1 Introduction
The Markowitz paradigm, also termed as the meanvariance paradigm, is often characterized as dealing with portfolio risk and (expected) return
[1, 2, 3]. A typical example of risk concerns in the current online market is the evolution of prices of the digital and cryptocurrencies (bitcoin, litecoin, ethereum, dash, and other altcoins etc). Variance plays a base model for many risk measures. Here, we address variance reduction problems when several decisionmaking entities are involved. When the decisions made by the entities influence each other, the decisionmaking is said to be interactive (interdependent). Such problems are termed as game problems. Game problems in which the state dynamics is given a linear stochastic system with a Brownian motion and a cost functional that is quadratic in the state and the control, is often called the linearquadratic Gaussian (LQG) games. For generic LQG game problems under perfect state observation, the optimal strategy of the decisionmaker is a linear statefeedback strategy which is identical to an optimal control for the corresponding deterministic linearquadratic game problem where the Brownian motion is replaced by the zero process. Moreover the equilibrium cost only differs from the deterministic game problem’s equilibrium cost by the integral of a function of time. However, when the diffusion (volatility) coefficient is state and controldependent, the structure of the resulting differential system as well as the equilibrium cost vector are modified. These results were widely known in both dynamic optimization, control and game theory literature. For both LQG control and LQG zerosum games, it can be shown that a simple square completion method, provides an explicit solution to the problem. It was successfully developed and applied by Duncan et al.
[4, 5, 6, 7, 8, 9] in the meanfieldfree case. Moreover, Duncan et al. have extended the direct method to more general noises including fractional Brownian noises and some nonquadratic cost functionals on spheres and torus. Inspired by applications in engineering (internet connexion, battery state etc) and in finance (price, stock option, multicurrency exchange etc) were not only Gaussians but also jump processes (Poisson, Lévy etc) play important features, the question of extending the framework to linearquadratic games under state dynamics driven by jumpdiffusion processes were naturally posed. Adding a Poisson jump and regime switching may allow to capture in particular larger jumps which may not be captured by just increasing diffusion coefficients. Several examples such as multicurrency exchange or cloudserver rate allocation on blockchains are naturally in matrix form.The main goal of this work is to investigate whether Direct Method can be used to solve matrixvalued risksensitive and adversarial robust meanfieldtype game problems which are nonstandard problems [10]. To do so, we modify the state dynamics to include conditional meanfield terms which are

the conditional expectation of the matrixvalued state with respect to the filtration of the common noise which is a regime switching process, is added to the drift, diffusion, and jump coefficient functionals.

the conditional expectation of the matrixvalued controlactions, is included in the drift, diffusion, jump coefficient functional.
We also modify the instant cost and terminal cost function to include

the square of the conditional expectation of the matrixvalued state and

the square of the conditional expectation of the matrixvalued control action.
Involving these features lead to matrixvalued meanfieldtype game theory which focuses on (matrixvalued) games with distributiondependent quantityofinterest such as payoff, cost and state dynamics. It can be seen as the multiple agent generalization of single agent (matrixvalued) meanfieldtype control problem [10].
If the state dynamics and or the cost functional involve a meanfield term (such as the expectation of the state and or the expectation of the control actions), the game is said to be a LQG game of meanfield type, or MFTLQG. For such game problems, various solution methods such as the Stochastic Maximum Principle (SMP) ([10]) and the Dynamic Programming Principle (DPP) with HamiltonJacobiBellmanIsaacs equation and FokkerPlanckKolmogorov equation have been proposed [11, 10, 12].
If the state dynamics and or the cost functional involve a conditional meanfield term (such as the conditional expectation of the matrixvalued state and or the conditional expectation of the matrixvalued control actions), the game is said to be a matrixvalued LQG game of conditional meanfield type, or cMFTLQG (or conditional McKeanVlasov matrixvalued LQG games). If in addition, the matrixvalued state dynamics is driven by a matrixvalued jumpdiffusion process, then the problem is termed as a cMFTLQJD matrixvalued game problem. We aim to study the behavior of such cMFTLQJD matrixvalued game problems when conditional meanfield terms are involved.
Games with global uncertainty and common noise have been widely suggested in the literature. Anonymous sequential and meanfield games with common noise can be considered as a natural generalization of the meanfield game problems (see [13, 14] and the references therein). The works in [15, 16] considered meanfield games with common noise and obtained optimality system that determine meanfield equilibria conditioned of the information. The work in [18, 19] provides sufficiency conditions for wellposedness of meanfield games with common noise and a major player. Existence of solutions of the resulting stochastic optimality systems are examined in [17]. A probabilistic approach to the master equation is developed in [43]. In order to determine the optimal strategies of the decisionmaker, the previous works used a maximum principle or a master equation which involves a stochastic FokkerPlanck equation (see [19, 20, 21, 22] and the references therein).
Most studies illustrated meanfield game methods in the linearquadratic game with infinite number of decisionmakers [23, 24, 25, 26, 27]. These works assume indistinguishability within classes and the cost functionals were assumed to be identical or invariant per permutation of decisionmakers indexes. Note that the indistinguishability assumption is not fulfilled for many interesting problems such as variance reduction or and risk quantification problems in which decisionmakers have different sensitivity towards the risk. One typical and practical example is to consider a multilevel building in which every resident has its own comfort zone temperature and aims to use the Heating, Ventilating, and Air Conditioning (HVAC) system to be in its comfort temperature zone and maintain it within its own comfort zone. This problem clearly does not satisfy the indistinguishability assumption used in the previous works on meanfield games. Therefore, it is reasonable to look at the problem beyond the indistinguishability assumption. Here we drop these assumptions and dealt with the problem directly with arbitrarily finite number of decisionmakers. In the LQmeanfield game problems the state process can be modeled by a set of linear stochastic differential equations of McKeanVlasov and the preferences are formalized by quadratic or exponential of integral of quadratic cost functions with meanfield terms. These game problems are of practical interests and a detailed exposition of this theory can be found in [10, 28, 29, 30, 31, 32]
. The popularity of these game problems is due to practical considerations in consensus problems, signal processing, pattern recognition, filtering, prediction, economics and management science
[20, 33, 34, 35].To some extent, most of the riskneutral versions of these optimal controls are analytically and numerically solvable [7, 29, 9, 4, 5]. On the other hand, the linear quadratic robust setting naturally appears if the decision makers’ objective is to minimize the effect of a small perturbation and related variance of the optimally controlled nonlinear process. By solving a linear quadratic game problem of meanfield type, and using the implied optimal control actions, decisionmakers can significantly reduce the variance (and the cost) incurred by this perturbation. The variance reduction and minmax problems have very interesting applications in risk quantification problems under adversarial attacks and in security issues in interdependent infrastructures and networks [37, 36, 38, 35, 39]. In [41], the control for the evacuation of a multilevel building is designed by means of meanfield games and meanfieldtype control. In [42], electricity price dynamics in the smart grid is analyzed using a meanfieldtype game approach under common noise which is of diffusion type. Riskneutral LinearQuadratic MFTLQJD games have been studied for the one dimensional case in [47].
Our contribution
In this paper, we use a simple argument that gives the riskneutral equilibrium strategy and robust adversarial meanfieldtype saddle point for a class of cMFTLQJD matrixvalued games without use of the wellknown solution methods (SMP and DPP). We apply a basic Itô’s formula following by a square completion method in the riskneutral/adversarial meanfieldtype matrixvalued game problems. It is shown that this method is well suited to cMFTLQJD riskneutral/robust games as well as to variance reduction performance functionals with jumpdiffusionregime switching common noise. Applying the solution methodology related to the DPP or the SMP requires involved (stochastic) analysis and convexity arguments to generate necessary and sufficient optimality criteria. We avoid all this with this method.
Zerosum stochastic differential games are important class of stochastic games. The optimality system leads to an HamiltonJacobiBellmanIsaacs (HJBI) system of equations, which is an extension of the HJB equation to stochastic differential games. When common noise is involved it becomes a stochastic HJBI system. Obviously, studying wellposedness, existence and uniqueness of such is a challenging task because of the minmax and maxmin operators. Usually, upper value and lower value equilibrium payoffs are investigated. In addition, when conditional meanfield terms are involved as it is the case here, the system is coupled with a stochastic FokkerPlanckKolmogorov system leading to a master system. Here we provide an easy way to solve such a system of means of a direct method.
Relationship between risksensitive and roust conditional meanfieldtype games are established in the case without jump and with a single regime.
Our contribution can be summarized as follows. We formulate and solve a matrixvalued linearquadratic meanfieldtype game described by a linear jumpdiffusion dynamics and a meanfielddependent quadratic cost functional that is conditioned a common noise which includes not only a Brownian motion but also a jump process and regime switching. Since the matrices are switching dependent, they can be seen as random coefficients. The optimal strategies for the decisionmakers are given semiexplicitly using a simple and direct method based on square completion, suggested in Duncan et al. in e.g. [6] for the meanfield free case. This approach does not use the wellknown solution methods such as the Stochastic Maximum Principle and the Dynamic Programming Principle with stochastic HamiltonJacobiBellmanIsaacs equation and stochastic FokkerPlanckKolmogorov equation. It does require extended stochastic backwardforward partial integrodifferential equations (PIDE) to solve the problem. In the riskneutral linearquadratic meanfieldtype game with perfect state observation and with common noise, we show that, generically there is a minmax strategy to the conditional mean of the state and provide a sufficient condition of existence of meanfieldtype saddle point. Sufficient conditions for existence and uniqueness of robust meanfield equilibria are obtained when the horizon length is small enough and the Riccati coefficients are almost surely positive.
In addition, this work extends the results in [40] in various ways:

Extension to matrixform of arbitrary dimensions.

the common noise which is a regime switching was not considered in [40].

The solution here involves a matrixvalued differential system which differs from the results in [40].
To solve the aforementioned problem in a semiexplicit way, we follow a direct method. The method starts by identifying a partial guess functional where the coefficient functionals are random and regime switching dependent. Then, it uses Itô’s formula for jumpdiffusionregime switching processes, followed by a completion of squares for both control and conditional mean control. Finally, the processes are identified using an orthogonal decomposition technique and stochastic differential equations are derived in a semiexplicit way. The procedure is summarized in Figure 1. The contributions of this work are summarized in Table 1.
Reference  
Cost function Terms 
[40]  this work 
riskneural noncooperation 
, 1D  , matrixvalued 
riskneural fullcooperation  , 1D  , matrixvalued 
riskneural adversarial/robust  , 1D  , matrixvalued 
risksensitive noncooperation  , matrixvalued  
risksensitive fullcooperation  , matrixvalued  
risksensitive adversarial/robust  , matrixvalued  
Drift Terms 



Diffusion Terms 

Jump Terms 

Switching Term 

Cost function Terms 

Number of Decision makers  
One single team  
Multiple players  
Two adversaries  
Two adversarial teams 
To the best of the authors knowledge this is the first work to consider regime switching in matrixvalued meanfieldtype game theory.
A brief outline of the rest of the paper follows. The next section introduces a generic game model. After that, the MFLQJD conditional meanfieldtype game problem is investigated and its solution is presented. The last section concludes the paper.
Notation and Preliminaries
We introduce the following notations. Let be a fixed time horizon and
be a given filtered probability space.The filtration
is the natural filtration of the union augmented by null sets of In practice, is used to capture smaller disturbance and is used for larger jumps of the system.An admissible control strategy of the decisionmaker is an adapted and squareintegrable process with values in a . We denote the set of all admissible controls by .
2 Problem Formulation
We consider decisionmakers over the time horizon Each decisionmaker chooses a matrixvalued strategy over the horizon The state satisfies the following matrixvalued linear jumpdiffusionregime switching system of meanfield type:
(1) 
where , , , , , , is a matrixvalued Brownian motion, , is a regime switching process with transition rates satisfying , . is a matrixvalued Poisson random process with compensated process is a matrix of Radon measure over the set of jump sizes is the filtration generated by the regime switching process By abuse of notation we omit the use of and for the left value of switching and the jump respectively.
To the state system (1), we associate the cost functional of decisionmaker
(2) 
where being the adjoint operator of (transposition), The coefficients are possibly time and regimeswitching dependent with values in
The reader may want to know why a matrixvalued dynamics instead of a vectorvalued dynamics. One typical example is the evolution of the rate of change of blockchain tokens and classical currencies. The exchange rate between token and token is given by the entries of with value Since these tokens are correlated one obtains a matrixvalued process
2.1 RiskNeutral
We provide basic definitions of the riskneutral problems and their solution concepts.
Definition 1 (MeanFieldType RiskNeutral BestResponse)
Given a riskneutral best response strategy of decisionmaker is a strategy that solves subject to (1). The set of riskneutral best responses of is denoted by
Definition 2 (MeanFieldType RiskNeutral Nash Equilibrium)
A meanfieldtype riskneutral Nash equilibrium is a strategy profile of all decisionmakers such that for every decisionmaker
Definition 3 (MeanFieldType RiskNeutral Full Cooperation)
A meanfieldtype riskneutral fully cooperative solution is a strategy profile of all decisionmakers such that
where
is the social (global) cost.
Definition 4 (MeanFieldType RiskNeutral Saddle Point Solution )
The set of decisionmakers is divided into two teams. A team of defenders and a team of attackers. The defenders set is
and the attackers set is
A meanfieldtype riskneutral saddle point is a strategy profile of the team of defenders and of the team of attackers such that
and is the value of the adversarial team (riskneutral) game, where
2.2 RiskSensitive
We provide basic definitions of risksensitive problems and their solution concepts.
Definition 5 (MeanFieldType RiskSensitive BestResponse)
Given a risksensitive best response strategy of decisionmaker is a strategy that solves
subject to (1). The set of risksensitive best responses of is denoted by
For
the risksensitive loss functional
includes not only the first moment
but also all the higher momentsDefinition 6 (MeanFieldType RiskSensitive Nash Equilibrium )
A meanfieldtype risksensitive Nash equilibrium is a strategy profile of all decisionmakers such that for every decisionmaker
Definition 7 (MeanFieldType RiskSensitive Full Cooperation)
A meanfieldtype risksensitive fully cooperative solution is a strategy profile of all decisionmakers such that
Definition 8 (MeanFieldType RiskSensitive Saddle Point Solution )
The set of decisionmakers is divided into two teams. A team of defenders and a team of attackers. The defenders set is and the attackers set is A meanfieldtype risksensitive saddle point is a strategy profile of the team of defenders and of the team of attackers such that
3 Main Results
This section presents the main results of the article.
3.1 RiskNeutral Case
We start with the riskneutral Nash equilibrium problem.
Theorem 1
Assume that are symmetric positive definite. Then the matrixvalued meanfieldtype (riskneutral) Nash equilibrium strategy and the (riskneutral) equilibrium cost are given by:
where and solve the following differential equations:
(3) 
whenever these differential equations have a unique solution that does not blow up within .
Under the symmetric matrix assumption above, it is easy to check that if is a solution then is also a solution. Therefore
From the state system (1), the conditional expected matrix where is the natural filtration of the regime switching process up to solves the following system:
which means that
which will be used for feedback in the optimal strategy. Next, we provide a semiexplicit solution to the fullcooperation case.
Corollary 1
Assume that are symmetric positive definite. The fully cooperative solution of the problem is given by
where , and solve the following differential equations:
(4) 
Notice that these Riccati equations have positive solution , and and there is no blow up in .
The proof of Corollary 1 is immediate from Theorem 1 by one single team and with a choice vector of matrices
Corollary 2
Assume that are symmetric positive definite for and are symmetric positive definite for We assume that The adversarial game problem of the team attackers and the team of defenders has a saddle and it is given by
where , and solve the following differential equations:
(5) 
The proof of Corollary 2 is immediate from Theorem 1 by considering two adversarial teams and with choice vector of matrices and respectively.
Notice that the Riccati equations in (9) have positive definite solution if in addition
which does not blow up within , and positive solution if in addition
within Next, we study the risksensitive case and point out some facts regarding the comparison of its solution with respect to the riskneutral case as the risksensitivity index vanishes.
3.2 RiskSensitive Case
A risk averse decisionmaker (with cost functional) is a decisionmaker who prefers higher cost with known risks rather than lower cost with unknown risks. In other words, among various control strategies giving the same cost with different level of risks, this decisionmaker always prefers the alternative with the lowest risk.
When the exponential martingale of compensated Poisson random process times a linear process yields to an exponential nonquadratic terms
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