Bellman System Research Articles

Time-consistent mean–variance proportional reinsurance and investment problem in a defaultable market

In this paper, we consider an optimal time-consistent reinsurance-investment problem incorporating a defaultable security for a mean–variance insurer under a constant elasticity of variance (CEV) model. In our model, the insurer’s surplus process is described by a jump-diffusion risk model, the insurer can purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset, a defaultable bond and a risky asset whose price process is assumed to follow a CEV model. Using a game theoretic approach, we establish the extended Hamilton–Jacobi–Bellman system for the post-default case and the pre-default case, respectively. Furthermore, we obtain the closed-from expressions for the time-consistent reinsurance-investment strategy and the corresponding value function in both cases. Finally, we provide numerical examples to illustrate the impacts of model parameters on the optimal time-consistent strategy.

Equilibrium Investment Strategy for a DC Plan With Partial Information and Mean–Variance Criterion

This paper studies a mean–variance portfolio selection problem with partial information for a defined-contribution (DC) pension scheme. We assume that the DC pension plan member can only observe the price of the risky asset but not the appreciation rate of it in the financial market. Moreover, inflation risk and salary risk are taken into account in our model. First, by virtue of the filtering theory, we transform the partially observable mean–variance portfolio selection problem to a completely observable one. Then, to look for an equilibrium investment strategy, we formulate and tackle the mean–variance problem within a game theoretic framework. By solving an extended Hamilton–Jacobi–Bellman (HJB) system of equations, closed-form expressions of the equilibrium investment strategy and corresponding equilibrium value function with partial information are derived for the DC pension plan. In addition, some numerical illustrations are provided to show the effects of parameters on the derived equilibrium investment strategies and the efficient frontier.

Parabolic Bellman-Systems with Mean Field Dependence

We consider the necessary conditions for Nash-points of Vlasov-McKean functionals $$\mathcal {J}^{i}[\mathbf{v}]=\int _{Q}mf^{i}(\cdot ,m,\mathbf{v})\,dx\,dt$$Ji[v]=?Qmfi(·,m,v)dxdt ($$i=1,...,N$$i=1,...,N). The corresponding payoffs $$f^{i}$$fi depend on the controls $$\mathbf{v}$$v and, in addition, on the field variable $$m=m(\mathbf{v})$$m=m(v). The necessary conditions lead to a coupled forward-backward system of nonlinear parabolic equations, motivated by stochastic differential games. The payoffs may have a critical nonlinearity of quadratic growth and any polynomial growth w.r.t. m is allowed as long as it can be dominated by the controls in a certain sense. We show existence and regularity of solutions to these mean-field-dependent Bellman systems by a purely analytical approach, no tools from stochastics are needed.

Time-consistent investment-reinsurance strategy for mean-variance insurers with a defaultable security

This paper considers an optimal investment and reinsurance problem involving a defaultable security for an insurer under the mean-variance criterion in a jump-diffusion risk model. The insurer can purchase proportional reinsurance or acquire new insurance business and invest in a financial market consisting of a risk-free asset, a stock and a defaultable bond. In particular, the correlation between the insurance risk model and the financial market is also considered. From a game theoretic perspective, the extended Hamilton–Jacobi–Bellman systems of equations are established for the post-default case and the pre-default case, respectively. In both cases, closed-form expressions for the optimal time-consistent investment-reinsurance strategies and the corresponding value functions are derived. Moreover, some properties of optimal strategies, value functions and efficient frontiers are discussed either analytically or numerically.

Robust equilibrium reinsurance-investment strategy for a mean–variance insurer in a model with jumps

This paper analyzes the equilibrium strategy of a robust optimal reinsurance-investment problem under the mean–variance criterion in a model with jumps for an ambiguity-averse insurer (AAI) who worries about model uncertainty. The AAI’s surplus process is assumed to follow the classical Cramér–Lundberg model, and the AAI is allowed to purchase proportional reinsurance or acquire new business and invest in a financial market to manage her risk. The financial market consists of a risk-free asset and a risky asset whose price process is described by a jump-diffusion model. By applying stochastic control theory, we establish the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations. Furthermore, we derive both the robust equilibrium reinsurance-investment strategy and the corresponding equilibrium value function by solving the extended HJB system of equations. In addition, some special cases of our model are provided, which show that our model and results extend some existing ones in the literature. Finally, the economic implications of our findings are illustrated, and utility losses from ignoring model uncertainty, jump risks and prohibiting reinsurance are analyzed using numerical examples.

Time-consistent investment strategy under partial information

This paper considers a mean–variance portfolio selection problem under partial information, that is, the investor can observe the risky asset price with random drift which is not directly observable in financial markets. Since the dynamic mean–variance portfolio selection problem is time inconsistent, to seek the time-consistent investment strategy, the optimization problem is formulated and tackled in a game theoretic framework. Closed-form expressions of the equilibrium investment strategy and the corresponding equilibrium value function under partial information are derived by solving an extended Hamilton–Jacobi–Bellman system of equations. In addition, the results are also given under complete information, which are need for the partial information case. Furthermore, some numerical examples are presented to illustrate the derived equilibrium investment strategies and numerical sensitivity analysis is provided.

Systems of integro-PDEs with interconnected obstacles and multi-modes switching problem driven by Lévy process

In this paper we show existence and uniqueness of the solution in viscosity sense for a system of nonlinear m variational integral-partial differential equations with interconnected obstacles whose coefficients $${(f_i)_{i=1,\ldots, m}}$$ depend on $${(u_j)_{j=1,\ldots,m}}$$ . From the probabilistic point of view, this system is related to optimal stochastic switching problem when the noise is driven by a Levy process. The switching costs depend on (t, x). As a by-product of the main result we obtain that the value function of the switching problem is continuous and unique solution of its associated Hamilton–Jacobi–Bellman system of equations. The main tool we used is the notion of systems of reflected BSDEs with oblique reflection driven by a Levy process.

A PDE Approach to Data-Driven Sub-Riemannian Geodesics in $SE$(2)

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external cost $\mathcal{C}:SE(2) \to [\delta,1]$, $\delta>0$, computed from image data. The method consists of a first step where an SR-distance map is computed as a viscosity solution of a Hamilton--Jacobi--Bellman system derived via Pontryagin's maximum principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For $\mathcal{C}=1$ we show that our method produces the global minimizers. Comparison with exact solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics. We present numerical computations of Maxwell points and cusp points, which we again verify for the uniform cost case $\mathcal{C}=1$. Regarding image analysis applications, trackings of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the SR-geometry.

OPTIMAL TIME-CONSISTENT PORTFOLIO AND CONTRIBUTION SELECTION FOR DEFINED BENEFIT PENSION SCHEMES UNDER MEAN–VARIANCE CRITERION

AbstractWe investigate two mean–variance optimization problems for a single cohort of workers in an accumulation phase of a defined benefit pension scheme. Since the mortality intensity evolves as a general Markov diffusion process, the liability is random. The fund manager aims to cover this uncertain liability via controlling the asset allocation strategy and the contribution rate. In order to have a more realistic model, we study the case when the risk aversion depends dynamically on current wealth. By solving an extended Hamilton–Jacobi–Bellman system, we obtain analytical solutions for the equilibrium strategies and value function which depend on both current wealth and mortality intensity. Moreover, results for the constant risk aversion are presented as special cases of our models.

Time-consistent investment and reinsurance strategies for mean–variance insurers with jumps

This paper studies an optimal investment and reinsurance problem incorporating jumps for mean–variance insurers within a game theoretic framework and aims to seek the corresponding time-consistent strategies. Specially, the insurers are allowed to purchase proportional reinsurance, acquire new business and invest in a financial market, where the surplus of the insurers is assumed to follow a jump–diffusion model and the financial market consists of one risk-free asset and one risky asset whose price process is modeled by a geometric Lévy process. By solving an extended Hamilton–Jacobi–Bellman system, the closed-form expressions for the time-consistent investment and reinsurance strategies and the optimal value function are derived. Moreover, some special cases of our model and results are presented, and some numerical illustrations and sensitivity analysis for our results are provided.

Existence and compactness for weak solutions to Bellman systems with critical growth

We deal with nonlinear elliptic and parabolic systems that are the Bellman systems associated to stochastic differential games as a main motivation. We establish the existence of weak solutions in any dimension for an arbitrary number of equations (players). The method is based on using a renormalized sub- and super-solution technique . The main novelty consists in the new structure conditions on the critical growth terms with allow us to show weak solvability for Bellman systems to certain classes of stochastic differential games.

ON NONLINEAR ELLIPTIC BELLMAN SYSTEMS FOR A CLASS OF STOCHASTIC DIFFERENTIAL GAMES IN ARBITRARY DIMENSION

We consider nonlinear elliptic Bellman systems which arise in the theory of stochastic differential games. The right-hand sides of the equations (which are called Hamiltonians) may have quadratic growth with respect to the gradient of the unknowns. Under certain assumptions on the Hamiltonians, that are satisfied for many types of stochastic games, we establish the existence of a regular solution. This partially generalizes recent works of Bensoussan, Frehse and Vogelgesang from two-dimensional setting onto arbitrary dimension. The main novelty of the paper consists of introducing a new (semi-continuity) method for obtaining the continuity of the solution and the corresponding estimates.

Characterization of Markovian equilibria in a class of differential games

We consider an alternative method to the classical one for the determination of Markov perfect Nash equilibria. The approach shown in the paper is based on the study of a quasi-linear system of partial differential equations instead of the Hamilton–Jacobi–Bellman system. The simpler structure of the former allows us to determine existence and uniqueness of Nash equilibria in non-renewable resource games under some assumptions. When closed-form solutions are not available, we give a method to obtain numerical solutions.

On Bellman systems without zero order term in the context of risk sensitive differential games

On Bellman systems without zero order term in the context of risk sensitive differential games

Ergodic Bellman systems for stochastic games in arbitrary dimension

Stochastic games with cost functionals J ( i ) ρ, x ( v ) = E ∫ ∞ 0 e – ρ t l i ( y, v ) d t , i = 1, 2 with controls v = v 1 , v 2 and state y ( t ) with y (0) = x are considered. Each player wants to minimize his (her) cost functional. E denotes the expected value and the state variables y are coupled with the controls v via a stochastic differential equation with initial value x . The corresponding Bellman system, which is used for the calculation of feedback controls v = v ( y ) and the solvability of the game, leads to a class of diagonal second-order nonlinear elliptic systems, which also occur in other branches of analysis. Their behaviour concerning existence and regularity of solutions is, despite many positive results, not yet well understood, even in the case where the l i , are simple quadratic functions. The objective of this paper is to give new insight to these questions for fixed ρ > 0, and, primarily, to analyse the limiting behaviour as the discount ρ → 0. We find that the modified solutions of the stochastic games converge, for subsequences, to the solution of the so-called ergodic Bellman equation and that the average cost converges. A former restriction of the space dimension has been removed. A reasonable class of quadratic integrands may be treated. More specifically, we consider the Bellman systems of equations – ∆ z + λ = H ( x, Dz ), where the space variable x belongs to a periodic cube (for the sake of simplifying the presentation). They are shown to have smooth solutions. If u ρ is the solution of – ∆ u ρ + ρ u ρ = H ( x, Du ρ ) then the convergence of u ρ — ῡ ρ to z , as ρ tends to 0, is established. The conditions on H are such that some quadratic growth in Du is allowed.