no regime switching.

Minimizing the Discounted Probability of Exponential Parisian Ruin via Reinsurance

We study the problem of minimizing the discounted probability of exponential Parisian ruin, that is, the discounted probability that an insurer's surplus exhibits an excursion below zero in excess of an exponentially distributed clock. The insurer controls its surplus via reinsurance priced according to the mean-variance premium principle, as in Liang, Liang, and Young (2019). We consider the classical risk model and apply stochastic Perron's method, as introduced by Bayraktar and Sirbu (2012,2013,2014), to show that the minimum discounted probability of exponential Parisian ruin is the unique viscosity solution of its Hamilton-Jacobi-Bellman equation with boundary conditions at $\pm \infty$. A major difficulty in proving the comparison principle arises from the discontinuity of the Hamiltonian.

A probabilistic approach to quasilinear parabolic PDEs with obstacle and Neumann problems

In this paper, by a probabilistic approach we prove that there exists a unique viscosity solution to obstacle problems of quasilinear parabolic PDEs combined with Neumann boundary conditions and algebra equations. The existence and uniqueness for adapted solutions of fully coupled forward-backward stochastic differential equations with reflections play a crucial role. Compared with existing works, in our result the spatial variable of solutions of PDEs lives in a region without convexity constraints, the second order coefficient of PDEs depends on the gradient of the solution, and the required conditions for the coefficients are weaker.

Stochastic impulse control Problem with state and time dependent cost functions

<p style='text-indent:20px;'>We consider stochastic impulse control problems when the impulses cost functions depend on <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ x $\end{document}</tex-math></inline-formula>. We use the approximation scheme and viscosity solutions approach to show that the value function is a unique viscosity solution for the associated Hamilton-Jacobi-Bellman equation (HJB) partial differential equation (PDE) of stochastic impulse control problems.