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This paper studies finite-time optimal consumption–investment problems with power, logarithmic and exponential utilities, in a regime switching market with random coefficients and possibly subject to non-convex constraints. Compared to the existing models, one distinguish feature of our model is that the trading constraints put on the consumption and investment strategies are coupled together in the cases of power and logarithmic utilities, leading to new technical challenges. We provide explicit optimal consumption–investment strategies and optimal values for these consumption–investment problems, which are expressed in terms of the solutions to some multi-dimensional diagonally quadratic backward stochastic differential equation (BSDE) and linear BSDE with unbound coefficients. These BSDEs are new in the literature and solving them is one of the main theoretical contributions of this paper. We accomplish the latter by applying the truncation, approximation technique to get some a priori uniformly lower and upper bounds for their solutions. [ABSTRACT FROM AUTHOR]

We study zero-sum stochastic games between a singular controller and a stopper when the (state-dependent) diffusion matrix of the underlying controlled diffusion process is degenerate. In particular, we show the existence of a value for the game and determine an optimal strategy for the stopper. The degeneracy of the dynamics prevents the use of analytical methods based on solution in Sobolev spaces of suitable variational problems. Therefore we adopt a probabilistic approach based on a perturbation of the underlying diffusion modulated by a parameter γ > 0 . For each γ > 0 the approximating game is non-degenerate and admits a value u γ and an optimal strategy τ ∗ γ for the stopper. Letting γ → 0 we prove convergence of u γ to a function v, which identifies the value of the original game. We also construct explicitly optimal stopping times θ ∗ γ for u γ , related but not equal to τ ∗ γ , which converge almost surely to an optimal stopping time θ ∗ for the game with degenerate dynamics. [ABSTRACT FROM AUTHOR]

We consider continuous-time mean-field stochastic games with strategic complementarities. The interaction between the representative productive firm and the population of rivals comes through the price at which the produced good is sold. Existence of minimal and maximal equilibria is shown via lattice-theoretical arguments, and sufficient conditions for deriving comparative statics of the equilibria are presented. A detailed numerical study, based on iterative schemes converging to the maximal and minimal equilibria, allows us to analyze in relevant financial examples of how the emergence of multiple equilibria is related to the intensity of strategic interactions. [ABSTRACT FROM AUTHOR]

Abstract.This article studies a robust optimal investment problem with an ambiguity-averse manager (AAM) for a defined contribution (DC) plan with multiple risks under the mean-variance criterion. In the pension accumulation stage, the interest rate, the volatility, and the salary level are considered to be stochastic. The financial market consists of a risk-free asset, a risky asset, and a rolling bond. We assume that the term structure of interest rates is driven by an affine interest rate model, while the stock price and the salary level are modeled by the stochastic volatility model with stochastic interest rate. The goal of an AAM is to find a robust optimal strategy to maximize the expectation of terminal wealth and minimize the variance of terminal wealth in the worst-case scenario. By applying the Lagrange dual theory and the robust optimal control approach, we obtain closed-form expressions of the robust precommitment strategy and the efficient frontier, and subsequently some special cases are derived. Finally, a numerical example is given to illustrate the results obtained. [ABSTRACT FROM AUTHOR]

Purpose: The main motivation of this paper is to present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L2-convergence rate is established. Design/methodology/approach: The authors establish a result concerning the L2-convergence rate of the solution of backward stochastic differential equation with jumps with respect to the Yosida approximation. Findings: The authors carry out a convergence rate of Yosida approximation to the semi-linear backward stochastic differential equation in infinite dimension. Originality/value: In this paper, the authors present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L2-convergence rate is established. [ABSTRACT FROM AUTHOR]

Can a principal still offer optimal dynamic contracts that are linear in end-of-period outcomes when the agent controls a process that exhibits memory? We provide a positive answer by considering a general Gaussian setting where the output dynamics are not necessarily semimartingales or Markov processes. We introduce a rich class of principal–agent models that encompasses dynamic agency models with memory. From a mathematical point of view, we show how contracting problems with Gaussian Volterra outcomes can be transformed into those of semimartingale outcomes by some change of variables to use the martingale optimality principle. Our main contribution is to show that for one-dimensional models, this setting always allows optimal linear contracts in end-of-period observable outcomes with a deterministic optimal level of effort. In higher dimensions, we show that linear contracts are still optimal when the effort cost function is radial, and we quantify the gap between linear contracts and optimal contracts for more general quadratic costs of efforts. [ABSTRACT FROM AUTHOR]

We consider the problem of pricing in a large market, which arises as a limit of small markets within which there are finitely many traded assets. We show that this framework allows accommodating both marginal-utility-based prices (for stochastic utilities) and arbitrage-free prices. Adopting a stochastic integration theory with respect to a sequence of semimartingales, we introduce the notion of marginal-utility-based prices for the large (post-limit) market and establish their existence, uniqueness and relation to arbitrage-free prices. These results rely on a theorem of independent interest on utility maximisation with a random endowment in a large market that we state and prove first. Further, we provide approximation results for the marginal-utility-based and arbitrage-free prices in the large market by those in small markets. In particular, our framework allows pricing asymptotically replicable claims, where we also show consistency in the pricing methodologies and provide positive examples. [ABSTRACT FROM AUTHOR]

When uncertainty is modelled by a non-dominated and non-compact set of probability measures, a notion of essential supremum for a family of real-valued functions is developed in terms of upper semi-analytic functions. We show how the properties postulated on the initial functions carry over to their quasi-sure essential supremum. We propose various applications to financial problems with frictions. We analyse superreplication and prove a bidual characterisation of the superhedging cost. We also study a weak no-arbitrage condition called absence of instantaneous profit (AIP ) under which prices are finite. This requires new results on the aggregation of quasi-sure statements. [ABSTRACT FROM AUTHOR]

This paper examines the robust time-consistent reinsurance-investment strategy for an ambiguity-averse insurer under the 4/2 stochastic volatility model. In this model, an ambiguity-averse insurer transfers the risk generated by insurance claims through purchasing proportional reinsurance and invests the remaining capital in a financial market composed of a risk-free and a risky asset to manage the risk. The claim process is described by the classical Cramér-Lundberg process, while the price process of the risky asset is driven by the 4/2 stochastic volatility model. Under the mean-variance criterion, by employing the stochastic optimal control theory, we establish the corresponding extended Hamilton-Jacobi-Bellman (HJB) equation, and derive the robust time-consistent reinsurance-investment strategy and the corresponding equilibrium value function. In addition, we also study the reinsurance-investment problem in the case of excess-of-loss reinsurance. Finally, a sensitivity analysis is given to examine the results obtained. [ABSTRACT FROM AUTHOR]

We consider a control problem for a mean-field coupled forward-backward stochastic differential equations, called also McKean–Vlasov equation (MF-FBSDE). For this type of equations, the coefficients depend not only on the state of the system, but also on its marginal distributions. They arise naturally in mean-field control problems and mean-field games. We consider the relaxed control problem where admissible controls are measure-valued processes. We prove the existence of a relaxed optimal control by using a suitable form of Skorokhod representation theorem and Jakubowski's topology, on the space of càdlàg functions. We use martingale measure to define the relaxed state process. Our results extend to MF-FBSDEs those already known for forward and backward stochastic equations of Itô type. [ABSTRACT FROM AUTHOR]

We introduce a non-zero-sum game between a government and a legislative body to study the optimal level of debt. Each player, with different time preferences, can intervene on the stochastic dynamics of the debt-to-GDP ratio via singular stochastic controls, in view of minimizing non-continuously differentiable running costs. We completely characterise Nash equilibria in the class of Skorokhod-reflection-type policies. We highlight the importance of different time preferences resulting in qualitatively different type of equilibria. In particular, we show that, while it is always optimal for the government to devise an appropriate debt issuance policy, the legislator should optimally impose a debt ceiling only under relatively low discount rates and a laissez-faire policy can be optimal for high values of the legislator’s discount rate. [ABSTRACT FROM AUTHOR]

This paper extends the results provided in Jasso-Fuentes et al. (Appl Math Optim 81(2):409–441, 2020b) and Jasso-Fuentes et al. (Pure Appl Funct Anal 9(3):675–704, 2024) regarding the study of discrete-time hybrid stochastic models with general spaces and total discounted payoffs. This extension incorporates the handling of negative and/or unbounded costs per stage. In particular, it encompasses interesting applications, such as scenarios where the controller optimizes net costs, social welfare costs, or distances between points. These situations arise when assumptions of both non-negativeness and boundedness on the cost per stage do not apply. Our proposal relies on Lyapunov-like conditions, enabling, among other aspects, the finiteness of the value function and the existence of solutions to the associated dynamic programming equation. This equation is crucial for deriving optimal control policies. To illustrate our theory, we include an example in inventory-manufacturing management, highlighting its evident hybrid nature. [ABSTRACT FROM AUTHOR]

To study the nonlinear properties of complex natural phenomena, the evolution of the quantity of interest can be often represented by systems of coupled nonlinear stochastic differential equations (SDEs). These SDEs typically contain several parameters which have to be chosen carefully to match the experimental data and to validate the effectiveness of the model. In the present paper the calibration of these parameters is described by nonlinear SDE-constrained optimization problems. In the optimize-before-discretize setting a rigorous analysis is carried out to ensure the existence of optimal solutions and to derive necessary first-order optimality conditions. For the numerical solution a Monte–Carlo method is applied using parallelization strategies to compensate for the high computational time. In the numerical examples an Ornstein–Uhlenbeck and a stochastic Prandtl–Tomlinson bath model are considered. [ABSTRACT FROM AUTHOR]

This paper considers the issue of optimal investment and reinsurance strategies for an insurer with regime-switching. In our mathematical model, a risk-free asset and a risky asset are assumed to rely on a continuous time-homogeneous, finite-state, observed Markov chain, and the latter's price dynamics is described by a general regime-switching jump-diffusion process. We are extending the classical claim process to a Markov-modulated compound Poisson process. The insurer faces the decision-making problem of choosing to invest his/her surplus in the financial market and purchase reinsurance such that the expected power utility of his/her terminal wealth is maximized. We apply dynamic programming principle to derive the regime-switching Hamilton–Jacobi–Bellman (HJB) equation. Then, by analysing the solutions of the HJB equation, the optimal investment and reinsurance strategies are obtained and given in the verification theorem. Finally, the numerical analysis based on a two-state Markov chain is presented to illustrate our results. [ABSTRACT FROM AUTHOR]

In this research, we investigate the maximum principle pertaining to risk-sensitive optimal control problems under partial observation, modeled by forward–backward stochastic differential equations (FBSDEs) of the general regularity McKean–Vlasov form. An important aspect of these equations is that their coefficients are nonlinearly influenced by both the state process and its distribution. The control variable consists of two components: a continuous control and an impulse control. The cost functional is an exponential of integral type based on the regularity McKean–Vlasov framework. By applying Girsanov’s theorem and taking derivatives with respect to the probability distribution, we establish the risk-sensitive maximum principle. This principle is formulated using variational inequalities, under the assumption that the control domain is convex. Moreover, the sufficient conditions of optimality is obtained under certain concavity assumptions. As an application, the main outcomes are used to solve a linear-quadratic risk-sensitive optimal control problem of the regularity McKean–Vlasov type, both under partial and full observation conditions. [ABSTRACT FROM AUTHOR]

This paper investigates Monte Carlo (MC) methods to estimate probabilities of rare events associated with the solution to the d-dimensional McKean–Vlasov stochastic differential equation (MV-SDE). MV-SDEs are usually approximated using a stochastic interacting P-particle system, which is a set of P coupled d-dimensional stochastic differential equations (SDEs). Importance sampling (IS) is a common technique for reducing high relative variance of MC estimators of rare-event probabilities. We first derive a zero-variance IS change of measure for the quantity of interest by using stochastic optimal control theory. However, when this change of measure is applied to stochastic particle systems, it yields a P × d -dimensional partial differential control equation (PDE), which is computationally expensive to solve. To address this issue, we use the decoupling approach introduced in (dos Reis et al. 2023), generating a d-dimensional control PDE for a zero-variance estimator of the decoupled SDE. Based on this approach, we develop a computationally efficient double loop MC (DLMC) estimator. We conduct a comprehensive numerical error and work analysis of the DLMC estimator. As a result, we show optimal complexity of O TOL r - 4 with a significantly reduced constant to achieve a prescribed relative error tolerance TOL r . Subsequently, we propose an adaptive DLMC method combined with IS to numerically estimate rare-event probabilities, substantially reducing relative variance and computational runtimes required to achieve a given TOL r compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics. [ABSTRACT FROM AUTHOR]

In this work, we present an efficient gradient projection method for solving a class of stochastic optimal control problem with expected integral state constraint. The first order optimality condition system consisting of forward-backward stochastic differential equations and a variational equation is first derived. Then, an efficient gradient projection method with linear drift coefficient is proposed where the state constraint is guaranteed by constructing specific multiplier. Further, the Euler method is used to discretize the forward-backward stochastic differential equations and the associated conditional expectations are approximated by the least square Monte Carlo method, yielding the fully discrete iterative scheme. Error estimates of control and multiplier are presented, showing that the method admits first order convergence. Finally, we present numerical examples to support the theoretical findings. [ABSTRACT FROM AUTHOR]

In this article, we study a nonlinear stochastic control problem perturbed by multiplicative Lévy noise, where the nonlinear operator (in divergence form) satisfies p-type growth with coercivity assumptions. By using Aldous tightness criteria and Jakubowski’s version of the Skorokhod theorem on non-metric spaces along with the standard L 1 -method, we establish the existence of a path-wise unique strong solution. Formulating the associated control problem, and using a variational approach together with the convexity property of cost functional (in control variable), we establish the existence of a weak optimal solution to the underlying problem. We use the technique of Maslowski and Seidler to prove an existence of an invariant measure for uncontrolled SPDE driven with multiplicative Lévy noise. [ABSTRACT FROM AUTHOR]

In this paper, we study a family of nonlinear (conditional) expectations that can be understood as a semimartingale with uncertain local characteristics, which are prescribed by a time and path-dependent set-valued function. We show that the associated control problem coincides with both its weak and relaxed counterparts. Furthermore, we establish regularity properties of the value function and discuss their relation to Feller properties of sublinear semigroups. In the Markovian case, we provide conditions that allow us to identify the corresponding semigroup as the unique viscosity solution to a nonlinear Hamilton–Jacobi–Bellman equation. To illustrate our results, we discuss a random G -double exponential Lévy setting. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the existence of optimal solutions for control problems involving nonlinear stochastic Schrödinger equations perturbed by a multiplicative fractional Brownian motion. Finite-dimensional approximations and a linearization method are employed and lead to ε-optimal solutions for the considered problems. [ABSTRACT FROM AUTHOR]

We consider the stochastic control problem of the shallow lake and continue the work of Kossioris, Loulakis, and Souganidis. First, we generalise the characterisation of the value function as the viscosity solution of a well-posed problem to include more general recycling rates. Then, we prove approximate optimality under bounded controls and we establish quantitative estimates. Finally, we implement a convergent and stable numerical scheme for the computation of the value function to investigate properties of the optimally controlled stochastic shallow lake. This approach permits to derive tails asymptotics for the invariant distribution and to extend results of Grass, Kiseleva and Wagener. ] beyond the small noise limit. [ABSTRACT FROM AUTHOR]

The collocation method is one of the most powerful and effective techniques to solve nonlinear differential equations. This method reduces the original problem to a system of algebraic equations. Awareness of the nature of the problem and the analytical behavior of the solution has an important influence on the choice of type and number of basis functions required by the collocation method for solving the different classes of differential equations. In general, the polynomial basis functions such as the Legendre polynomials are not always an appropriate choice to approximate the functions that can be expanded based on monomial basis functions with real powers. In such cases, a combination of polynomial basis functions with some maps to replace the natural powers with arbitrary real powers can be a suitable alternative. This paper presents a collocation method based on the shifted fractional-order Legendre functions to solve the linear and nonlinear Hamilton–Jacobi–Bellman PDEs in stochastic optimal control problems. The convergence analysis of the proposed method has also been investigated. Moreover, the method is used to solve some practical problems with different solution structures such as resource extraction and Merton's portfolio selection models. Numerical results for four different examples indicated that the collocation method based on the fractional-order Legendre functions are provided. Due to the non differentiability of the solution of HJB PDEs at x = 0 for resource extraction and Merton's portfolio selection models, the Legendre collocation method cannot produce solutions with the desired accuracy. In fact, it is observed that the collocation method based on the fractional-order Legendre functions can be more efficient than the standard Legendre collocation method. Also, the sample paths of the control and state processes are obtained together with the estimate given by the Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

This paper is concerned with both nonzero-sum and zero-sum Stackelberg stochastic differential games on a finite horizon in feedback information pattern, where the control variables enter into the diffusion term. For the nonzero-sum case, a system of parabolic partial differential equations is obtained to give the verification theorem of the feedback Stackelberg equilibrium. As an example, a linear-quadratic nonzero-sum feedback Stackelberg stochastic differential game is investigated, where the diffusion term is also control-dependent. Coupled Riccati equations are introduced to express the feedback Stackelberg equilibrium, and sufficient conditions for the existence of their solutions are obtained. For the zero-sum case, a verification theorem that contains only one Hamilton–Jacobi–Bellman equation is given. In addition, the solvability of the Riccati equation, which is derived from the linear-quadratic zero-sum case with the leader's control appearing in the diffusion term, is further discussed, and a sufficient and necessary condition for the existence and uniqueness of the solution is given. [ABSTRACT FROM AUTHOR]

In the present paper, we study the existence and optimal controllability of a multi-term time-fractional stochastic system with non-instantaneous impulses. Using semigroup theory, stochastic techniques, and Krasnoselskii's fixed point theorem, we first establish the existence of a mild solution. Further, we obtain that there exists an optimal state-control pair for the system under certain assumptions. Some examples are given to illustrate the abstract results. [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]

We consider a class of restless bandit problems that finds a broad application area in reinforcement learning and stochastic optimization. We consider N independent discrete-time Markov processes, each of which had two possible states: 1 and 0 ('good' and 'bad'). Only if a process is both in state 1 and observed to be so does reward accrue. The aim is to maximize the expected discounted sum of returns over the infinite horizon subject to a constraint that only M (< N) processes may be observed at each step. Observation is error-prone: there are known probabilities that state 1 (0) will be observed as 0 (1). From this one knows, at any time t, a probability that process i is in state 1. The resulting system may be modeled as a restless multi-armed bandit problem with an information state space of uncountable cardinality. Restless bandit problems with even finite state spaces are PSPACE-HARD in general. We propose a novel approach for simplifying the dynamic programming equations of this class of restless bandits and develop a low-complexity algorithm that achieves a strong performance and is readily extensible to the general restless bandit model with observation errors. Under certain conditions, we establish the existence (indexability) of Whittle index and its equivalence to our algorithm. When those conditions do not hold, we show by numerical experiments the near-optimal performance of our algorithm in the general parametric space. Furthermore, we theoretically prove the optimality of our algorithm for homogeneous systems. [ABSTRACT FROM AUTHOR]

The authors give a stochastic maximum principle for square-integrable optimal control of linear stochastic systems. The control domain is not necessarily convex and the cost functional can have a quadratic growth. In particular, they give a stochastic maximum principle for the linear quadratic optimal control problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the stochastic maximum principle for a partially observed optimal control problem of forward-backward stochastic differential equations (FBSDEs for short) of McKean–Vlasov type driven by both a family of Teugels martingales and an independent Brownian motion. The coefficients of the system and the cost functional depending on the state of the solution process as well as its probability law and the control variable. We establish partially observed necessary conditions of optimality for this system under assumption that the control domain is supposed to be convex. Our main result is based on Girsavov's theorem and the derivatives with respect to probability law. As an application of the general theory, a partially observed linear-quadratic control problem of McKean–Vlasov type is studied in terms of stochastic filtering. [ABSTRACT FROM AUTHOR]

In this paper we study a system of decoupled forward-backward stochastic differential equations driven by a G-Brownian motion (G-FBSDEs) with non-degenerate diffusion. Our objective is to establish the existence of a relaxed optimal control for a non-smooth stochastic optimal control problem. The latter is given in terms of a decoupled G-FBSDE. The cost functional is the solution of the backward stochastic differential equation at the initial time. The key idea to establish existence of a relaxed optimal control is to replace the original control problem by a suitably regularised problem with mollified coefficients, prove the existence of a relaxed control, and then pass to the limit. [ABSTRACT FROM AUTHOR]

We explore a new kind of second-order impulsive stochastic differential system with delays. By using the Hausdorff measure of noncompactness and M $ \ddot {o} $ o ¨ nch fixed point theorem, we investigated the existence of mild solution for the considered systems. Further, we proved the existence of optimal control of the proposed system by using minimizing sequence, Lagrange functional, and Marzur lemma. An example is provided to validate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the optimal investment and reinsurance strategies of an insurer who wishes to maximize the expected power utility of its terminal wealth on finite time horizon. We are also interested in the problem of maximizing the growth rate of expected power utility per unit time on the infinite time horizon. The risk process of the insurer is described by an approximation of the classical Cramér–Lundberg process. The insurer invests in a market consisting of a bank account and multiple risky assets. The mean returns of the risky assets depend linearly on economic factors that are formulated as the solutions of linear stochastic differential equations. With this setting, Hamilton–Jacobi–Bellman equations that are derived via a dynamic programming approach have explicit solution obtained by solving a matrix Riccati equation. Hence, the optimal investment and reinsurance strategies can be constructed explicitly. Finally, we present some numerical results related to properties of our optimal strategy and the ruin probability using the optimal strategy. [ABSTRACT FROM AUTHOR]

AbstractIn this article, we analyze a robust optimal reinsurance and investment problem in a model with default risks and jumps for a general company which holds shares of an insurer and a reinsurer. The insurer’s surplus is characterized by a diffusion model and the insurer has the opportunity to buy proportional reinsurance from the reinsurer. Moreover, the insurer can invest in a risk-free asset (bank account), a risky asset (stock), and a defaultable bond. Meanwhile, the reinsurer can also invest in a risk-free asset and another risky asset. The price dynamics of the risky asset is assumed to follow a jump-diffusion model. What is more, the general company is concerned about the ambiguity and aims to find a robust optimal reinsurance and investment strategy. The objective of the general company is to maximize the minimal expected exponential utility of the weighted sums of the insurer’s and the reinsurer’s terminal wealth. Applying stochastic dynamic programming approach, we first establish the robust HJB (Hamilton-Jacobi-Bellman) equations for the post-default case and the pre-default case, respectively. Furthermore, we obtain the robust optimal strategy explicitly and present the corresponding verification theorem. Finally, we give some numerical examples to demonstrate the influences of model parameters on the robust optimal strategy. [ABSTRACT FROM AUTHOR]

We consider a method for solving the problem of optimal control of a composite dynamic system with stochastic resource constraints. The method is based on the method of hierarchical optimization, which includes iterative solution of local problems and the coordination problem. To solve local problems, we apply the model predictive control. Taking into account stochastic resource constraints leads to the solution of one-stage problems of stochastic programming. [ABSTRACT FROM AUTHOR]

This paper is devoted to a Stackelberg stochastic differential game for a linear mean-field type stochastic differential system with a mean-field type quadratic cost functional over a finite horizon. Coefficients in the state equation and weighting matrices in the cost functional are all deterministic. Closed-loop Stackelberg equilibrium strategies are introduced that are independent of initial states. It begins by solving the follower’s stochastic linear quadratic optimal control problem. By transforming the original problem into a new one with a known optimal control, the closed-loop optimal strategy of the follower is characterized by two coupled Riccati equations and a linear mean-field type backward stochastic differential equation. Then the leader turns to solve a stochastic linear quadratic optimal control problem for a mean-field type forward-backward stochastic differential equation. Necessary conditions for the existence of closed-loop optimal strategies for the leader are given by the existence of two coupled Riccati equations with a linear mean-field type backward stochastic differential equation. The solvability of Riccati equations of the leader’s problem is discussed, particularly in cases where the diffusion term of the state equation does not contain the control process of the follower. Moreover, the leader’s value function is expressed via two backward stochastic differential equations and two Lyapunov equations. Finally, a numerical example is given to show the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

This paper studies constrained Markov decision processes under the total expected discounted cost optimality criterion, with a state-action dependent discount factor that may take any value between zero and one. Both the state and the action space are assumed to be Borel spaces. By using the linear programming approach, consisting in stating the control problem as a linear problem on a set of occupation measures, we show the existence of an optimal stationary Markov policy. Our results are based on the study of both weak-strong topologies in the space of occupation measures and Young measures in the space of Markov policies. [ABSTRACT FROM AUTHOR]

We consider a stochastic differential game, where each player continuously controls the diffusion intensity of her own state process. The players must all choose from the same diffusion rate interval [ σ 1 , σ 2 ] , and have individual random time horizons that are independently drawn from the same distribution. The players whose states at their respective time horizons are among the best p ∈ (0 , 1) of all terminal states receive a fixed prize. We show that in the mean field version of the game there exists an equilibrium, where the representative player chooses the maximal diffusion rate when the state is below a given threshold, and the minimal rate else. The symmetric n-fold tuple of this threshold strategy is an approximate Nash equilibrium of the n-player game. Finally, we show that the more time a player has at her disposal, the higher her chances of winning. [ABSTRACT FROM AUTHOR]

We consider a population of finite size M, where the current number N of entities, N ∈ { 0 , 1 , 2 , ... , M } , determines its states. We geometrically analyze the amounts of information iN and iM−N, carried by the random variables N and (M−N), respectively, and population entropy S = E (i N) = E (i (M − N)). The population is modeled by a family of Birth-Death Processes of size M+1, indexed by the population utilization parameter ρ (i.e., birth/death ratio), which determines its macrostates. Considering the quantities: x = (N − E (N)) & y = (i N (ρ) − S (ρ)) and M x = ((M − N) − E (M − N)) & M y = (i M − N (ρ) − S (ρ)) as vectors, it is shown that the angles: φ = ∠ (x ; y) and ψ = ∠ ( M x ; M y) are supplementary ones, that is φ (ρ) + ψ (ρ) = π , ρ > 0. Expressions for their inner products < x , y > and < M x , M y > , being the covariances of N&iN(ρ) and (M − N) & i M − N (ρ) , respectively, which sum equals zero, with respect to the parameter ρ are also obtained. Further, what is especially important, it is revealed that φ = ψ = π / 2 , that is both inner products equal zero, at the point ρMmax, which is the value of parameter ρ at which the entropy has maximum value. Finally, for an information linear population only, it is shown that N obeys uniform distribution at the point ρM, max, that is that S (ρ M , max ) = ln ⁡ (M + 1). These results are further illustrated on populations described by truncated geometrical, binomial, and truncated Poisson distribution. [ABSTRACT FROM AUTHOR]

This paper studies the mixed zero-sum stochastic differential game problem. We allow the functionals and dynamics to be of polynomial growth. The problem is formulated as an extended doubly reflected BSDEs with a specific generator. We show the existence of solution for this doubly reflected BSDEs and we prove the existence of a saddle-point of the game. Moreover, in the Markovian framework we prove that the value function is the unique viscosity solution of the associated Hamilton–Jacobi–Bellman (HJB) equation. [ABSTRACT FROM AUTHOR]

We start with a stochastic control problem where the control process is of finite variation (possibly with jumps) and acts as integrator both in the state dynamics and in the target functional. Problems of such type arise in the stream of literature on optimal trade execution pioneered by Obizhaeva and Wang (J. Financ. Mark. 16:1–32, 2013) (models with finite resilience). We consider a general framework where the price impact and the resilience are stochastic processes. Both are allowed to have diffusive components. First we continuously extend the problem from processes of finite variation to progressively measurable processes. Then we reduce the extended problem to a linear–quadratic (LQ) stochastic control problem. Using the well-developed theory on LQ problems, we describe the solution to the obtained LQ one and translate it back to the solution for the (extended) initial trade execution problem. Finally, we illustrate our results by several examples. Among other things, the examples discuss the Obizhaeva–Wang model with random (terminal and moving) targets, the necessity to extend the initial trade execution problem to a reasonably large class of progressively measurable processes (even going beyond semimartingales), and the effects of diffusive components in the price impact process and/or the resilience process. [ABSTRACT FROM AUTHOR]

We consider an optimal liquidation problem with instantaneous price impact and stochastic resilience for small instantaneous impact factors. Within our modelling framework, the optimal portfolio process converges to the solution of an optimal liquidation problem with general semimartingale controls when the instantaneous impact factor converges to zero. Our results provide a unified framework within which to embed the two most commonly used modelling frameworks in the liquidation literature and provide a foundation for the use of semimartingale liquidation strategies and the use of portfolio processes of unbounded variation. Our convergence results are based on novel convergence results for BSDEs with singular terminal conditions and novel representation results of BSDEs in terms of uniformly continuous functions of forward processes. [ABSTRACT FROM AUTHOR]

This study investigates an optimal consumption–investment problem in which the unobserved stock trend is modulated by a hidden Markov chain that represents different economic regimes. In the classic approach, the hidden state is estimated using historical asset prices, but recent technological advances now enable investors to consider alternative data in their decision-making. These data, such as social media commentary, expert opinions, COVID-19 pandemic data and GPS data, come from sources other than standard market data sources but are useful for predicting stock trends. We develop a novel duality theory for this problem and consider a jump-diffusion process for alternative data series. This theory helps investors identify "useful" alternative data for dynamic decision-making by providing conditions for the filter equation that enable the use of a control approach based on the dynamic programming principle. We apply our theory to provide a unique smooth solution for an agent with constant relative risk aversion once the distributions of the signals generated from alternative data satisfy a bounded likelihood ratio condition. In doing so, we obtain an explicit consumption–investment strategy that takes advantage of different types of alternative data that have not been addressed in the literature. [ABSTRACT FROM AUTHOR]