Stochastic Riccati Equation Research Articles

Optimal Feedback for Stochastic Linear Quadratic Control and Backward Stochastic Riccati Equations in Infinite Dimensions

It is a longstanding unsolved problem to characterize the optimal feedbacks for general SLQs (i.e., stochastic linear quadratic control problems) with random coefficients in infinite dimensions; while the same problem but in finite dimensions was just addressed very recently. This paper is devoted to giving a solution to this problem under some assumptions which can be verified for interesting concrete models. More precisely, under these assumptions, we establish the equivalence between the existence of optimal feedback operator for infinite dimensional SLQs and the solvability of the corresponding operator-valued, backward stochastic Riccati equations. A key contribution of this work is to introduce a suitable notion of solutions (i.e., transposition solutions to the aforementioned Riccati equations), which plays a crucial role in both the statement and the proof of our main results.

Constrained mean-variance investment-reinsurance under the Cramér–Lundberg model with random coefficients

In this paper, we study an optimal mean-variance investment-reinsurance problem for an insurer (she) under the Cramér–Lundberg model with random coefficients. At any time, the insurer can purchase reinsurance or acquire new business and invest her surplus in a security market consisting of a risk-free asset and multiple risky assets, subject to a general convex cone investment constraint. We reduce the problem to a constrained stochastic linear-quadratic control problem with jumps whose solution is related to a system of partially coupled stochastic Riccati equations (SREs). Then we devote ourselves to establishing the existence and uniqueness of solutions to the SREs by pure backward stochastic differential equation (BSDE) techniques. We achieve this with the help of approximation procedure, comparison theorems for BSDEs with jumps, log transformation and BMO martingales. The efficient investment-reinsurance strategy and efficient mean-variance frontier are explicitly given through the solutions of the SREs, which are shown to be a linear feedback form of the wealth process and a half-line, respectively.

Mean-variance portfolio selection with non-linear wealth dynamics and random coefficients

This paper studies the continuous time mean-variance portfolio selection problem with one kind of non-linear wealth dynamics. To deal with the expectation constraint, an auxiliary stochastic control problem is firstly solved by two new generalized stochastic Riccati equations from which a candidate portfolio in feedback form is constructed, and the corresponding wealth process will never cross the vertex of the parabola. In order to verify the optimality of the candidate portfolio, the convex duality (requires the monotonicity of the cost function) is established to give another more direct expression of the terminal wealth level. The variance-optimal martingale measure and the link between the non-linear financial market and the classical linear market are also provided. Finally, we obtain the efficient frontier in closed form. From our results, people are more likely to invest their money in riskless asset compared with the classical linear market.

The Equivalence Conditions of Optimal Feedback Control-Strategy Operators for Zero-Sum Linear Quadratic Stochastic Differential Game with Random Coefficients

From the previous work, when solving the LQ optimal control problem with random coefficients (SLQ, for short), it is remarkably shown that the solution of the backward stochastic Riccati equations is not regular enough to guarantee the robustness of the feedback control. As a generalization of SLQ, interesting questions are, “how about the situation in the differential game?”, “will the same phenomenon appear in SLQ?”. This paper will provide the answers. In this paper, we consider a closed-loop two-person zero-sum LQ stochastic differential game with random coefficients (SDG, for short) and generalize the results of Lü–Wang–Zhang into the stochastic differential game case. Under some regularity assumptions, we establish the equivalence between the existence of the robust optimal feedback control strategy operators and the solvability of the corresponding backward stochastic Riccati equations, which leads to the existence of the closed-loop saddle points. On the other hand, the problem is not closed-loop solvable if the solution of the corresponding backward stochastic Riccati equations does not have the needed regularity.

Maximum principle for partially observed leader–follower stochastic differential game

AbstractThis paper deals with the optimal control problem for partially observed leader–follower stochastic differential game. By virtue of the classical variational method and Girsanov's theorem, the stochastic maximum principles for the follower under one type of partially observed case and for the leader under the complete information structure are derived. As applications, two partially observed cases are considered for the linear–quadratic models. Then by the stochastic filtering technique, the optimal feedback controls for the follower and the leader are represented by the new stochastic Riccati equations.

Stochastic linear quadratic optimal control problems with expectation-type linear equality constraints on the terminal states

A stochastic linear quadratic (LQ) optimal control problem with an expectation-type linear equality constraint on the terminal state is considered. Under the solvability condition on a stochastic Riccati equation and a surjectivity condition on the linear constraint mapping, the constrained stochastic LQ problem is solved completely by the Lagrangian duality theory. Some equivalent characterizations of the surjectivity condition are discussed by the controllability theory of linear control systems. Especially, the equivalence between the surjectivity condition and a Kalman-type rank condition is proved under proper conditions.

Optimal feedback controls of stochastic linear quadratic control problems in infinite dimensions with random coefficients

It is a longstanding unsolved problem to characterize the optimal feedback controls for general linear quadratic optimal control problem of stochastic evolution equation with random coefficients. A solution to this problem is given in [22] under some assumptions which can be verified for interesting concrete models, such as controlled stochastic wave equations, controlled stochastic Schrödinger equations, etc. More precisely, the authors establish the equivalence between the existence of an optimal feedback operator and the solvability of the corresponding operator-valued, backward stochastic Riccati equations. However, their result cannot cover some important stochastic partial differential equations, such as stochastic heat equations, stochastic stokes equations, etc. A key contribution of the current work is to relax the C0-group assumption of unbounded linear operator A in [22] and using the contraction semigroup assumption instead. Therefore, our result can be well applicable in the linear quadratic problem of stochastic parabolic equations. To this end, we introduce a suitable notion to the aforementioned Riccati equation, and some delicate techniques which are even new in the finite dimensional case.

Stochastic linear-quadratic control with a jump and regime switching on a random horizon

In this paper, we study a stochastic linear-quadratic control problem with random coefficients and regime switching on a random horizon $ [0,T\wedge\tau] $, where $ \tau $ is a given random jump time for the underlying state process and $ T $ a constant. We obtain the explicit optimal feedback control and explicit optimal value of the problem by solving a system of stochastic Riccati equations (SREs) with jumps on the random horizon $ [0,T\wedge\tau] $. By the decomposition approach stemming from filtration enlargement theory, we express the solution to the system of SREs with jumps in terms of another system of SREs involving only Brownian filtration on the deterministic horizon $ [0,T] $. Solving the latter system is the key theoretical contribution of this paper and we accomplish this under three different conditions, one of which seems to be new in the literature. The above results are then applied to study a mean-variance hedging problem with random parameters that depend on both Brownian motion and Markov chain. The optimal portfolio and optimal value are presented in closed forms with the aid of a system of linear backward stochastic differential equations with jumps and unbounded coefficients in addition to the SREs with jumps.

Infinite horizon Stackelberg differential games with random coefficients under control input constraint

In this paper, we study Stackelberg stochastic differential game driven by stochastic differential equation (SDE for short) with random coefficients on infinite horizon under convex control constraint. The control enters into the drift and diffusion terms simultaneously. By employing the first-order adjoint equation (backward stochastic differential equation, BSDE for short), we are able to announce Pontryagin's maximum principle for the leader's global Stackelberg solution, within adapted open-loop structure, where the term global means that the leader's domination over the entire game duration. Due to the follower's adjoint equation develops into a BSDE, the leader will be confronted with a control problem where the state equation is a kind of infinite horizon fully coupled forward-backward stochastic differential equation (FBSDE for short). We shed light on an important application in a class of infinite horizon linear-quadratic (LQ for short) Stackelberg games in which the control process of leader's is constrained in a closed convex subset Γ 1 of full space R m 1 . Whenever the control domain is postulated to be full space, we derive a new kind of infinite horizon backward stochastic Riccati equation (BSRE for short) with asymmetric coefficients.

Construction of special soliton solutions to the stochastic Riccati equation

Abstract A scheme for the analytical stochastization of ordinary differential equations (ODEs) is presented in this article. Using Itô calculus, an ODE is transformed into a stochastic differential equation (SDE) in such a way that the analytical solutions of the obtained equation can be constructed. Furthermore, the constructed stochastic trajectories remain bounded in the same interval as the deterministic solutions. The proposed approach is in a stark contrast to methods based on the randomization of solution trajectories and is not focused on the analysis of martingales. This article extends the theory of Itô calculus by directly implementing it into analytical schemes for the solution of differential equations based on the generalized operator of differentiation. The efficacy of the presented analytical stochastization techniques is demonstrated by deriving stochastic soliton solutions to the Riccati differential equation. The presented semi-analytical stochastization scheme is relevant for the investigation of the global dynamics of different biological and biomedical processes where the variation interval of the stochastic solution is predetermined by the rationale of the model.

Optimal Reinsurance and Investment Strategies Under Mean-Variance Criteria: Partial and Full Information

This paper is concerned with an optimal reinsurance and investment problem for an insurance firm under the criterion of mean-variance. The driving Brownian motion and the rate in return of the risky asset price dynamic equation cannot be directly observed. And the short-selling of stocks is prohibited. The problem is formulated as a stochastic linear-quadratic control problem where the control variables are constrained. Based on the separation principle and stochastic filtering theory, the partial information problem is solved. Efficient strategies and efficient frontier are presented in closed forms via solutions to two extended stochastic Riccati equations. As a comparison, the efficient strategies and efficient frontier are given by the viscosity solution to the HJB equation in the full information case. Some numerical illustrations are also provided.

Constrained stochastic LQ control with regime switching and application to portfolio selection

This paper is concerned with a stochastic linear-quadratic optimal control problem with regime switching, random coefficients and cone control constraint. The randomness of the coefficients comes from two aspects: the Brownian motion and the Markov chain. Using Itô’s lemma for Markov chain, we obtain the optimal state feedback control and optimal cost value explicitly via two new systems of extended stochastic Riccati equations (ESREs). We prove the existence and uniqueness of the two ESREs using tools including multidimensional comparison theorem, truncation function technique, log transformation and the John–Nirenberg inequality. These results are then applied to study mean-variance portfolio selection problems with and without short-selling prohibition with random parameters depending on both the Brownian motion and the Markov chain. Finally, the efficient portfolios and efficient frontiers are presented in closed forms.

Constrained stochastic LQ control on infinite time horizon with regime switching

This paper is concerned with a stochastic linear-quadratic (LQ) optimal control problem on infinite time horizon, with regime switching, random coefficients, and cone control constraint. To tackle the problem, two new extended stochastic Riccati equations (ESREs) on infinite time horizon are introduced. The existence of the nonnegative solutions, in both standard and singular cases, is proved through a sequence of ESREs on finite time horizon. Based on this result and some approximation techniques, we obtain the optimal state feedback control and optimal value for the stochastic LQ problem explicitly. Finally, we apply these results to solve a lifetime portfolio selection problem of tracking a given wealth level with regime switching and portfolio constraint.

Global solutions of stochastic Stackelberg differential games under convex control constraint

This paper is concerned with a Stackelberg stochastic differential game, where the systems are driven by stochastic differential equation (SDE for short), in which the control enters the randomly disturbed coefficients (drift and diffusion). The control region is postulated to be convex. By making use of the first-order adjoint equation (backward stochastic differential equation, BSDE for short), we are able to establish the Pontryagin’s maximum principle for the leader’s global Stackelberg solution, within adapted open-loop structure and closed-loop memoryless information one, respectively, where the term global indicates that the leader’s domination over the entire game duration. Since the follower’s adjoint equation turns out to be a BSDE, the leader will be confronted with a control problem where the state equation is a kind of fully coupled forward–backward stochastic differential equation (FBSDE for short). As an application, we study a class of linear–quadratic (LQ for short) Stackelberg games in which the control process is constrained in a closed convex subset Γ of full space R m . The state equations are represented by a class of fully coupled FBSDEs with projection operators on Γ . By means of monotonicity condition method, the existence and uniqueness of such FBSDEs are obtained. When the control domain is full space, we derive the resulting backward stochastic Riccati equations.

Strong error estimates for a space-time discretization of the linear-quadratic control problem with the stochastic heat equation with linear noise

Abstract We propose a time-implicit, finite-element-based space-time discretization of the necessary and sufficient optimality conditions for the stochastic linear-quadratic optimal control problem with the stochastic heat equation driven by linear noise of type $[X(t)+\sigma (t)]\,\,\textrm{d}W(t)$ and prove optimal convergence w.r.t. both space and time discretization parameters. In particular, we employ the stochastic Riccati equation as a proper analytical tool to handle the linear noise, and thus extend the applicability of the earlier work by Prohl & Wang (2021, Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation. ESAIM Control Optim. Calc. Var., 27, 54), where the error analysis was restricted to additive noise.

Control óptimo estocásticio LQR con ruido blanco y colorado: La técnica de programación dinámica

This work deals with LQR optimal control problems where the randomness of the dynamic systems evolves according to: (a) multiplicative and additive white noise, (b) mixture white noise and (c) white noise and colored noise. Total cost quadratic and discounted cost quadratic are studied. The white noise is represented by the Wiener process whereas the colored noise is represented by the solution of the Ornstein-Uhlenbeck’s stochastic dierential equation, whit initial condition is constant or normally distributed. In order to find the value functions and optimal policies, as well as algebraic and stochastic Riccati equations the dynamic programming technique was used. The theoretical results are illustrated by two applications. Numerical simulations are carried out using Matlab.

A BSDE approach to stochastic linear quadratic control problem

AbstractIn this article, we study a kind of linear quadratic optimal control problem driven by forward–backward stochastic differential equations (FBSDEs in short) with deterministic coefficients. The cost functional is defined by the solution of FBSDEs. By means of the Girsanov transformation, the original issue is turned equivalently into the classical LQ problem. By functional analysis approach, some necessary and sufficient conditions for the existence of optimal controls have been obtained. Moreover, we investigate the relationship between two groups of first‐order and second‐order adjoint equations. A new stochastic Riccati equation is derived, which leads to the state feedback form of optimal control. By introducing a new Hamiltonian function with an exponential factor, we establish the stochastic maximum principle to deal with the stochastic linear quadratic problem for forward–backward stochastic system with nonconvex control domain using first‐order adjoint equation. An illustrative example is given as well.

Indefinite stochastic linear-quadratic optimal control problems with random coefficients: Closed-loop representation of open-loop optimal controls

This paper is concerned with a stochastic linear-quadratic optimal control problem in a finite time horizon, where the coefficients of the control system are allowed to be random, and the weighting matrices in the cost functional are allowed to be random and indefinite. It is shown, with a Hilbert space approach, that for the existence of an open-loop optimal control, the convexity of the cost functional (with respect to the control) is necessary; and the uniform convexity, which is slightly stronger, turns out to be sufficient, which also leads to the unique solvability of the associated stochastic Riccati equation. Further, it is shown that the open-loop optimal control admits a closed-loop representation. In addition, some sufficient conditions are obtained for the uniform convexity of the cost functional, which are strictly more general than the classical conditions that the weighting matrix-valued processes are positive (semi-) definite.

A kind of optimal investment problem under inflation and uncertain time horizon

In this paper, we study a kind of optimal investment problem under inflation and uncertain time horizon. It can be generally formulated into a stochastic optimal control problem. In particular for the constant relative risk aversion utility, we employ the method of completion of squares to give an explicit form of optimal portfolio and maximum utility by the solution of a stochastic Riccati equation, whose wellposedness is obtained and also of significance in its own right. The most distinguishing result of our work is that the randomness of exit time actually affects not only the optimal portfolio but also the maximum utility in the case of stochastic market parameters. Moreover, we present several numerical examples to show the application of theoretical results and further discuss the influence of inflation and random time horizon from the economic viewpoint.

Backward Stochastic Riccati Equation with Jumps Associated with Stochastic Linear Quadratic Optimal Control with Jumps and Random Coefficients

In this paper, we investigate the solvability of matrix valued backward stochastic Riccati equations with jumps (BSREJ), which are associated with a stochastic linear quadratic (SLQ) optimal control problem with random coefficients and driven by both Brownian motion and a Poisson process. By the dynamic programming principle, Doob--Meyer decomposition, and inverse flow technique, the existence and uniqueness of the solution for the BSREJ are established. The difficulties addressed in this issue not only are brought from the high nonlinearity of the generator of the BSREJ like the case driven only by Brownian motion, but also from that (i) the inverse flow of the controlled linear stochastic differential equation driven by Poisson process may not exist without additional technical conditions, and (ii) showing that the inverse matrix term involving a jump process in the generator is well-defined. Utilizing the structure of the optimal control problem, we overcome these difficulties and establish the existence of the solution. In addition, we show the construction of the optimal feedback control with the help of the Riccati equation and the relation between the solution of the Riccati equation and the value function of the SLQ problem, which also implies the uniqueness of BSREJ.