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In this paper we explore optimal liquidation in a market populated by a number of heterogeneous market makers that have limited inventory-carrying and risk-bearing capacity. We derive a reduced form model for the dynamics of their aggregated inventory considering a proper scaling limit. The resulting price impact profile is shown to depend on the characteristics and relative importance of their inventories. The model is flexible enough to reproduce the empirically documented power law behavior of the price impact function. For any choice of the market makers characteristics, optimal execution within this modeling approach can be recast as a linear-quadratic stochastic control problem. The value function and the associated optimal trading rate can be obtained semi-explicitly subject to solving a differential matrix Riccati equation. Numerical simulations are conducted to illustrate the performance of the resulting optimal liquidation strategy in relation to standard benchmarks. Remarkably, they show that the increase in performance is determined by a substantial reduction of higher order moment risk. [ABSTRACT FROM AUTHOR]

We consider optimal control problems for a system governed by a stochastic differential equation driven by a d-dimensional Brownian motion where both the drift and the diffusion coefficient are controlled. It is well known that without additional convexity conditions the strict control problem does not admit an optimal control. To overcome this difficulty, we consider the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is governed by a stochastic differential equation driven by a continuous orthogonal martingale measure. This relaxed model admits an optimal control that can be approximated by a sequence of strict controls by the so-called chattering lemma. We establish optimality necessary conditions, in terms of two adjoint processes, extending Peng's maximum principle to relaxed control problems. We show that relaxing the drift and diffusion martingale parts directly as in deterministic control does not lead to a true relaxed model as the obtained controlled dynamics is not continuous in the control variable. [ABSTRACT FROM AUTHOR]

We propose a refined dynamic programming model based on a hidden Markov chain formulation and a nonlinear filtering technique to calculate the optimal debt ratio for public and private sectors for different scenarios. We then conduct the empirical analysis of the U.S. markets in real estate and equities during 1991.Q1 and 2020.Q1, comparing them with the theoretical results. It indicates that U.S. households and governments spent more than they can afford. While households reduced their debt ratio during times of economic distress, the public sector hiked its debt ratio to stimulate the economy. The policy effect took a long time to accumulate, and the outcome was lower than expected to revitalize the economy in time. [ABSTRACT FROM AUTHOR]

We consider the diffusive limit of a typical pure-jump Markovian control problem as the intensity of the driving Poisson process tends to infinity. We show that the convergence speed is provided by the Hölder exponent of the Hessian of the limit problem, and explain how correction terms can be constructed. This provides an alternative efficient method for the numerical approximation of the optimal control of a pure-jump problem in situations with very high intensity of jumps. We illustrate this approach in the context of a display advertising auction problem. [ABSTRACT FROM AUTHOR]

This paper considers an optimal control problem for a time-invariant linear stochastic system with discrete time, scalar unbounded control, additive noise, and a probabilistic criterion for retaining its trajectories in a given neighborhood of zero. We use dynamic programming and two-sided Bellman function estimates to derive analytical expressions for the optimal control at two time steps and a suboptimal control on any control horizon. The effectiveness of these controls is illustrated on a numerical example. [ABSTRACT FROM AUTHOR]

Stochastic optimal principle leads to the resolution of a partial differential equation (PDE), namely the Hamilton–Jacobi–Bellman (HJB) equation. In general, this equation cannot be solved analytically, thus numerical algorithms are the only tools to provide accurate approximations. The aims of this paper is to introduce a novel fitted finite volume method to solve high dimensional degenerated HJB equation from stochastic optimal control problems in high dimension ( n ≥ 3 ). The challenge here is due to the nature of our HJB equation which is a degenerated second-order partial differential equation coupled with an optimization problem. For such problems, standard scheme such as finite difference method losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. We discretize the HJB equation using the fitted finite volume method, well known to tackle degenerated PDEs, while the time discretisation is performed using the Implicit Euler scheme.. We show that matrices resulting from spatial discretization and temporal discretization are M-matrices. Numerical results in finance demonstrating the accuracy of the proposed numerical method comparing to the standard finite difference method are provided. [ABSTRACT FROM AUTHOR]

This paper investigates a continuous-time portfolio optimization problem with the following features: (i) a no-short selling constraint; (ii) a leverage constraint, that is, an upper limit for the sum of portfolio weights; and (iii) a performance criterion based on the lower mean square error between the investor's wealth and a predetermined target wealth level. Since the target level is defined by a deterministic function independent of market indices, it corresponds to the criterion of absolute return funds. The model is formulated using the stochastic control framework with explicit boundary conditions. The corresponding Hamilton–Jacobi–Bellman equation is solved numerically using the kernel-based collocation method. However, a straightforward implementation does not offer a stable and acceptable investment strategy; thus, some techniques to address this shortcoming are proposed. By applying the proposed methodology, two numerical results are obtained: one uses artificial data, and the other uses empirical data from Japanese organizations. There are two implications from the first result: how to stabilize the numerical solution, and a technique to circumvent the plummeting achievement rate close to the terminal time. The second result implies that leverage is inevitable to achieve the target level in the setting discussed in this paper. [ABSTRACT FROM AUTHOR]

This paper analyzes the limiting behavior of stochastic linear-quadratic optimal control problems in finite time-horizon [0, T] as T → ∞. The so-called turnpike properties are established for such problems, under stabilizability condition which is weaker than the controllability, normally imposed in the similar problem for ordinary differential systems. In dealing with the turnpike problem, a crucial issue is to determine the corresponding static optimization problem. Intuitively mimicking the deterministic situations, it seems to be natural to include both the drift and the diffusion expressions of the state equation to be zero as constraints in the static optimization problem. However, this would lead us to a wrong direction. It is found that the correct static problem should contain the diffusion as a part of the objective function, which reveals a deep feature of the stochastic turnpike problem. [ABSTRACT FROM AUTHOR]

In this paper we study optimal trading strategies in a financial market in which stock returns depend on a hidden Gaussian mean reverting drift process. Investors obtain information on that drift by observing stock returns. Moreover, expert opinions in the form of signals about the current state of the drift arriving at fixed and known dates are included in the analysis. Drift estimates are based on Kalman filter techniques. They are used to transform a power utility maximization problem under partial information into an optimization problem under full information where the state variable is the filter of the drift. The dynamic programming equation for this problem is studied and closed-form solutions for the value function and the optimal trading strategy of an investor are derived. They allow to quantify the monetary value of information delivered by the expert opinions. We illustrate our theoretical findings by results of extensive numerical experiments. [ABSTRACT FROM AUTHOR]

This paper is to establish a sufficient maximum principle for one kind of stochastic optimal control problem with three types of delays: a discrete delay, a moving-average delay and a noisy memory. The main features of this research include the introduction of a unified adjoint equation and a simple method to get the adjoint process. One kind of optimal consumption problem and its special cases are studied as illustrative examples, for which the adjoint equations are solved with two different approaches and the optimal consumption strategies are obtained. [ABSTRACT FROM AUTHOR]

This article is devoted to studying the topological structure of a solution set for Clarke subdifferential type fractional non-instantaneous impulsive differential inclusion driven by Poisson jumps. Initially, for proving the solvability result, we use a nonlinear alternative of Leray–Schauder fixed point theorem, Gronwall inequality, stochastic analysis, a measure of noncompactness, and the multivalued analysis. Furthermore, the mild solution set for the proposed problem is demonstrated with nonemptyness, compactness, and, moreover, R δ -set. By employing Balder's theorem, the existence of optimal control is derived. At last, an application is provided to validate the developed theoretical results. [ABSTRACT FROM AUTHOR]

This article contributes to research dealing with the optimal dividend policy problem of a firm whose goal is to maximize the expected total discounted dividend payments before bankruptcy. We consider a model of a firm whose cash surplus exhibits regime switching, but unlike the existing literature, we exclude diffusion from our model. We assume that firm's cash surplus follows the telegraph process, which leads to a problem of singular stochastic control. Surprisingly, this problem turns out to be more complicated than the ones arising in the models involving diffusion. We solve this problem by using the method of variational inequalities and show that the optimal dividend policy can be of three significantly different types depending on the parameters of the model. [ABSTRACT FROM AUTHOR]

This paper considers an optimal control problem for a discrete-time stochastic system with the probability of first reaching the boundaries of a given domain as the optimality criterion. Dynamic programming-based sufficient conditions of optimality are formulated and proved. The isobells of levels 1 and 0 of the Bellman function are used for obtaining two-sided estimates of the right-hand side of the dynamic programming equation, two-sided estimates of the Bellman function, and two-sided estimates of the optimal-value function of the probabilistic criterion. A suboptimal control design method is proposed. The conditions of equivalence to an optimal control problem with a probabilistic terminal criterion are established. An illustrative example is given. [ABSTRACT FROM AUTHOR]

This paper examines an optimal control problem with a probabilistic criterion for retaining the trajectories of a discrete-time stochastic system in given sets. The dynamic programming method is employed for obtaining the isobells of levels 1 and 0 of the Bellman function, two-sided estimates for the right-hand side of the dynamic programming equation, two-sided estimates for the Bellman function, and the optimal-value function of the probabilistic criterion. These results are then used for deriving an approximate formula for the optimal control. As an illustrative example the problem of keeping an inverted pendulum in the neighborhood of an unstable equilibrium is considered. [ABSTRACT FROM AUTHOR]

In this paper, we present a Longstaff-Schwartz-type algorithm for optimal stopping time problems based on the Brownian motion filtration. The algorithm is based on Leão et al. (??2019) and, in contrast to previous works, our methodology applies to optimal stopping problems for fully non-Markovian and non-semimartingale state processes such as functionals of path-dependent stochastic differential equations and fractional Brownian motions. Based on statistical learning theory techniques, we provide overall error estimates in terms of concrete approximation architecture spaces with finite Vapnik-Chervonenkis dimension. Analytical properties of continuation values for path-dependent SDEs and concrete linear architecture approximating spaces are also discussed. [ABSTRACT FROM AUTHOR]

We prove a general existence result in stochastic optimal control in discrete time, where controls, taking values in conditional metric spaces, depend on the current information and past decisions. The general form of the problem lies beyond the scope of standard techniques in stochastic control theory, the main novelty is a formalization in conditional metric space and the use of conditional analysis. We illustrate the existence result by several examples such as wealth-dependent utility maximization under risk constraints and utility maximization with a conditional dimension. We also provide a discussion as to how our methods compare to techniques based on random sets. [ABSTRACT FROM AUTHOR]

This paper presents reduced-order nonlinear filtering schemes based on a theoretical framework that combines stochastic dimensional reduction and nonlinear filtering. Here, dimensional reduction is achieved for estimating the slow-scale process in a multiscale environment by constructing a filter using stochastic averaging results. The nonlinear filter is approximated numerically using the ensemble Kalman filter and particle filter. The particle filter is further adapted to the complexities of inherently chaotic signals. In particle filters, an ensemble of particles is used to represent the distribution of the state of the hidden signal. The ensemble is updated using observation data to obtain the best representation of the conditional density of the true state variables given observations. Particle methods suffer from the "curse of dimensionality," an issue of particle degeneracy within a sample, which increases exponentially with system dimension. Hence, particle filtering in high dimensions can benefit from some form of dimensional reduction. A control is superimposed on particle dynamics to drive particles to locations most representative of observations, in other words, to construct a better prior density. The control is determined by solving a classical stochastic optimization problem and implemented in the particle filter using importance sampling techniques. [ABSTRACT FROM AUTHOR]

In this paper, we describe the new OPTCON3 algorithm, which serves to determine approximately optimal policies for stochastic control problems with a quadratic objective function and nonlinear dynamic models. It includes active learning and the dual effect of optimizing policies, whereby optimal policies are used to learn about the stochastics of the dynamic system in addition to their immediate effect on the performance of the system. The OPTCON3 algorithm approximates the nonlinear model with a time-varying linear model and applies a procedure similar to that of Kendrick to the series of linearized models to calculate approximately optimal policies. The results for two simple economic models serve to test the OPTCON3 algorithm and compare it to previous solutions of the stochastic control problem. Initial evaluations show that the OPTCON3 approach may be promising to enhance our understanding of the adaptive economic policy problem under uncertainty. [ABSTRACT FROM AUTHOR]

We study the optimal control problem for a descriptor system whose evolution is described by Ito's differential-algebraic equation. The quadratic cost functional is considered. The main constraint is that the characteristic matrix pencil corresponding to the equation is regular. We establish the conditions for the existence and uniqueness of the optimal control and the corresponding optimal state. The results are illustrated on an example of a descriptor system that describes transient states in a radio engineering filter with random perturbations in the form of white noise. [ABSTRACT FROM AUTHOR]

The paper overviews stochastic optimization models of actuarial mathematics and methods for their solution from the point of view of the methodology of multicriteria stochastic programming and optimal control. The evolution of the capital of an insurance company is considered in discrete time. The main random parameters of the models are insurance payouts, i.e., the ratios of paid insurance claims to the corresponding premiums per unit time. Optimization variables are the structure of the insurance portfolio (gross premium structure) and amount of dividends. As efficiency criteria, indicators of the profitability of the insurance business are used, and, as risk indicators the ruin probability and the recourse capital necessary to prevent the ruin are taken. The goal of the optimization is to find Pareto-optimal solutions. Methods for finding these solutions are proposed. [ABSTRACT FROM AUTHOR]

An optimal production flow control problem of a make-to-stock manufacturing firm with price, cost, and demand uncertainty is studied. The objective of the flow rate control problem is maximizing the average profit that is the difference between the expected revenue and the expected production, inventory holding, and backlog costs. The uncertainties in the system are captured jointly in discrete environment states. In each environment state, the price, cost, and demand take different levels. The transitions between different environment states evolve according to a time-homogenous Markov chain. By using a continuous flow model, the optimal production policy is stated as a state-dependent hedging policy. The performance of the system where the production cost alternates between a high and a low cost level and the demand is either constant or also alternates between a high and a low level is analyzed under the double-hedging policy. According to this policy, the producer produces only when the cost is low and the surplus is between the two hedging levels. However when the backlog is below the lower hedging level, the producer produces with the maximum capacity regardless of the cost. The effects of production cost, production capacity, demand variability, and the dependence of the demand and the cost on the performance of the system are analyzed analytically and numerically. It is shown that controlling the production rate optimally allows producers respond to the fluctuations in price, cost, and demand in an effective way and maximize their profits. [ABSTRACT FROM AUTHOR]

We consider the problem of determining an optimal strategy for electricity injection that faces an uncertain power demand stream. This demand stream is modeled via an Ornstein–Uhlenbeck process with an additional jump component, whereas the power flow is represented by the linear transport equation. We analytically determine the optimal amount of power supply for different levels of available information and compare the results to each other. For numerical purposes, we reformulate the original problem in terms of the cost function such that classical optimization solvers can be directly applied. The computational results are illustrated for different scenarios. [ABSTRACT FROM AUTHOR]

We consider the problem of governing systemic risk in a banking system model. The banking system model consists in an initial value problem for a system of stochastic differential equations whose dependent variables are the log-monetary reserves of the banks as functions of time. The banking system model considered generalizes previous models studied in Fouque and Sun (in: Fouque and Langsam (eds) Handbook of systemic risk, Cambridge University Press, Cambridge, pp 444–452, 2013), Carmona et al. (Commun Math Sci 13(4):911–933, 2015), Garnier et al. (SIAM J Financ Math 4:151–184, 2013) and describes an homogeneous population of banks. Two distinct mechanisms are used to model the cooperation among banks and the cooperation between banks and monetary authority. These mechanisms are regulated respectively by the parameters α and γ . A bank fails when its log-monetary reserves go below an assigned default level. We call systemic risk or systemic event in a bounded time interval the fact that in that time interval at least a given fraction of the banks fails. The probability of systemic risk in a bounded time interval is evaluated using statistical simulation. A method to govern the probability of systemic risk in a bounded time interval is presented. The goal of the governance is to keep the probability of systemic risk in a bounded time interval between two given thresholds. The governance is based on the choice of the log-monetary reserves of a kind of "ideal bank" as a function of time and on the solution of an optimal control problem for the mean field approximation of the banking system model. The solution of the optimal control problem determines the parameters α and γ as functions of time, that is defines the rules of the borrowing and lending activity among banks and between banks and monetary authority. Some numerical examples are discussed. In particular during a two year period we consider the governance of systemic risk in the next year when governance decisions are taken quarterly. The systemic risk governance is tested in absence and in presence of positive and negative shocks acting on the banking system. [ABSTRACT FROM AUTHOR]

In this paper we consider a discrete-time formulation of dynamic transaction cost problems. We examine applicability of numerical discrete probability approximation as an alternative simplistic approach to solve dynamic transaction cost problems. We provide a computational study of a lattice-based heuristic method on simple transaction cost models and highlight its many advantages. The solution of these problems provides a dynamic investor with important insights as to how the portfolio should be re-balanced when faced with transaction costs. [ABSTRACT FROM AUTHOR]

In this paper, the optimal control problem for a discrete-time stochastic system with a general-form probabilistic criterion is considered. Using dynamic programming and the properties of the Bellman function, new two-sided bounds of this function that refine the earlier results are constructed. The derived bounds are then adopted to justify the application of the modified strategy that is optimal in the two-step investment portfolio management problem under risk to the corresponding multistep problem. An example that illustrates the advantages of such a strategy over other well-known strategies is given. [ABSTRACT FROM AUTHOR]

In this paper, we consider the optimal impulse control problem of a currency with exchange rate dynamics whose state follows a geometric Lévy process. The objective of the Central Bank is to keep the exchange rate as close as possible to a given target. Running costs associating with the difference between the actual exchange rate and the target are continuously incurred to the system. We suppose that, when the Central Bank intervenes in the system, it requires the fixed and proportional costs. Applying the theory of stochastic impulse controls, we find the optimal exchange rate at which interventions should be performed and the optimal sizes of the interventions, so as to minimize the expected total discounted sum of the intervention costs and running costs incurred over the infinite time horizon. Furthermore, numerical comparisons for the optimal intervention strategy between the geometric Lévy process model and the geometric Brownian motion model are investigated in details. [ABSTRACT FROM AUTHOR]

By means of stochastic optimal control, this paper aims at studying the shadow pricing of renewable natural resources in uncertainty. Two cases are considered, respectively centralized and decentralized control processes. The decentralized control is in the form of a stochastic control of the state vector distributed among several agents. In both cases, the optimal control path minimizing the cost function, which is a decreasing function of time, corresponds to the real option valuation. The latter is a cost-effective optional investment in the resource stock preservation in uncertainty. The results obtained from numerical simulations show coherence with those encountered in the literature on option pricing. [ABSTRACT FROM AUTHOR]

Consideration was given to the design of the optimal control of the general discrete stochastic system with a criterion as the probability of the state vector sojourn in the given sets at each time instant. Derived were relations of the dynamic programming enabling one to establish an optimal solution in the class of Markov strategies without extension of the state vector with subsequent reduction to an equivalent problem with the probabilistic terminal criterion. Consideration was given to the problem of one-parameter correction of the flying vehicle trajectory. An analytical solution was established. [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]

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]

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]

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]

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]

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]

In this work we consider a stochastic optimal control problem with either convex control constraints or finitely many equality and inequality constraints over the final state. Using the variational approach, we are able to obtain first and second order expansions for the state and cost function, around a local minimum. This fact allows us to prove general first order necessary condition and, under a geometrical assumption over the constraint set, second order necessary conditions are also established. We end by giving second order optimality conditions for problems with constraints on expectations of the final state. [ABSTRACT FROM AUTHOR]

In this article, we study the situation, where the opportunity is given to invest into a government-owned business by partial privatization to a private company. After payment of an initial installment cost, the private company's investments are flexible within a range [0, k] until the business is completed. We model the problem in a real option framework, using geometric mean reversion to describe the dynamics of the business. We determine the optimal time for the private company to enter and pay the initial installment cost as well as the optimal dynamic investment strategy that it follows afterward. For the latter, analytic solution cannot be obtained. We use quadratic splines in order to solve the corresponding dynamic programming problem. Finally, we determine the optimal degree of privatization in our model from the government's perspective. [ABSTRACT FROM AUTHOR]

In this paper we propose and study a continuous-time stochastic model of optimal allocation for a defined contribution pension fund with a minimum guarantee. We adopt the point of view of a fund manager maximizing the expected utility from the fund wealth over an infinite horizon. In our model the dynamics of wealth takes directly into account the flows of contributions and benefits, and the level of wealth is constrained to stay above a 'solvency level.' The fund manager can invest in a riskless asset and in a risky asset, but borrowing and short selling are prohibited. We concentrate the analysis on the effect of the solvency constraint, analyzing in particular what happens when the fund wealth reaches the allowed minimum value represented by the solvency level. The model is naturally formulated as an optimal stochastic control problem with state constraints and is treated by the dynamic programming approach. We show that the value function of the problem is a regular solution of the associated Hamilton-Jacobi-Bellman equation. Then we apply verification techniques to get the optimal allocation strategy in feedback form and to study its properties. We finally give a special example with explicit solution. [ABSTRACT FROM AUTHOR]

We construct the explicit connection existing between a solvable model of the discrete velocities non-linear Boltzmann equation and the Hamilton–Bellman–Jacobi equation associated with a simple optimal control of a piecewise deterministic process. This study extends the known relation that exists between the Burgers equation and a simple controlled diffusion problem. In both cases the resulting partial differential equations can be linearized via a logarithmic transformation and hence offer the possibility to solve physically relevant non-linear field models in full generality. [ABSTRACT FROM AUTHOR]