Bang–bang-type Nash equilibrium point for Markovian nonzero-sum stochastic differential game

In this Note, we solve a nonzero-sum stochastic differential game (NZSDG) with bang–bang-type equilibrium controls by using backward stochastic differential equations (BSDEs). The generator is multi-dimensional and discontinuous with respect to z.     

A Fokker–Planck control framework for multidimensional stochastic processes

An efficient framework for the optimal control of probability density functions (PDFs) of multidimensional stochastic processes is presented. This framework is based on the Fokker–Planck equation that governs the time evolution of the PDF of stochastic processes and on tracking objectives of terminal configuration of the desired PDF. The corresponding optimization problems are formulated as a sequence of open-loop optimality systems in a receding-horizon control strategy. Many theoretical results concerning the forward and the optimal control problem are provided. In particular, it is shown that under appropriate assumptions the open-loop bilinear control function is unique. The resulting optimality system is discretized by the Chang–Cooper scheme that guarantees positivity of the forward solution. The effectiveness of the proposed computational framework is validated with a stochastic Lotka–Volterra model and a noised limit cycle model.     

Mean–variance target-based optimisation for defined contribution pension schemes in a stochastic framework

We solve a mean–variance optimisation problem in the accumulation phase of a defined contribution pension scheme. In a general multi-asset financial market with stochastic investment opportunities and stochastic contributions, we provide the general forms for the efficient frontier, the optimal investment strategy, and the ruin probability. We show that the mean–variance approach is equivalent to a “user-friendly” target-based optimisation problem which minimises a quadratic loss function, and provide implementation guidelines for the selection of the target. We show that the ruin probability can be kept under control through the choice of the target level. We find closed-form solutions for the special case of stochastic interest rate following the Vasiček (1977) dynamics, contributions following a geometric Brownian motion, and market consisting of cash, one bond and one stock. Numerical applications report the behaviour over time of optimal strategies and non-negative constrained strategies.     

Weak second-order stochastic Runge–Kutta methods for non-commutative stochastic differential equations

A new explicit stochastic Runge–Kutta scheme of weak order 2 is proposed for non-commutative stochastic differential equations (SDEs), which is derivative-free and which attains order 4 for ordinary differential equations. The scheme is directly applicable to Stratonovich SDEs and uses 2m-1$2m-1$ random variables for one step in the m-dimensional Wiener process case. It is compared with other derivative-free and weak second-order schemes in numerical experiments.     

An MPCC approach for stochastic Stackelberg–Nash–Cournot equilibrium

In this article, we investigate a Stochastic Stackelberg–Nash–Cournot Equilibrium problem by reformulating it as a Mathematical Program with Complementarity Constraints (MPCC). The complementarity constraints are further reformulated as a system of nonsmooth equations. We characterize the followers’ Nash–Cournot equilibria by studying the implicit solution of a system of equations. We outline numerical methods for the solution of a stochastic Stackelberg–Nash–Cournot Equilibrium problem with finite distribution of market demand scenarios and propose a discretization approach based on implicit numerical integration to deal with stochastic Stackelberg–Nash–Cournot Equilibrium problem with continuous distribution of demand scenarios. Finally, we discuss the two-leader Stochastic Stackelberg–Nash–Cournot Equilibrium problem.     

Runge–Kutta Lawson schemes for stochastic differential equations

In this paper, we present a framework to construct general stochastic Runge–Kutta Lawson schemes. We prove that the schemes inherit the consistency and convergence properties of the underlying Runge–Kutta scheme, and confirm this in some numerical experiments. We also investigate the stability properties of the methods and show for some examples, that the new schemes have improved stability properties compared to the underlying schemes.

     

On the stability of the equilibrium states for Hamiltonian dynamical systems arising in non–equilibrium thermodynamics

In this paper we consider a thermodynamic system with an internal state variable, and study the stability of its equilibrium states by exploiting the reduced entropy inequality. Remarkably, we derive a Hamiltonian dynamical system ruling the evolution of the system in a suitable thermodynamic phase space. The use of the Hamiltonian formalism allows us to prove the equivalence of the asymptotic stability at constant temperature, at constant entropy and at constant energy, thus extending some classical results by Coleman and Gurtin (J. Chem. Phys., 47, 597–613, 1967).     

On the pathwise exponential stability of non–linear stochastic partial differential equations

Sufficient conditions to get exponential stability for the sample paths (with probability one) of a non–linear monotone stochastic Partial Differential Equation are proved. In fact, we improve a stability criterion established in Chow [3] since, under the same hypotheses, we get pathwise exponential stability instead of stability of sample paths     

A maximum principle for Markov regime-switching forward–backward stochastic differential games and applications

In this paper, we present an optimal control problem for stochastic differential games under Markov regime-switching forward–backward stochastic differential equations with jumps. First, we prove a sufficient maximum principle for nonzero-sum stochastic differential games problems and obtain equilibrium point for such games. Second, we prove an equivalent maximum principle for nonzero-sum stochastic differential games. The zero-sum stochastic differential games equivalent maximum principle is then obtained as a corollary. We apply the obtained results to study a problem of robust utility maximization under a relative entropy penalty and to find optimal investment of an insurance firm under model uncertainty.     

Numerical solution of stochastic differential equations by second order Runge–Kutta methods

In this paper we propose the numerical solutions of stochastic initial value problems via random Runge–Kutta methods of the second order and mean square convergence of these methods is proved. A random mean value theorem is required and established. The concept of mean square modulus of continuity is also introduced. Expectation and variance of the approximating process are computed. Numerical examples show that the approximate solutions have a good degree of accuracy.     

A sufficient condition for the existence of a positive solution for a nonlinear fractional differential equation with the Riemann–Liouville derivative

In this work, by means of the fixed point theorem in a cone, we establish the existence result for a positive solution to a kind of boundary value problem for a nonlinear differential equation with a Riemann–Liouville fractional order derivative. An example illustrating our main result is given. Our results extend previous work in the area of boundary value problems of nonlinear fractional differential equations [C. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010) 1050–1055].     

Properties of ε-value functions for a two person differential game

We consider the problem of approximate minimax for the Bolza problem of optimal control. Starting from the method of dynamic programming (Bellman) we define the ?-value function to be the approximation for the value function being a solution to the Hamilton–Jacobi equation.     

The truncated Euler–Maruyama method for stochastic differential delay equations

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao (Appl. Math. Comput. 217, 5512–5524 2011), and the theory there showed that the Euler–Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in L ^p ) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao (J. Comput. Appl. Math. 290, 370–384 2015) to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.     

Existence of positive solution for a sixth–order differential system with variable parameters

This paper investigates the existence of positive solutions for a sixth-order differential system with three variable parameters. Using a fixed point theorem and an operator spectral theorem we give some new existence results.     

Does cooperation in auditing matter? A comparison of a non-cooperative and a cooperative game model of auditing

A game model of auditing including internal control investigation and substantive testing is analysed as a non-cooperative game. It is shown that in order for the presumed socially desirable outcome of high and honest effort by all to be obtained, it is necessary to adjust the cost structure to ameliorate the costs of not-qualifying erroneous accounts if the auditor can prove he or she has worked hard. Comparison with a cooperative game analysis of the model shows that there is a region of parameters where both cooperative and non-cooperative versions of the game lead to this socially desirable outcome. The significance of this result is that whilst society expects an ‘independent’ auditor not to cooperate with the auditee, the practical realities of auditing require a considerable degree of cooperation. This leads to an ‘expectations gap’ between what society expects and what actually happens, except in those cost regions where both the cooperative and non-cooperative versions of the audit game lead to the same solution.     

A Legendre-based computational method for solving a class of Itô stochastic delay differential equations

Numerical Algorithms - This paper provides a numerical method for solving a class of Itô stochastic delay differential equations (SDDEs). The method’s novelty is its use of the spectral...     

The Π-strategy in a differential game with linear control constraints

The concept of a linear constraint on the controls of the players in a differential game of pursuit, which, in a certain sense, generalizes both integral and geometrical constraints, is introduced. The optimal parallel pursuit strategy (Π-strategy) is constructed for the corresponding problem.     

Asset allocation for a DC pension fund with stochastic income and mortality risk: A multi-period mean–variance framework

This paper investigates an asset allocation problem for defined contribution pension funds with stochastic income and mortality risk under a multi-period mean–variance framework. Different from most studies in the literature where the expected utility is maximized or the risk measured by the quadratic mean deviation is minimized, we consider synthetically both to enhance the return and to control the risk by the mean–variance criterion. First, we obtain the analytical expressions for the efficient investment strategy and the efficient frontier by adopting the Lagrange dual theory, the state variable transformation technique and the stochastic optimal control method. Then, we discuss some special cases under our model. Finally, a numerical example is presented to illustrate the results obtained in this paper.