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This paper analyzes an augmented political business cycle model taking into account the effect of employment creation decisions by the ruling party jointly on inflation and growth. The objective is to maximize voter support in the next election that depends on the rate of unemployment as well as that of growth and inflation. We allow for stochasticity in the New Keynesian Phillips Curve model for the relationship between inflation and unemployment as well as in a benchmark labour productivity function for analyzing the growth rate. We provide explicit solution paths of the affine Markov control problem that results from our formulation. We also provide numerical illustrations with plausible parametric configurations to generate more insight into our model. Our results are broadly in line with the conventional wisdom of Phillips curve, with inflation and unemployment being roughly negatively related. The growth rate is, as expected, negatively related to the unemployment. We observe that, in the sequel, it is the lowering of the rate of inflation that provides support for the ruling party. Thus, electoral pressures drive the government to engage in cost control rather than productive investment (e.g., boosting employment or output growth). [ABSTRACT FROM AUTHOR]

We consider the optimal control problem for quasilinear stochastic systems with continuous time whose coefficients have a generally non-linear dependence on the program control. We establish sufficient conditions for a strong and weak relative minimum. We give examples of using the resulting conditions for constructing optimal control in a nonlinear onedimensional problem and in a two-dimensional linear problem with information constraints and analyze the possible results. [ABSTRACT FROM AUTHOR]

Existence and uniqueness of solutions to stochastic differential equation dX-diva(∇X)dt=∑j=1N(bj·∇X)∘dβj in (0,T)×O; X(0,ξ)=x(ξ), ξ∈O, X=0 on (0,T)×∂O is studied. Here O is a bounded and open domain of Rd, d≥1, {bj} is a divergence free vector field, a:[0,T]×O×Rd→Rd is a continuous and monotone mapping of subgradient type and {βj} are independent Brownian motions in a probability space (Ω,F,P). The weak solution is defined via stochastic optimal control problem. [ABSTRACT FROM AUTHOR]

In this work we study an interior penalty method for a finite-dimensional large-scale linear complementarity problem (LCP) arising often from the discretization of stochastic optimal problems in financial engineering. In this approach, we approximate the LCP by a nonlinear algebraic equation containing a penalty term linked to the logarithmic barrier function for constrained optimization problems. We show that the penalty equation has a solution and establish a convergence theory for the approximate solutions. A smooth Newton method is proposed for solving the penalty equation and properties of the Jacobian matrix in the Newton method have been investigated. Numerical experimental results using three non-trivial test examples are presented to demonstrate the rates of convergence, efficiency and usefulness of the method for solving practical problems. [ABSTRACT FROM AUTHOR]

In this article, we study and compare two approaches to solving stochastic optimal control problems with an expectation constraint on the final state. The case of a probability constraint is included in this framework. The first approach is based on a dynamic programming principle and the second one uses Lagrange relaxation. In this article, we focus on discrete-time problems, but the two discussed approaches can be applied to discretized continuous-time problems. [ABSTRACT FROM AUTHOR]

We control ferromagnetic N-spin dynamics in the presence of thermal fluctuations by minimizing a quadratic functional subject to the stochastic Landau–Lifshitz–Gilbert equation. Existence of a weak solution of the stochastic optimal control problem is shown. The related first order optimality conditions consist of a coupled forward–backward SDE system, which is numerically solved by a structure-inheriting discretization, the least squares Monte-Carlo method to approximate related conditional expectations, and the new stochastic gradient method. Computational experiments are reported which motivate optimal controls in the case of interacting anisotropy, stray field, exchange energies, and acting noise. [ABSTRACT FROM AUTHOR]

We investigate the optimal management of a portfolio consisting of a risk asset and a risk-free bond. The asset dynamics is specified by the M-CEV model. Assuming that the investor has a power utility function, we derive an explicit analytical portfolio-management formula that contains confluent hypergeometric functions. The asymptotic form of the proposed portfolio management strategy is obtained, as well as approximate formulas consisting only of elementary functions. We additionally present an application of our results in the context of algorithmic statistical-arbitrage strategies. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of optimal management of renewable resources such as agricultural commodities and fishery production. For that purpose, we consider the population associated with such commodities and assume that its size evolves according to a logistic growth model with a predation term given by a Holling type- n functional response. Additionally, we assume that such population is subject to random fluctuations, modeled by a diffusive term driven by a one-dimensional Brownian motion and having a power-type coefficient, thus endowing the model under consideration with the property of having constant elasticity of variance. Since the stochastic differential equation associated with this model does not fit the standard assumptions in the stochastic optimal control literature, namely sublinear growth, we develop an appropriate version of the dynamic programming principle for the problem under consideration herein, proceeding also to provide a characterization of the optimal harvesting strategies and discuss some qualitative properties of the corresponding value function. [ABSTRACT FROM AUTHOR]

We present an efficient method for solving optimal stopping problems with a probabilistic constraint. The goal is to optimize the expected cumulative cost, but constrained by an upper bound on the probability that the cost exceeds a specified threshold. This probabilistic constraint causes optimal policies to be time-dependent and randomized, however, we show that an optimal policy can always be selected with 'piecewise-monotonic' time-dependence and 'nearly-deterministic' randomization. We prove these properties using the Bellman optimality equations for a Lagrangian relaxation of the original problem. We present an algorithm that exploits these properties for computational efficiency. Its performance and the structure of optimal policies are illustrated on two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the discrete optimality system of an optimal harvesting problem. While maximizing a combination of the total expected utility of the consumption and of the terminal size of a population, as a dynamic constraint, we assume that the density of the population is modeled by a stochastic quasi-linear heat equation. Finite-difference and symplectic partitioned Runge-Kutta (SPRK) schemes are used for space and time discretizations, respectively. It is the first time that a SPRK scheme is employed for the optimal control of stochastic partial differential equations. Monte-Carlo simulation is applied to handle expectation appearing in the cost functional. We present our results together with a numerical example. The paper ends with a conclusion and an outlook to future studies, on further research questions and applications. [ABSTRACT FROM AUTHOR]

Recently, the optimization-by-inference approach has been proposed as a new means for solving high-dimensional optimization problems quickly. Approximate Inference COntrol (AICO) is one of the most successful and promising methods that implement the optimization-by-inference approach. AICO is able to solve stochastic optimal control problems and has already been successfully used in many applications. However, it is known that the iterative inference of AICO sometimes fails to converge to the optimal solution. To make the optimization more robust, in this paper, we propose to take model uncertainty into account. In AICO, the cost function to be minimized is accurate around a particular state of a given stochastic system, but the accuracy is uncertain in regions far from that state. Because using such an uncertain function is harmful to the convergence, we modify AICO, so that it does not use the function in uncertain regions. Our method is easy to implement and does not add much computational time to the original AICO. Experiments using two different scenarios show that our method substantially improves AICO in terms of the rate at which the algorithm produces convergent results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of a bilinear optimal control problem subject to the Fokker-Planck equation. The control function depends on time and space and acts as a coefficient of the advection term. For this reason, suitable integrability properties of the control function are required to ensure well posedness of the state equation. Under these low regularity assumptions and for a general class of objective functionals, we prove the existence of optimal controls. Moreover, for common quadratic cost functionals of tracking and terminal type, we derive the system of first-order necessary optimality conditions. [ABSTRACT FROM AUTHOR]

A new procedure for designing optimal bounded control of stochastically excited multi-degree-of-freedom (MDOF) nonlinear viscoelastic systems is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the system is formulated as a quasi-integrable Hamiltonian system with viscoelastic terms and each viscoelastic term is replaced approximately by an elastically restoring force and a visco-damping force based on the randomly periodic behavior of the motion of quasi-integrable Hamiltonian system. Thus, a stochastically excited MDOF nonlinear viscoelastic system is converted to an equivalent quasi-integrable Hamiltonian system without viscoelastic terms. Then, by applying stochastic averaging, the system is further reduced to a partially averaged system of less dimension. The adjoint equation and maximum condition for the optimal control problem of the partially averaged system are derived by using the stochastic maximum principle, and the optimal bounded control force is determined from the maximum condition. Finally, the probability and statistics of the stationary response of optimally controlled system are obtained by solving the Fokker-Plank-Kolmogorov equation (FPK) associated with the fully averaged Itô equation of the controlled system. An example is worked out to illustrate the proposed procedure and its effectiveness. [ABSTRACT FROM AUTHOR]

This paper discusses an eigenvalue problem for a singular ergodic control. The eigenvalue has a probabilistic interpretation which can be regarded as the least, long-time averaged (ergodic) cost for a singular control problem. The existence and uniqueness of positive radial solutions of an equation with constraints involving gradient which is related to a stochastic optimal control problem under certain conditions on the nonlinearity of the equation are examined. [ABSTRACT FROM AUTHOR]

In many practical applications, stability with respect to part of the system's states is often necessary with finite-time convergence to the equilibrium state of interest. Finite-time partial stability involves dynamical systems whose part of the trajectory converges to an equilibrium state in finite time. In this paper, we address finite-time partial stability in probability and uniform finite-time partial stability in probability for nonlinear stochastic dynamical systems. Specifically, we provide Lyapunov conditions involving a Lyapunov function that is positive definite and decrescent with respect to part of the system state and satisfies a differential inequality involving fractional powers for guaranteeing finite-time partial stability in probability. In addition, we show that finite-time partial stability in probability leads to uniqueness of solutions in forward time and we establish necessary and sufficient conditions for almost sure continuity of the settling-time operator of the nonlinear stochastic dynamical system. Finally, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control design for finite-time partial stochastic stability and finite-time, partial-state stochastic stabilization. Finite-time partial stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that is positive definite and decrescent with respect to part of the system state and can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton-Jacobi-Bellman equation guaranteeing both finite-time, partial-state stability and optimality. The overall framework provides the foundation for extending stochastic optimal linear-quadratic controller synthesis to nonlinear-nonquadratic optimal finite-time, partial-state stochastic stabilization. [ABSTRACT FROM AUTHOR]

A modification was proposed for the relations of the method of dynamic programming in the problems of optimal stochastic control of the discrete systems by the probabilistic performance criterion. It enabled one to simplify the process of finding the optimal Markov strategy and obtain a suboptimal solution. Its efficiency was verified by the examples of maneuver optimization of the stationary satellite in the neighborhood of a geostationary orbit. An explicit form of the optimal control for the bilinear system with probabilistic terminal criterion was determined using the results obtained. [ABSTRACT FROM AUTHOR]

This paper deals with an offset control of traffic signals using a cellular automaton traffic model. A stochastic optimal control method for distributed traffic signals is modified to achieve coordinated traffic signal control with the proposed offset control method. In the proposed coordinated traffic signal control method, splits of each cycle and common cycle length are calculated using a modified stochastic optimal control method, and then the offset is calculated using an estimation method based on a modified CA traffic model at intervals. Also, simulations are carried out at multiple intersections using a micro traffic simulator. The effectiveness of the proposed coordinated control method is proved by comparing with other traffic signal control methods such as pre-timed signal control, two types of the traditional coordinated control and distributed control. [ABSTRACT FROM AUTHOR]

We consider the management of a single hydroelectric dam, subject to uncertain inflows and electricity prices and to a so-called 'tourism constraint': the water storage level must be high enough during the tourist season with high enough probability. We cast the problem in the stochastic optimal control framework: we search at each time t the optimal control as a function of the available information at t. We lay out two approaches. First, we formulate a chance-constrained stochastic optimal control problem: we maximize the expected gain while guaranteeing a minimum storage level with a minimal prescribed probability level. Dualizing the chance constraint by a multiplier, we propose an iterative algorithm alternating additive dynamic programming and update of the multiplier value 'à la Uzawa'. Our numerical results reveal that the random gain is very dispersed around its expected value; in particular, low gain values have a relatively high probability to materialize. This is why, to put emphasis on these low values, we outline a second approach. We propose a so-called stochastic viability approach that focuses on jointly guaranteeing a minimum gain and a minimum storage level during the tourist season. We solve the corresponding problem by multiplicative dynamic programming. To conclude, we discuss and compare the two approaches. [ABSTRACT FROM AUTHOR]

In the dual risk model, the surplus process of a company is a Lévy process with sample paths that are skip-free downwards. In this paper, the authors assume that the surplus process is the sum of a compound Poisson process and an independent Wiener process. The dual of the jump-diffusion risk model under a threshold dividend strategy is discussed. The authors derive a set of two integro-differential equations satisfied by the expected total discounted dividend until ruin. The cases where profits follow an exponential or mixtures of exponential distributions are solved. Applying the key method of the Laplace transform, the authors show how the integro-differential equations are solved. The authors also discuss the conditions for optimality and show how an optimal dividend threshold can be calculated as well. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of the generalized Lyapunov and Stein equations in $\mathbb {R}^{n}$, arising respectively from stochastic optimal control in continuous- and discrete-time. Generalizing the Smith method, our algorithms converge quadratically and have an O( n) computational complexity per iteration and an O( n) memory requirement. For large-scale problems, when the relevant matrix operators are 'sparse', our algorithm for generalized Stein (or Lyapunov) equations may achieve the complexity and memory requirement of O( n) (or similar to that of the solution of the linear systems associated with the sparse matrix operators). These efficient algorithms can be applied to Newton's method for the solution of the rational Riccati equations. This contrasts favourably with the naive Newton algorithms of O( n) complexity or the slower modified Newton's methods of O( n) complexity. The convergence and error analysis will be considered and numerical examples provided. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to derive some pointwise second-order necessary conditions for stochastic optimal controls in the general case that the control variable enters into both the drift and the diffusion terms. When the control region is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established; while when the control region is allowed to be nonconvex, we obtain a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle. It is found that, quite different from the first-order necessary conditions, the correction part of the solution to the second-order adjoint equation appears in the pointwise second-order necessary conditions whenever the diffusion term depends on the control variable, even if the control region is convex. [ABSTRACT FROM AUTHOR]

This paper is devoted to the optimal control problem of switching system in which constraints on the state variable are given by inclusions. Using Ekeland's variational principle, second-order necessary condition of optimality for stochastic switching systems with constraints is obtained, and transversality conditions for switching law are established. [ABSTRACT FROM AUTHOR]

We consider the optimal investment and consumption behavior of an agricultural sector that generates income from productive capital and interest payments and that has expenditures due to interest payments for debt. The productivity of capital, the interest rate, and the capital depreciation rate are uncertain. We adapt an existing model from the literature and generalize it to a stochastic depreciation of capital stock. Moreover, we assume a discontinuous evolution of the productivity of capital. The uncertainties are modeled as stochastic processes with discontinuous paths using well-known Wiener processes and simple Poisson processes. Accordingly, the investor's net wealth follows a jump diffusion stochastic differential equation. The goal of maximizing the utility of consumption over an infinite time horizon leads to a stochastic optimal control problem. We choose a hyperbolic absolute risk aversion utility function. To determine the optimal investment and consumption rate, the associated Hamilton-Jacobi-Bellman equation is solved by a separation method, and a verification theorem is applied. [ABSTRACT FROM AUTHOR]

We consider a controlled stochastic system in presence of state-constraints. Under the assumption of exponential stabilizability of the system near a target set, we aim to characterize the set of points which can be asymptotically driven by an admissible control to the target with positive probability. We show that this set can be characterized as a level set of the optimal value function of a suitable unconstrained optimal control problem which in turn is the unique viscosity solution of a second order PDE which can thus be interpreted as a generalized Zubov equation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the optimality of a hedging-point control policy in a multistate Markovian failure-prone manufacturing system with a risk-averse criterion that is defined as the conditional value-at-risk (CVaR) of the steady-state instantaneous running cost, where the system is subject to a constant single-product demand rate. An explicit expression for the optimal control policy is also obtained for the two-state case. The results are important from both theoretical and practical viewpoints. Indeed, the paper extends the well-known classical theoretical result on the optimality of hedging-point control policies under risk-neutral criteria, which are typically given by long-run average costs, and it develops a flexible and practical method for incorporating risk aversion into cost criteria. The approach presented here can be used to specify optimal control policies in similar manufacturing systems with CVaR criteria. [ABSTRACT FROM AUTHOR]

This paper deals with the coordination of production, replenishment and inspection decisions for a manufacturing-oriented supply chain with a failure-prone transformation stage, random lead time and imperfect delivered lots. Upon reception of the lot, the manufacturer executes an acceptance sampling plan with a zero non-conforming criterion. If the sample does not contain non-conforming items, the lot is accepted; otherwise, it is rejected. In this work, two strategies regarding the refused sampled lot are studied. The first one involves a return of the lot to the supplier, who commits to improving the quality of the lot, while the second assumes that the manufacturer performs a 100 % inspection and rectification operation. This work presents two main objectives. The first one is to jointly optimize, in a stochastic and dynamic context, the ordering point of raw material, the lot size of raw material, the final product inventory threshold and the severity of the sampling plan using a simulation-based optimization approach. The second one is to determine the best of the two quality control strategies. The in-depth study has shown that no strategy could be preferred in all the cases. For this reasons, we present an easy decision-making tool (indifference curves) to help the manager select the best quality control strategy when considering the entire supply chain. [ABSTRACT FROM AUTHOR]

This paper presents an existence and uniqueness result for a kind of forward-backward stochastic differential equations (FBSDEs for short) driven by Brownian motion and Poisson process under some monotonicity conditions. By virtue of the conclusion of FBSDEs, we solve a linear-quadratic stochastic optimal control problem for forward-backward stochastic systems with random jumps. Moreover, we also solve a linear-quadratic nonzero-sum stochastic differential game problem. We obtain explicit forms of the unique optimal control and the unique Nash equilibrium point, respectively. [ABSTRACT FROM AUTHOR]

We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton-Jacobi-Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the value function of the controlled equation and that the feedback law is verified. [ABSTRACT FROM AUTHOR]

In this paper, a real time stochastic optimal control method for traffic signals of asymmetrical intersection is proposed. A modified cellular automaton (CA) traffic model and Bayesian network (BN) model are used to predict the traffic jams. Here, the calculation for priori probabilistic of outflows at different traffic signals is modified based on the actual situation. In addition, PSO algorithm is used to search optimal traffic signals based on the stochastic model. Finally, the effectiveness of the proposed method is shown through simulations at an asymmetrical intersection using a micro-traffic simulator. [ABSTRACT FROM AUTHOR]

This paper is concerned with optimal control of neutral stochastic functional differential equations (NSFDEs). The Pontryagin maximum principle is proved for optimal control, where the adjoint equation is a linear neutral backward stochastic functional equation of Volterra type (VNBSFE). The existence and uniqueness of the solution are proved for the general nonlinear VNBSFEs. Under the convexity assumption of the Hamiltonian function, a sufficient condition for the optimality is addressed as well. [ABSTRACT FROM AUTHOR]

We consider the single-leg airline revenue management problem in continuous time with Poisson arrivals. Earlier work on this problem generally uses the Hamilton-Jacobi-Bellman equation to find an optimal policy whenever the value function is differentiable and is a solution to this equation. In this paper, we employ a different probabilistic approach, which does not rely on the smoothness of the value function. Instead, we use a continuous-time discrete-event dynamic programming operator to construct the value function and study its properties. A by-product of this approach is the analysis of the differentiability of the value function. We show that differentiability may break down for example with discontinuous arrival intensities. Therefore, one should exercise caution in using arguments based on the differentiability of the value function and the Hamilton-Jacobi-Bellman equation in general. [ABSTRACT FROM AUTHOR]

A strategy for stochastic optimal control of multi-degrees-of-freedom (MDOF) strongly nonlinear systems under combined harmonic and wide-band noise excitations is proposed. First, a stochastic averaging procedure is developed to obtain the partially averaged Itô stochastic differential equations for weakly controlled strongly nonlinear systems under combined harmonic and wide-band noise excitations. Then, the dynamical programming equation for stochastic optimal control problem is derived and solved to yield the optimal control law. Finally, the responses of optimally controlled MDOF nonlinear systems are predicted by solving the Fokker-Planck-Kolmogorov equation associated with the fully averaged Itô stochastic differential equations. As an example, the stochastic optimal control of two coupled Duffing oscillators under combined harmonic and wide-band noise excitations is worked out in detail to illustrate the strategy. The effectiveness of the proposed control strategy is verified using the results from Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic recursive optimal control problem in which the objective functional is described by the solution of a backward stochastic differential equation driven by $$G$$ -Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related Hamilton-Jacobi-Bellman (HJB) equation in the framework of $$G$$ -expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation. [ABSTRACT FROM AUTHOR]

This paper studies the stochastic optimal control problem of finding optimal dividend policies of an insurance company in discrete time with the use of general Lipschitz payoff functions involving indicators of profitability and risk. To construct positional optimal controls and to evaluate the performance indicators, the dynamic programming method is validated. The convergence rate of the successive approximation method in finding generally unbounded Bellman functions is estimated. The Pareto-optimal set of the problem is numerically approximated by so-called barrier-proportional control strategies. [ABSTRACT FROM AUTHOR]

In this paper, a real-time stochastic optimal control method of traffic signal is modified. In addition, H-GA-PSO algorithm is proposed to search optimal traffic signals based on the stochastic model. The H-GA-PSO algorithm is a modified Hierarchical Particle Swarm Optimization (H-PSO) algorithm based on Genetic Algorithm (GA) processing. Finally, the effectiveness of the stochastic optimal control method with H-GA-PSO algorithm is shown through simulations at multiple intersections using a micro-traffic simulator. [ABSTRACT FROM AUTHOR]

An efficient usage of available resources is a substantial requirement for the successful design of networked control systems. Recent results indicate major benefits of event-based control compared to conventional designs, when resources such as communication, energy, and computation, are sparse. This paper considers multiple entities of heterogeneous control systems whose feedback loops are coupled through a common communication medium. The design of the decentralized event-triggering control system is formulated as an average-cost problem that aims at the minimization of a social cost criterion. A state aggregation technique is used to develop a bi-level design method, which divides into a local average-cost problem within every subsystem and a global resource allocation problem assigning optimal transmission rates to every subsystem. Stability conditions are derived that guarantee stochastic stability of the aggregate system. Under these conditions, it is shown that the design approach is asymptotically optimal as the number of subsystems increases. [ABSTRACT FROM AUTHOR]

A new procedure for designing optimal bounded control of quasi-nonintegrable Hamiltonian systems with actuator saturation is proposed based on the stochastic averaging method for quasi-nonintegrable Hamiltonian systems and the stochastic maximum principle. First, the stochastic averaging method for controlled quasi-nonintegrable Hamiltonian systems is introduced. The original control problem is converted into one for a partially averaged equation of system energy together with a partially averaged performance index. Then, the adjoint equation and the maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The bounded optimal control forces are obtained from the maximum condition and solving the forward-backward stochastic differential equations (FBSDE). For infinite time-interval ergodic control, the adjoint variable is stationary process, and the FBSDE is reduced to an ordinary differential equation. Finally, the stationary probability density of the Hamiltonian and other response statistics of optimally controlled system are obtained by solving the Fokker-Plank-Kolmogorov equation associated with the fully averaged Itô equation of the controlled system. For comparison, the bang-bang control is also presented. An example of two degree-of-freedom quasi-nonintegrable Hamiltonian system is worked out to illustrate the proposed procedure and its effectiveness. Numerical results show that the proposed control strategy has higher control efficiency and less discontinuous control force than the corresponding bang-bang control at the price of slightly less control effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we study the stochastic maximum principle for optimal control problem of anticipated forward-backward system with delay and Lévy processes as the random disturbance. This control system can be described by the anticipated forward-backward stochastic differential equations with delay and Lévy processes (AFBSDEDLs), we first obtain the existence and uniqueness theorem of adapted solutions for AFBSDEDLs; combining the AFBSDEDLs' preliminary result with certain classical convex variational techniques, the corresponding maximum principle is proved. [ABSTRACT FROM AUTHOR]

A stochastic minimax optimal control strategy for uncertain quasi-Hamiltonian systems is proposed based on the stochastic averaging method, stochastic maximum principle and stochastic differential game theory. First, the partially completed averaged Itô stochastic differential equations are derived from a given system by using the stochastic averaging method for quasi-Hamiltonian systems with uncertain parameters. Then, the stochastic Hamiltonian system for minimax optimal control with a given performance index is established based on the stochastic maximum principle. The worst disturbances are determined by minimizing the Hamiltonian function, and the worst-case optimal controls are obtained by maximizing the minimal Hamiltonian function. The differential equation for adjoint process as a function of system energy is derived from the adjoint equation by using the Itô differential rule. Finally, two examples of controlled uncertain quasi-Hamiltonian systems are worked out to illustrate the application and effectiveness of the proposed control strategy. [ABSTRACT FROM AUTHOR]

In this paper, we study infinite-period mean-variance formulations for portfolio selections with an uncertain exit time. We employ the convergence control method together with the dynamic programming algorithm to derive analytical expressions for the optimal portfolio policy and the mean-variance efficient frontier under certain conditions. We illustrate these results by an numerical example. [ABSTRACT FROM AUTHOR]

Market makers continuously set bid and ask quotes for the stocks they have under consideration. Hence they face a complex optimization problem in which their return, based on the bid-ask spread they quote and the frequency at which they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, we consider a stochastic control problem similar to the one introduced by Ho and Stoll (J Fin Econ 9(1): 47-73, ) and formalized mathematically by Avellaneda and Stoikov (Quant Fin 8(3):217-224, ). The market is modeled using a reference price S following a Brownian motion with standard deviation σ, arrival rates of buy or sell liquidity-consuming orders depend on the distance to the reference price S and a market maker maximizes the expected utility of its P&L over a finite time horizon. We show that the Hamilton-Jacobi-Bellman equations associated to the stochastic optimal control problem can be transformed into a system of linear ordinary differential equations and we solve the market making problem under inventory constraints. We also shed light on the asymptotic behavior of the optimal quotes and propose closed-form approximations based on a spectral characterization of the optimal quotes. [ABSTRACT FROM AUTHOR]

We obtain a maximum principle for stochastic control problem of general controlled stochastic differential systems driven by fractional Brownian motions (of Hurst parameter H>1/2). This maximum principle specifies a system of equations that the optimal control must satisfy (necessary condition for the optimal control). This system of equations consists of a backward stochastic differential equation driven by both fractional Brownian motions and the corresponding underlying standard Brownian motions. In addition to this backward equation, the maximum principle also involves the Malliavin derivatives. Our approach is to use conditioning and Malliavin calculus. To arrive at our maximum principle we need to develop some new results of stochastic analysis of the controlled systems driven by fractional Brownian motions via fractional calculus. Our approach of conditioning and Malliavin calculus is also applied to classical system driven by standard Brownian motions while the controller has only partial information. As a straightforward consequence, the classical maximum principle is also deduced in this more natural and simpler way. [ABSTRACT FROM AUTHOR]

In this paper, nonlinear dynamic characteristics and optimal control of giant magnetostrictive laminated plate (GMLP) subjected to in-plane stochastic excitation were studied. Von del Pol nonlinear item was introduced to interpret the hysteresis phenomenon of the strain-magnetic field intensity curve of giant magnetostrictive material, and the nonlinear dynamic model of GMLP subjected to in-plane stochastic excitation was developed. Local and global stochastic stabilities were analyzed according to largest Lyapunov exponent theory and singular boundary theory. The functions of steady-state probability density and joint probability density were obtained, and the condition of stochastic Hopf bifurcation was analyzed. The reliability function was solved from backward Kolmogorov equation, and the probability density of the first-passage time was obtained. Finally, the optimal control strategy was proposed in stochastic dynamic programming method. Numerical simulation shows that the stability of the solution varies with parameter, and stochastic Hopf bifurcation appears in the process; the reliability of the system was improved by optimal control, and the first-passage time was delayed. The result is helpful to engineering applications of GMLP. [ABSTRACT FROM AUTHOR]

For a sequence of dynamic optimization problems, we aim at discussing a notion of consistency over time. This notion can be informally introduced as follows. At the very first time step t, the decision maker formulates an optimization problem that yields optimal decision rules for all the forthcoming time steps t, t,..., T; at the next time step t, he is able to formulate a new optimization problem starting at time t that yields a new sequence of optimal decision rules. This process can be continued until the final time T is reached. A family of optimization problems formulated in this way is said to be dynamically consistent if the optimal strategies obtained when solving the original problem remain optimal for all subsequent problems. The notion of dynamic consistency, well-known in the field of economics, has been recently introduced in the context of risk measures, notably by Artzner et al. (Ann. Oper. Res. 152(1):5-22, ) and studied in the stochastic programming framework by Shapiro (Oper. Res. Lett. 37(3):143-147, ) and for Markov Decision Processes (MDP) by Ruszczynski (Math. Program. 125(2):235-261, ). We here link this notion with the concept of 'state variable' in MDP, and show that a significant class of dynamic optimization problems are dynamically consistent, provided that an adequate state variable is chosen. [ABSTRACT FROM AUTHOR]

A new procedure for designing optimal control of quasi non-integrable Hamiltonian systems under stochastic excitations is proposed based on the stochastic averaging method for quasi non-integrable Hamiltonian systems and the stochastic maximum principle. First, the control problem consisting of 2 n-dimensional equations governing the controlled quasi non-integrable system and performance index is converted into a partially averaged one consisting of one-dimensional equation of the controlled system and performance index by using the stochastic averaging method. Then, the adjoint equation and the maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The optimal control forces are determined from the maximum condition and solving the forward-backward stochastic differential equations (FBSDE). For infinite time-interval ergodic control, the adjoint variable is a stationary process and the FBSDE is reduced to a partial differential equation. Finally, the response statistics of optimally controlled system is predicted by solving the Fokker-Plank equation (FPE) associated with the fully averaged Itô equation of the controlled system. An example of two degree-of-freedom (DOF) quasi non-integrable Hamiltonian system is worked out to illustrate the proposed procedure and its effectiveness. [ABSTRACT FROM AUTHOR]

This paper deals with a theoretical stochastic dynamic optimization model for the external financing of firms. We aim at searching for the best intensity of payment that a financier has to apply to a company in order to have a loan repaid. The techniques involved are related to the optimal control theory with exit time. We follow a dynamic programming approach. Our model also presents a distinction between the legal and the illegal financier, and a theoretical comparison analysis of the results is presented. Some numerical examples provide further validation of the theoretical results. [ABSTRACT FROM AUTHOR]