Charles-Albert Lehalle
email: c.lehalle@imperial.ac.uk

Charafeddine Mouzouni
email: mouzouni@math.univ-lyon1.fr

Univ Lyon

A MEAN FIELD GAME OF PORTFOLIO TRADING AND ITS CONSEQUENCES ON PERCEIVED CORRELATIONS

Keywords: 2010 Mathematics Subject Classification. 91G80 mean field games, market microstructure, crowding, multi-asset portfolio, optimal trading, optimal stochastic control

. It starts by extending it to portfolios of correlated instruments. This leads to several original contributions: first that hedging strategies naturally stem from optimal liquidation schemes on portfolios. Second we show the influence of trading flows on naive estimates of intraday volatility and correlations. Focussing on this important relation, we exhibit a closed form formula expressing standard estimates of correlations as a function of the underlying correlations and the initial imbalance of large orders, via the optimal flows of our mean field game between traders. To support our theoretical findings, we use a real dataset of 176 US stocks from January to December 2014 sampled every 5 minutes to analyze the influence of the daily flows on the observed correlations. Finally, we propose a toy model based approach to calibrate our MFG model on data.

INTRODUCTION

Optimal liquidation emerged as an academic field with two seminal papers: one [START_REF] Almgren | Value under liquidation[END_REF] focussed on the balance between trading fast (to minimize the uncertainty of the obtained price) and trading slow (to minimize the "market impact", i.e. the detrimental influence of the trading pressure on price moves) for one representative instrument; while the other [START_REF] Bertsimas | Optimal control of execution costs[END_REF] focussed on a portfolio of tradable instruments, shedding light on the interplay with correlations of price returns and market impact. The last twenty years have seen a lot of proposals to sophisticate the single instrument case (see these reference books [START_REF] Burgot | Market Microstructure in Practice[END_REF][START_REF] Cartea | Algorithmic and High-Frequency Trading[END_REF][START_REF] Guéant | The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making[END_REF] for typical models and references) but very few on extending it to portfolios of multiple assets (with the notable exception of [START_REF] Guéant | A convex duality method for optimal liquidation with participation constraints[END_REF]). Moreover, the usual framework for optimal execution is the one of one large privileged agent facing a "mean-field" or a "background noise" made of the sum of behaviours of other market participants, and academic literature seldom tackles the strategic interaction of many market participants seeking to execute large orders.

More recently, game theory has been introduced in this field. First around cases with few agents, like in [START_REF] Schied | High-frequency limit of Nash equilibria in a market impact game with transient price impact[END_REF], and then by [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF][START_REF] Firoozi | Mean Field Game epsilon-Nash equilibria for partially observed optimal execution problems in finance[END_REF][START_REF] Jaimungal | Mean-Field Game Strategies for a Major-Minor Agent Optimal Execution Problem[END_REF] relying on Mean Field Games (MFG) to get rid of the combinatorial complexity of games with few players, considering a lot of agents, such that their aggregated behaviour reduces to a "anonymous mean field of liquidity", shared by all of them.

In this paper, we clearly start within the framework and results obtained by [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF] and extend them to the case of a portfolio of tradable instruments. Our agents are the same as in this paper: optimal traders seeking to buy and sell positions given at the start of the day. That for, they rely on the stochastic control problem well defined for one instrument in [START_REF] Cartea | Algorithmic and High-Frequency Trading[END_REF], which result turns to be deterministic because of its linear-quadratic nature: minimize the cost of the trading under risk-averse conditions and a terminal cost. This framework can be compared to the one used by [START_REF] Bertsimas | Optimal control of execution costs[END_REF] in their section on portfolio, with a diagonal matrix for the market impact, and in a game played by a continuum of agents. Note that in all these papers, including ours, the time scale is large enough to not take into account orderbook dynamics, and small enough to be used by traders and dealing desks; our typical terminal time goes from one hour to several days, and time steps have to be read in minutes. In their paper, Cardaliaguet and Lehalle have shown how a continuum of such agents with heterogenous preferences can emulate a mix of typical brokers (having a large risk aversion and terminal cost), and opportunistic traders (with a low risk aversion). It will be the same for us. But while their paper only addresses the strategic behavior of investors on a one single financial instrument this one handles the case of a portfolio of correlated assets. In the real applications, a financial instrument is rarely traded on its own; most investors construct diversified or hedged portfolios or index trackers by simultaneously buying and selling a large number of assets.

This has motivated the present work in which we introduce an extension of the initial Cardaliaguet-Lehalle framework to the case of a multi-asset portfolio. On the one hand, this extension allows to cover a new type of trading strategies, such as Program Trading (executing large baskets of stocks), Arbitrage Strategies (which aims to benefit from discrepancies in the dynamics of two or more assets), Hedging Strategies (where a round trip on a second asset -typically a very liquid one -can be used to partially hedge the price risk in the execution process of a given asset), and Index Tracking (i.e. following the composition given by a formula, like in factor investing, or simply following the market capitalization of a list of instruments). On the other hand, it enables us to understand the dependence structure between the market orders flows at "equilibrium", and assess their influence on standard estimates of the covariance (or correlation) matrix of asset returns. These questions were independently raised by some authors and studied in seldom empirical and theoretical works (see e.g. [START_REF] Benzaquen | Dissecting cross-impact on stock markets: An empirical analysis[END_REF][START_REF] Boulatov | Informed trading and portfolio returns[END_REF][START_REF] Cont | Running for the exit: distressed selling and endogenous correlation in financial markets[END_REF][START_REF] Hasbrouck | Common factors in prices, order flows, and liquidity[END_REF][START_REF] Mastromatteo | Trading lightly: Cross-impact and optimal portfolio execution[END_REF] and references therein).

Following the seminal paper [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF], we assume that the market impact is either instantaneous or permanent, and that the public prices -of all assets -are influenced by the permanent market impact of all market participants. Conversely, since the agents are affected by the public prices, they aim to anticipate the "market mean field" (i.e. the market trend due to the market impact of the mean field of all agents) by using all the information they have in order to minimize their exposition to the other agents' impact. As explained in [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF] this leads to a Nash equilibrium configuration of MFG type, in which all agents anticipate the average trading speed of the population and adjust their execution accordingly. We refer the reader to Section 2 for a more detailed explanation of the Mean Field Game model. In the context of a MFG with multi-asset portfolio, the strategic interaction between the agents during a trading day leads to a non-trivial relationship between the assets' order flows, which in turn generates a non-trivial impact on the intraday covariance (or correlation) matrix of asset returns. In Section 3, we provide an exact formula for the excess covariance matrix of returns that is endogenously generated by the trading activity, and we show that the magnitude of this effect is more significant when the market impact is large. This means for an highly crowded market, illiquid products or large initial orders (cf. Section 3). These results can be related to the ones of [START_REF] Cont | Running for the exit: distressed selling and endogenous correlation in financial markets[END_REF], except that in this paper we do not focus our attention on distressed sells only; we are able to capture the influence of the usual variations of trading flows to deformations of the naive estimate of the covariance matrix of a portfolio of assets that are simultaneously traded. We also carry out several numerical simulations and apply our results in an empirical analysis which is conducted on a database of market data from January to December 2014 for a pool of 176 US stocks. At first, we exhibit the theoretical relation between the intraday covariance matrix of net traded flows and the standard intraday covariance matrix is increasing, then we use this relation to estimate some parameters of our model, including the market impact coefficients (cf. Section 3). Next, we normalize the covariance matrix of returns to compute the intraday median diagonal pattern (across diagonal terms), and the intraday median off-diagonal pattern (across off-diagonal terms) (cf. Section 3.3), as a way of characterizing the typical intraday evolution for diagonal and off-diagonal terms. It allows us to obtain empirically the well-known intraday pattern of volatility that is in line with our model, and we show that it flattens out as the typical size of transactions diminishes. In such a case the empirical volatility is close to its "fundamental" value (cf. Figure 4). Finally, we propose a toy model based approach to calibrate our MFG model on data.

This paper is structured as follows: in Section 2 we formulate the problem of optimal execution of a multi-asset portfolio inside a Mean Field Game. We derive a MFG system of PDEs and prove uniqueness of solutions to that system for a general Hamiltonian function. Then we construct a regular solution in the quadratic framework, which will be considered throughout the rest of the paper. Next, we provide a convenient numerical scheme to compute the solution of the MFG system, and present several examples of an agent's optimal trading path, and the average trading path of the population. Section 3 is devoted to the analysis of the crowd's trading impact on the intraday covariance matrix of returns. At the MFG equilibrium configuration, we derive a formula for the impact of assets' order flows on the dependence structure of asset returns. Next, we carry out numerical simulations to illustrate this fact, and apply our results inan empirical analysis on a pool of 176 US stocks.

OPTIMAL PORTFOLIO TRADING WITHIN THE CROWD

2.1. The Mean Field Game Model. Consider a continuum of investors (agents), which are indexed by a parameter a. Each agent has to trade a portfolio corresponding to instructions given by a portfolio manager. Think about a continuum of brokers or dealing desks executing large orders given by their clients. The portfolios are made of desired positions in a universe of d different stocks (or any financial assets). The initial position of any agent a is denoted by q a 0 := (q 1,a 0 , ..., q d,a 0 ). For any i, when the initial inventory q i,a 0 is positive, it means the agent has to sell this number of shares (or contracts) whereas when it is negative, the agent has to sell this amount. Given a common horizon T > 0, we suppose that all the investors have to sell or buy within the trading period [0, T ]. This means the agent has to sell this number of shares (or contracts) whereas when it is negative, the agent has to buy this amount.

The intraday position of each investor a is modeled by a R d -valued process (q a t ) t∈[0,T ] which has the following dynamics:

dq a t = v a t dt, q a (0) = q a 0 .
The investor controls its trading speed (v a t ) t := (v 1,a t , ..., v d,a t ) t through time, in order to achieve its trading goal. Following the standard optimal liquidation literature, we assume that, for each stock, the dynamics of the mid-price can be written as:

(2.1)

dS i t = σ i dW i t + α i µ i t dt, i = 1, ..., d;
where σ i > 0 is the arithmetic volatility of the i th stock, and α 1 , ..., α d are nonnegative scalars modeling the magnitude of the permanent market impact. Here (W 1 t , ..., W d t ) t 0 are d correlated Wiener processes, and the process (µ t ) t∈[0,T ] := (µ 1 t , ..., µ d t ) t corresponds to the average trading speed of all investors across the portfolio of assets. Throughout, we shall denote by Σ the covariance matrix of the d-dimensional process (W t ) t∈[0,T ] := (σ 1 W 1 t , ..., σ d W d t ) t∈[0,T ] and suppose that Σ is not singular. The performance of any investor a is related to the amount of cash generated throughout the trading process. Given the price vector (S t ) t∈[0,T ] := (S 1 t , ..., S d t ) t∈[0,T ] , the amount of cash (X a t ) t∈[0,T ] on the account of the trader a is given by:

X a t = - t 0 v a s • S s ds - d i=1 t 0 V i L i v i,a s V i ds,
where the positive scalars V 1 , ..., V d denotes the magnitude of daily market liquidity (in practice the average volume traded each day can be used as a proxy for this parameter) of each asset. Here L 1 , ..., L d are the execution cost functions (similar to the ones of [START_REF] Guéant | A convex duality method for optimal liquidation with participation constraints[END_REF]), modelling the instantaneous component of market impact, which takes part in the average cost of trading. The family of functions L i : R → R are assumed to fulfil the following set of assumptions:

• L i (0) = 0;
• L i is strictly convex and nonnegative;

• L i is asymptotically super-linear, i.e. lim |p|→+∞

L i (p) |p| = +∞.
The initial Cardaliaguet-Lehalle model [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF], corresponds to d = 1, and a quadratic liquidity function of the form L(p) = κ|p| 2 .

In this paper, we consider a reward function that is similar to [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF], and corresponding to Implementation Shortfall (IS) orders. In this specific case the reward function of any investor a is given by: (2.2)

U a (t, x, s, q; µ) := sup v E x,s,q X a T + q a T • (S T -A a q a T ) - γ a 2 T t q a s • Σq a s ds ,
where A a := diag(A a 1 , ..., A a d ), A a i > 0, and γ a is a non-negative scalar which quantifies the investor's risk aversion. That is when γ a = 0 the investor is indifferent about holding inventories through time, while when γ a is large the investor attempt to liquidate as fast as possible. The quadratic term q a T • (S T -A a q a T ) penalizes non-zero terminal inventories. One should note that the expression of the reward function (2.2) is derived by considering that agents are risk-averse with CARA utility function. We omit the details and refer the reader to [START_REF] Guéant | The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making[END_REF]Chapter 5].

The Hamilton-Jacobi equation associated to (2.2) is

0 = ∂ t U a - γ a 2 q • Σq + 1 2 Tr ΣD 2 s U a + Aµ • ∇ s U a + sup v v • ∇ q U a -v • s + d i=1 V i L i v i V i ∇ x U a ,
with the terminal condition U a (T , x, s, q; µ) = x + q • (s -A a q).

In all this paper we set A := diag(α 1 , ..., α d ). Due to the simplifications that we will obtain afterwards, we suppose that µ = (µ t ) t∈[0,T ] is a deterministic process, so that the HJB equation above is deterministic. When µ is a random process, that is adapted to the natural filtration of (W t ) t∈[0,T ] , we obtain a stochastic backward HJB equation which requires a specific treatment (cf. [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF]). Following the approach of [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF], we consider the following ersatz:

U a (t, x, s, q; µ) = x + q • s + u a (t, q; µ), which entails the following HJB equation for u a :

(2.3) γ a 2 q • Σq = ∂ t u a + Aµ • q + sup v v • ∇ q u a - d i=1 V i L i v i V i ,
endowed with the terminal condition:

u a T = -A a q • q. For any i = 1, ..., d, let H i be the Legendre-Fenchel transform of the function L i that is given by

H i (p) := sup ρ pρ -L i (ρ).
Since the maps (L i ) 1 i d are strictly convex, (H i ) 1 i d are functions of class C 1 , and the optimal feedback strategies associated to (2.3) are given by v i,a (t, q) := V i Ḣi (∂ q i u a (t, q)) ,

where Ḣi denotes the first derivative of H i . Therefore, the Mean Field Game system associated to the above problem reads:

(2.4)

                                             γ a 2 q • Σq = ∂ t u a + Aµ • q + d i=1 V i H i (∂ q i u a (t, q)) ∂ t m + d i=1 V i ∂ q i m Ḣi (∂ q i u a (t, q)) = 0 µ i t =
(q,a)

V i Ḣi (∂ q i u a (t, q)) m(t, dq, da) m(0, dq, da) = m 0 ( dq, da), u a T = -A a q • q. The Mean Field Game system (2.4) describes a Nash equilibrium configuration, with infinitely many well-informed market investors: any individual player anticipates the right average trading flow on the trading period [0, T ], and computes his optimal strategy accordingly. Observe that we make a strong assumption by supposing that the considered group of investors has a precise knowledge of market mean field. In reality this knowledge is only partial or approximate.

Well-posedness for system (2.4) is investigated in [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF] within the general framework of Mean Field Games of Controls. In this work, we provide simpler arguments to deal with the specific cases of our study. We shall suppose that (H i ) 1 i d are of class C 2 and satisfy the following condition:

(2.5) ∀i = 1, ..., d, ∀p ∈ R, C -1 0 Ḧi (p) C 0 ,
for some C 0 > 0, and m 0 is a probability density with a finite second order moment. Moreover, we suppose that the investors' index varies in a closed subset D ⊂ R.

We say that (u a , m) a∈D is a solution to the MFG system (2.4) if the following hold:

• u a ∈ C 1,2 ([0, T ] × R), for a.e a ∈ D, and m in C([0, T ]; L 1 (R × D));
• the equation for u a holds in the classical sens, while the equation for m holds in the sense of distribution;

• for any t ∈ [0, T ], (2.6) R×D |q| dm(t, dq, da) < ∞, and |∇ q u a (t, q)| C 1 (1 + |q|), for some C 1 > 0.
Let us start with the following remark on the uniqueness of solutions to (2.4). Proposition 2.1. Under the above assumptions, system (2.4) has at most one solution.

Proof. Let (u a 1 , m 1 ) a∈D and (u a 2 , m 2 ) a∈D be two solutions to (2.4), and set ūa := u a 1 -u a 2 , m := m 1 -m 2 . At first, let us assume that m 1 , m 2 are smooth so that the computations below holds. By using system (2.4), we have:

(2.7) d dt (q,a) ūa m = - (q,a) m d i=1 V i (H i (∂ q i u 1 ) -H i (∂ q i u 2 )) + A(µ 1 -µ 2 ) • q - (q,a) ū d i=1 V i ∂ q i m 1 Ḣi (∂ q i u 1 ) -∂ q i m 2 Ḣi (∂ q i u 2 ) ,
where µ 1 , µ 2 correspond respectively to (u a 1 , m 1 ) a∈D and (u a 2 , m 2 ) a∈D . On the one hand, note that

(q,a) mA(µ 1 -µ 2 ) • q = 1 2 d dt A Ē • Ē, where Ē(t) := (q,a) q d m(t).
This follows from

d dt Ē = µ 1 -µ 2 ,
which is in turn obtained from system (2.4) after an integration by parts.

On the other hand, by virtue of (2.5) we have

d i=1 V i m (H i (∂ q i u 1 ) -H i (∂ q i u 2 )) -∂ q i ū m 1 Ḣi (∂ q i u 1 ) -m 2 Ḣi (∂ q i u 2 ) = - d i=1 V i m 1 H i (∂ q i u 2 ) -H i (∂ q i u 1 ) -Ḣi (∂ q i u 1 ) ∂ q i (u 2 -u 1 ) - d i=1 V i m 2 H i (∂ q i u 1 ) -H i (∂ q i u 2 ) -Ḣi (∂ q i u 2 ) ∂ q i (u 1 -u 2 )
min

1 i d V i (q,a) (m 1 + m 2 ) 2C 0 |∇ q u 1 -∇ q u 2 | 2 .
Therefore, (2.7) provides

(2.8) min

1 i d V i T 0 (q,a) |∇ q u 1 (s) -∇ q u 2 (s)| 2 d(m 1 + m 2 ) ds + C 2 A Ē(T ) • Ē(T ) = 0.
By using a standard regularization process, identity (2.8) holds true for any solutions (u a 1 , m 1 ) a∈D and (u a 2 , m 2 ) a∈D of (2.4). Thus, one can use this identity to deduce that ∇ q u 1 ≡ ∇ q u 2 on {m 1 > 0} ∪ {m 2 > 0}, so that m 1 , m 2 solve the same transport equation:

∂ t ν + d i=1 V i ∂ q i ν Ḣi (∂ q i u a 1 (t, q)) = 0, ν t=0 = m 0 .
This entails m 1 ≡ m 2 and so u 1 ≡ u 2 , by virtue of our regularity assumptions.

Quadratic Liquidity Functions.

In practice the liquidity function is often chosen as strictly convex power function of the form: L(p) = η|p| 1+φ + ω|p|, with η, φ, ω > 0. The additional term ω|p| captures proportional costs such as the bid-ask spread, taxes, fees paid to brokers, trading venues and custodians [START_REF] Guéant | The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making[END_REF]. The quadratic case (φ = 1) -that is also considered in [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF] -is particularly interesting because it induces some considerable simplifications and allows to compute the solutions at a relatively low cost. Throughout the rest of this paper, we suppose that the liquidity functions take the following simple form:

(2.9)

L i (p) = η i |p| 2 where η i > 0, i = 1, ..., d.
Following the approach of [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF], we start by setting m0 ( da) := q m 0 ( dq, da). We shall suppose that (2.10) m0 (a) = 0, for a.e a ∈ D, and that investors do not change their preference parameter a over time. Thus, we always have q m(t, dq, da) = m0 ( da), so that we can disintegrate m into

m(t, dq, da) = m a (t, dq) m0 ( da),
where m a (t, dq) is a probability measure in q for m0 -almost any a. Let us now define the following process which plays an important role in our analysis:

E a (t) := q q m a (t, dq) ∀t ∈ [0, T ], for a.e a ∈ D,
and we shall denote by E a,1 , ..., E a,d the components of E a . By virtue of the PDE satisfied by m, observe that E a satisfies the following:

Ėa (t) = q q ∂ t m a (t, dq) (2.11) = q V i 2η i ∂ q i u a (t, q) 1 i d m a (t, dq), so that (2.12) µ t = a Ėa (t) d m0 (a).
Due to the existence of linear and quadratic terms in the equation satisfied by u a , we expect the solution to have the following form:

(2.13) u a (t, q) = h a (t) + q • H a (t) + 1 2 q • H a (t) • q where h a (t) is R-valued function, H a (t) := (H i a (t)) 1 i d is R d -valued function,
and the map H a (t) := (H i,j a (t)) 1 i,j d take values in the set of R d×d -symmetric matrices. Inserting (2.13) in the HJB equation of (2.4) and collecting like terms in q leads to the following coupled system of BODEs:

(2.14)              ḣa = -VH a • H a Ḣa = -Aµ -2H a VH a Ḣa = -2H a VH a + γ a Σ h a (T ) = 0, H a (T ) = 0, H a (T ) = -2A a , where V := diag V 1 4η 1 , ..., V d 4η d .
In order to solve completely (2.14) we need to know µ, or the process Ėa thanks to (2.12). Thus, one needs an additional equation to completely solve the problem.

By virtue of (2.11), we have

(2.15) Ėa = 2VH a + 2VH a E a .
By combining this equation with system (2.14) one obtains the following FBODE:

(2.16)

             Ëa = -2VA a Ėa d m0 (a) + 2γ a VΣE a E a (0) = E a 0 := q qm 0 (q, a)/ m0 (a) Ėa (T ) + 4VA a E a (T ) = 0.
This system is a generalized form of the one that is studied in [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF], and summarizes the whole market mean field. Observe that the permanent market impact acts as a friction term while the market risk terms act as a pushing force toward a faster execution. The investors heterogeneity is taken into account in the first derivative term, which means that the contribution of all the market participants to the average trading flow is already anticipated by all agents. System (2.16) is our starting point to solve the MFG system (2.4) in the quadratic case. Due to the forward-backward structure of system (2.16), we need a smallness condition on A in order to construct a solution. This assumption is also considered in [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF], and is not problematic from a modeling standpoint since |A| is generally small in applications (cf. Section 3.3). Let us present the construction of solutions to system (2.16). Proposition 2.2. Suppose that A a , γ a ∈ L ∞ (D), then there exists α 0 > 0 such that, for |A| α 0 , the following hold:

(i) there exists a unique process E a in L 1 m0 (D; C 1 ([0, T ])) which solves system (2.16); (ii) there exists a constant C 2 > 0, such that (2.17) sup

0 w T |µ w | C 2 1 + a |E a 0 | d m0 e C 2 T ,
where (µ t ) t∈[0,T ] is given by (2.12).

Proof

H a (t) = T t exp w t 2H a (s)V ds A a Ėa (w) d m0 (a) dw, so that E a (t) = exp t 0 2VH a (w) dw E a 0 + 2V t 0 exp t τ 2VH a (w) dw T τ exp w τ 2H a (s)V ds A a Ėa (w) d m0 (a) dw dτ.
By combining this relation with (2.15), we deduce that Ėa satisfies the following fixed point relation:

(2.20) x a (t) = Φ A (x a )(t) := 2VH a (t) exp t 0 2VH a (w) dw E a 0 +4VH a (t)V t 0 exp t τ 2VH a (w) dw T τ exp w τ 2H a (s)V ds A a x a (w) d m0 (a) dw dτ + 2V T t exp w t 2H a (w)V dw A a x a (w) d m0 (a) dw.
Conversely, one checks that if x a is a solution to the fixed point relation (2.20), for a.e. a ∈ D, then E a (t) = E a 0 + t 0 x a (s) ds is a solution to system (2.16). To solve the fixed point relation (2.20), one just uses Banach fixed point Theorem on Φ A : X → X, where

X := L 1 m0 (D; C([0, T ])).
It is clear that Φ A is a contraction for |A| small enough: indeed, given x, y ∈ X, it holds that:

|Φ A (x a )(t) -Φ A (y a )(t)| C|A| x -y X
where C > 0 depends only on T , γ ∞ , A ∞ , |V| and |Σ|. Thus, given the solution x a to (2.20), the function E a (t) = E a 0 + t 0 x a (s) ds solves (2.16), and belongs to L 1 m0 (D; C([0, T ])) given that m 0 have a finite first order moment. Estimate (2.17) ensues from Gr önwall's Lemma.

We are now in position to solve the MFG system (2.4) in the case of quadratic liquidity functions (2.9). Theorem 2.3. Under (2.9), (2.10), and assumptions of Proposition 2.2, the Mean Field Game system (2.4) has a unique solution.

Proof. Since (2.16) is solvable thanks to Proposition 2.2, we can now solve completely system (2.14) and deduce u a (t, q; µ) thanks to (2.13). In fact, owing to (2. [START_REF] Curato | Optimal execution with non-linear transient market impact[END_REF]) we know that (cf. [START_REF] Martin | On the exponential representation of solutions of linear differential equations[END_REF]):

         H a (t) = T t exp w t 2H a (s)V ds Aµ w dw h a (t) = T t VH a (w) • H a (w) dw,
so that the function u a (t, q; µ), that is given by (2.13), is C 1,2 ([0, T ] × R). Furthermore, by virtue of (2.17)-(2.18), note that

(2.21) |∇ q u a (t, q)| C(1 + |q|)
for some constant C > 0 which depends only on T and data. Now, as u a is regular and satisfies (2.21), we know that the transport equation

∂ t m a + d i=1 V i ∂ q i (m a ∂ q i u a (t, q)) = 0, m a (0, dq) = m 0 ( dq, da)/ m0 (a)
has a unique weak solution m a ∈ C([0, T ]; L 1 (R)) for a.e a ∈ D, so that m := m a m0 solves, in the weak sense, the following Cauchy problem:

∂ t m + d i=1
V i ∂ q i (m∂ q i u a (t, q)) = 0, m(0, dq, da) = m 0 ( dq, da).

In addition, one easily checks that m belongs to C([0, T ]; L 1 (R × D)).

By invoking the uniqueness of solutions to (2.16), we have

E a (t) =
q qm a (t, q) dq for a.e a ∈ D.

Thus through the same computations as in (2.11) we obtain

µ i t =
(q,a)

V i ∂ q i u a (t, q)m a (t, q) m0 (a) da dq, i = 1, ..., d, so that (u a , m) a∈D solves the MFG system (2.4). By virtue of Proposition 2.1, any constructed solution is unique. So to conclude the proof, it remains to show that: R×D |q|m(t, q, a) dq da < ∞.

For that purpose, let us set Ψ(t) := R×D |q| 2 m(t, q, a) dq da. After differentiating Ψ and integrating by parts, we obtain the following ODE that is satisfied by Ψ:

Ψ(t) = Ψ(0) + 2 t 0 a H a (w) • E a (w) m0 ( da) dw + 2 t 0 a (H a (w)q • q) m(t, dq, da) dw, so that |Ψ(t)| |Ψ(0)| + C sup 0 w T E a (w) L 1 m0 + t 0 |Ψ(w)| dw
holds thanks to (2.17)-(2.18). Hence, as m 0 has a finite second order moment, we deduce from Gr önwall's Lemma that for any t ∈ [0, T ] R×D |q| 2 m(t, q, a) dq da < ∞, which in turn entails the desired result.

Stylized Facts & Numerical Simulations.

Let us now comment our results and highlight several stylized facts of the system. By virtue of (2.13), the optimal trading speed v * a is given by:

v * a (t, q) = 2VH a (t)q + 2VH a (t) (2.22) = 2VH a (t)q + 2V T t exp w t 2H a (s)V ds Aµ w dw =: v 1, * a (t, q) + v 2, * a (t; µ).
The above expression shows that the optimal trading speed is divided into two distinct parts v 1, * a , v 2, * a . The first part v 1, * a corresponds to the classical Almgren-Chriss solution in the case of a complexe portfolio (cf. [START_REF] Guéant | The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making[END_REF]). The second part v 2, * a adjusts the speed based on the anticipated future average trading on the remainder of the trading window [t, T ]. Since the matrix H a is negative, note that the strategy gives more weight to the current expected average trading. Moreover, the contribution of the corrective term decreases as we approach the end of the trading horizon. The correction term aims to take advantage of the anticipated market mean field.

Let us set

(2.23) G a (t, w) := exp w t 2H a (s)V ds A.
Note that the matrix G a is not necessarily symmetric and could have a different structure than H a . In view of the market price dynamics, the trading speed expression shows that an action of an individual investor or trader on asset i could have a direct impact on the price of asset j, at least when the two assets are fundamentally correlated, i.e. Σ i,j = 0. This phenomenon of cross impact is related to the fact that other traders already anticipates the market mean field and aim to take advantage from that information, especially when asset j is more liquid than asset i (or vice versa). Thus, if an investor is trading as the crowd is expecting her to trade, then she is more likely to get a "cross-impact" through the action of the other traders. This fact is empirically addressed in [START_REF] Benzaquen | Dissecting cross-impact on stock markets: An empirical analysis[END_REF][START_REF] Hasbrouck | Common factors in prices, order flows, and liquidity[END_REF].

Another expression of the optimal trading speed can also be derived thanks to (2.15). In fact, we have that:

(2.24) v * a (t, q) = Ėa + 2VH a (t)(q -E a ).
The above formulation shows that an individual investor should follow the market mean field but with a correction term which depends on the situation of her inventory relative to the population average inventory.

In order to simplify the presentation, we ignore from now on investors heterogeneity and assume that market participants have identical preferences. Under this assumption, system (2.16) simply reads:

(2.25)

Ë = -2VA Ė + 2γVΣE E(0) = E 0 , Ė(T ) + 4VAE(T ) = 0.
Given a discretization step δt = N -1 , the solution of (2.25) is approached by a sequence (x k , y k ) 0 k N according to the following implicit scheme:

           x 0 = E 0 x k -x k-1 -δty k-1 = 0, k = 1, ..., N y k -y k-1 -δt (2γVΣx k -2VAy k ) = 0, k = 1, ..., N 4VAx N + y N = 0.
Hence, computing an approximate solution to system (2.25) reduces to solving a straightforward linear system. One checks that under conditions of Proposition 2.2, the above numerical scheme converges and is stable. Now, we can present some examples by using the above numerical method. We consider a portfolio containing three assets (Asset 1, Asset 2, Asset 3) with the following characteristics:

• σ 1 = σ 3 = 0.3 $.day -1/2 .share -1 , σ 2 = 1 $.day -1/2 .share -1 ; • V 1 = 2, 000, 000 share.day -1 , V 2 = V 3 = 5, 000, 000 share.day -1 ; • η 1 = η 2 = 0.1 $.share -1 , η 3 = 0.4 $.share -1 , A 1 = A 2 = 2.5 $.day -1 .share -1 ; • α 1 = α 2 = 8 × 10 -4 $.share -1 , α 3 = 6 × 10 -4 $.share -1 .
In Figure 1(a)-1(d), we consider a market with the initial average inventories E 1 0 = 100, 000, E 2 0 = 50, 000, and E 3 0 = -25, 000 shares, for Asset 1, Asset 2, and Asset 3 respectively. In this example, we suppose that the correlation between the price increments of Asset 1 and Asset 2 is 80 %, and we set γ = 5 × 10 -5 $ -1 except for Figure 1(c).

Figure 1(a) shows that changing the permanent market impact prefactors (α k ) 1 k 3 has a significant influence on the average execution speed. This fact was pointed out in [START_REF] Cardaliaguet | Mean field game of controls and an application to trade crowding[END_REF], and is essentially related to the fact that the higher the permanent market impact parameter the more the anticipated influence of the other market participants become important. Namely, when α k is large, traders anticipate a more significant pressure on the price of Asset k, and adjust their trading speed. On the other hand, dynamics of Asset 2 shows that the higher the market liquidity the faster is the execution. This is expected since the more liquid the faster assets are traded. Finally, dynamics of Asset 3 shows that traders accelerate their execution on volatile asset. It corresponds to a natural reaction due to risk aversion; a trader will try to reduce his exposure to the more risky (hence volatile) assets in priority. (b) Optimal trading of an individual investor with: q 1 0 = 40, 000, q 2 0 = 0, and q 3 0 = 110, 000 (d) Optimal trading of an individual investor with: q 1 0 = 100, 000, and q 2 0 = q 3 0 = 0 FIGURE 1. Simulated examples of the dynamics of E and optimal trading curves of an individual investor. The dashed lines in Figure 1(a) correspond to: α 1 = 6 × 10 -4 $.share -1 , V 2 = 7, 000, 000 share.day -1 , and σ 3 = 5 $.day -1/2 .share -1 .

Figure 1(c) illustrates the behavior of the crowd of investors with an increasing risk aversion (higher γ). In the two presented scenarios, one can observe that Asset 2 is liquidated very quickly, then a short position is built (around t = 0.05 for γ = 5 × 10 -2 $ -1 ) and it is finally progressively unwound. This exhibits the emergence of a Hedging Strategy: indeed, since Asset 1 and Asset 2 are highly correlated, investors can slow down the execution process for the less liquid asset (Asset 1) to reduce the transaction costs, by using the more liquid asset (Asset 2) to hedge the market risk associated to Asset 1. The trader has an incentive to use such a strategy as soon as the cost of the roundtrip in Asset 2 is smaller than the corresponding reduction of the risk exposure (seen from its reward function U a (t, x, s, q; µ) defined by equality (2.2)). Now, we provide examples of individual players' optimal strategies. We consider two examples: an individual investor with initial inventory q 1 0 = 40, 000, q 2 0 = 0, and q 3 0 = 110, 000 in Figure 1(b); and an individual investor with initial inventory q 1 0 = 100, 000, and q 2 0 = q 3 0 = 0 in Figure 1(d). In Figure 1(d) the considered investor starts from q 1 0 = E 1 0 . Hence, by virtue of (2.24) her liquidation curve follows exactly the market mean field. Moreover, the investor takes advantage of the anticipated evolution of the market by building favorable positions on Asset 2 and Asset 3: building a short (resp. a long) position on Asset 2 (resp. Asset 3), and buying (resp. selling) back in order to take advantage of price drop (resp. raise) induced by the massive liquidation (resp. purchase). The trading strategies on Asset 2 and Asset 3 are related to the term v 2, * in (2.22). This strategy can be described as a "Liquidity Arbitrage Strategy".

Figure 1(b) shows two interesting facts: on the one hand, the individual player builds a short position on Asset 1 after achieving her goal (complete liquidation) in order to take advantage of the market selling pressure; on the other hand, by taking into account the market buying pressure on Asset 1, the investor slows down her liquidation to reduce execution costs since she anticipates no sustainable price decline.

THE DEPENDENCE STRUCTURE OF ASSET RETURNS

The main purpose of this section is to analyze the impact of large transactions on the observed covariance matrix between asset returns, by using the Mean Field Game framework of Section 2. For that purpose, we assume a simple model where a continuum of players trade a portfolio of assets on each day, and where the initial distribution of inventories across the investors m 0 changes randomly from one day to another according to some given law of probability. We assume that the price dynamics is given by (2.1), and we consider the problem of estimating the covariance matrix of asset returns given a large dataset of intraday observations of the price. For the sake of simplicity, we ignore investors heterogeneity and assume that market participants have identical preferences. Next, we compare our findings with an empirical analysis on a pool of 176 US stocks sampled every 5 minutes over year 2014 and calibrate our model to market data.

Throughout this section, we denote by X 2 the variance of X, and X, Y the covariance between X and Y, for any two random variables X, Y. Moreover, we will call a "bin" a slice of 5 minutes. We focused on continuous trading hours because the mechanism of call auctions (i.e. opening and closing auctions is specific). Since US markets open from 9h30 to 16h, our database has 78 bins per day. They will be numbered from 1 to M and indexed by k.

3.1. Estimation using Intraday Data. We suppose that E 0 is a random variable with a given realization on each trading period [0, T ], where T = 1 day (trading day); and we consider the problem of estimating the covariance matrix of asset returns given the following observations of the price: , n = 1, ..., d where S n t ,k is the price of asset n in bin k of day . We suppose that t ,1 = 0, t ,M = T , for any 1 N, and t ,k = t ,k = t k for any 1 k M, 1 , N. For simplicity, we suppose that the covariance matrix of asset returns between t k and t k+1 is estimated form data by using the following "naive" estimator :

S n t 1
(3.1) C i,j [t k ,t k+1 ] := 1 N -1 N l=1 δS i,k,l -δS i,k δS j,k,l -δS j,k ,
where δS n,k,l = S n t l,k+1 -S n t l,k and δS n,k = N -1 N l=1 δS n,k,l , n = i, j. We define the correlation matrix as follows:

(3.2) R i,j [t k ,t k+1 ] := C i,j [t k ,t k+1 ] C i,i [t k ,t k+1 ] C j,j [t k ,t k+1 ] 1/2 .
Suppose that the price dynamics is given by (2.1), then the following proposition provides an exact computation of C i,j [t k ,t k+1 ] . Proposition 3.1. Assume that E 0 is independent from the process (W t ) t∈[0,T ] , then for any 1 k M -1 and 1 i, j d, the following hold:

(3.3) C i,j [t k ,t k+1 ] = (t k+1 -t k )Σ i,j + α i α j η i η j 4V i V j Λ i,j k + N ,
where

N → 0 as N → ∞, Λ i,j k := 1 , d θ i, k , θ j, k + 1 , d π i, k , θ j, k + 1 , d θ i, k , π j, k + 1 , d π i, k , π j, k , and
π n, k := t k+1 t k H n, (s)E (s) ds, θ n, k := t k+1 t k T s
G n, (s, w)µ (w) dw ds.

Proof. Use the exact expression of the price dynamics (2.1), the law of large numbers, and the independence between E 0 and (W t ) t∈[0,T ] to obtain:

(3.4) C i,j [t k ,t k+1 ] = N + (t k+1 -t k )Σ i,j + α i α j 1 (N -1) N l=1 t k+1 t k µ i,l s -μi s ds t k+1 t k µ j,l s -μj s ds ,
where μn u = N -1 N l=1 µ n,l u , and µ l , E l are respectively the realizations of µ, E in day l. Now, owing to (2.22)-(2.23), we know that

µ l t = 2VH(t)E l (t) + 2V T t G(t, w)µ l w dw =: ν 1,l (t) + ν 2,l (t).
Thus by setting νn,l k :=

t k+1 t k ν n,l (s) -N -1 N l=1 ν n,l (s) ds, n = 1, 2,
we deduce that

t k+1 t k µ i, s -μi s ds t k+1 t k µ j, s -μj s ds = ν1,l,i k + ν2,l,i k ν1,l,j k + ν2,l,j k .
The desired result ensues by noting the existence of estimation noises 1 N , 2 N , 3 N and 4 N , such that:

(N -1) -1 N l=1 ν1,l,i k ν1,l,j k = η i η j 4V i V j 1 , d π i, k , π j, k + 1 N ; (N -1) -1 N l=1 ν2,l,i k ν2,l,j k = η i η j 4V i V j 1 , d θ i, k , θ j, k + 2 N ; (N -1) -1 N l=1 ν1,l,i k ν2,l,j k = η i η j 4V i V j 1 , d π i, k , θ j, k + 3 N ; (N -1) -1 N l=1 ν1,l,j k ν2,l,i k = η i η j 4V i V j 1 , d θ i, k , π j, k + 4 N .
The proof is complete.

Remark 3.2. One can easily derive an analogous result for C i,j

[0,T ] 1 i,j d
. Namely, it holds that:

(3.5) C i,j [0,T ] = T Σ i,j + α i α j η i η j 4V i V j Λ i,j + N ,
where N → 0 as N → ∞,

Λ i,j := 1 , d θ i, , θ j, + 1 , d π i, , θ j, + 1 , d θ i, , π j, , + 1 , d π i, , π j, , and
π n, := T 0 H n, (s)E (s) ds, θ n, := T 0 T s G n, (s, w)µ (w) dw ds.
Identities (3.3) and (3.5) show that the realized covariance matrix is the sum of the fundamental covariance and an excess realized covariance matrix generated by the impact of the crowd of institutional investors' trading strategies. Note on the one hand that the diagonal terms C i,i are always deviated from fundamentals because of the contribution of (π i,i ) 2 and (θ i,i ) 2 . On the other hand, since H and G inherit a structure similar to Σ, the excess of realized covariance in the off-diagonal terms is non-zero as soon as oneor both -of the conditions below is satisfied:

• there exists i 0 = j 0 such that Σ i 0 ,j 0 = 0;

• there exists i 0 = j 0 such that E i 0 0 , E j 0 0 = 0.

Moreover, (3.3) and (3.5) show that the excess realized covariance can deviate significantly from fundamentals when: the market impact is large (high crowdedness), the considered assets are highly liquid (small η i /V i ), the risk aversion coefficient γ is high, or when the standard deviation of E 0 is large. In addition, since the contribution of θ n, k and π n, k vanishes as we approach the end of the trading horizon, observe that

(3.6) C i,j [t k ,t k+1 ] ∼ (t k+1 -t k )Σ i,j
, as t k+1 → T , which means that one converges to market fundamentals at the end of the trading period. This is due to the fact that, in our model, all traders have high enough risk aversions so that their trading speeds go to zero close to the terminal time T .

By virtue of (3.3), one can also explain the realized correlation matrix in terms of the fundamental correlations ρ i,j := Σ i,j /(Σ i,i Σ j,j ) 1/2 . Namely, it holds that:

(3.7) R i,j [t k ,t k+1 ] = ρ i,j   (t k+1 -t k ) 2 Σ i,i Σ j,j C i,i [t k ,t k+1 ] C j,j [t k ,t k+1 ]   1/2 + α i α j η i η j Λ i,j k 4V i V j C i,i [t k ,t k+1 ] C j,j [t k ,t k+1 ] 1/2 + N =: ρ i,j A i,j k + B i,j k + N for any 1 i < j d.
This expression shows that the deviation of the realized correlation from fundamentals is a linear map. The numerator of the multiplicative part A i,j k does not depend on the off-diagonal terms of H while it is the case for the additive part B i,j k .

Numerical Simulations.

In this part, we conduct several numerical experiments in order to illustrate the impact of trading on the structure of the covariance matrix of asset returns.

We consider the example of Section 2.3 by choosing ρ 1,2 = 60%, ρ 1,3 = 30% and ρ 2,3 = 5%. For simplicity, we suppose that E 0 is a centered Gaussian random vector with a covariance matrix Γ that is given by:

Γ := λ 2 .   1 0.2 -0.1 0.2 1 0.3 -0.1 0.3 1   ,
where λ = 10, 000 share. We fix a time step δt = 10 -2 day (∼ 4 min), set t k+1 -t k = δt,

and estimate C i,j [t k ,t k +δt] 1 i j 3 1 k M-1 and R i,j [t k ,t k +δt] 1 i<j 3 1 k M-1
by generating a sample of N = 10, 000 observations using the numerical method of Section 2.3. Figures 2(a)-2(d) show that the observed covariance and correlation matrices are significantly deviated from fundamentals and especially at the beginning of the trading day. Figures 2(b)-2(d) also illustrates the sensitivity of the deviation relative to the change of the standard deviation of initial inventories: as λ diminishes, the impact of trading is lower and the covariance and correlation matrices converge toward fundamentals.

On the other hand, we observe that the beginning of the trading period is dominated by the dependence structure of initial inventories. This is due to the domination of the additive terms (B i,j k ) 1 i<j 3 ; in fact, given the relative high magnitude of denominator terms, (A i,j k ) 1 i<j 3 are very small when t k → 0. Furthermore, we note that all the observed quantities converge toward fundamentals at the end of the trading period, which is in line with (3.6).

C [t+ t] 1,3 ( smaller )
C [t+ t] 2,3 ( smaller )
3.3. Empirical Application. Now, we carry out an empirical analysis on a pool of d = 176 stocks. The data consists of five-minute binned trades (δt = 5 min) and quotes information from January 2014 to December 2014, extracted from the primary market of each stock (NYSE or NASDAQ). We only focus on the beginning of the continuous trading session removing 30 min after the open and the last 90 min before the close, in order to avoid the particularities of trading activity in these periods and target close strategies. Thus, the number of days is N = 252 and the number of bins per day is M = 55. Days will be labelled by l = 1, ..., N, bins by k = 1, ..., M, and for simplicity we note C i,j k instead of C i,j [t k -δt,t k ] for any 1 i, j d.

Our goal is to empirically assess the the influence of trading activity on the intraday covariance matrix of asset returns, and then compare the obtained models with our previous theoretical observations. Given our analysis in Sections 3.1 and 3.2, we expect an excess in the observed covariance matrix of asset returns and especially at the beginning of the trading period. Moreover, we expect the magnitude of this effect to be an increasing function of the typical size of market orders as it is noticed in Figures 2(b) and 2(d).

3.3.1. Market Impact. Let us start by assessing the relationship between the intraday variance of asset returns and the intraday variance of net exchanged flows (F i,i k ) 1 i d , that is defined by:

F i,i k := 1 N -1 N l=1 ν i k,l -νi k ν i k,l -νi k
for any 1 i d and k = 1, ..., M; where ν i k,l is the net sum of exchanged volumes between t k -δt and t k for stock i in day l, and νi k = N -1 N l=1 ν i k,l (i.e. νi k is an estimate of the expectation of ν i k,l regardless of the day). As a by-product, we obtain estimates for the permanent market impact factors. Though ν does not represent exactly the same quantity as the variable µ of Section 3.1, both variables are an indicator of market order flows and for simplicity we shall use ν as a proxy for µ.

(C i,i k ) 1 k M and (F i,i k ) 1 k M for GOOG. Figure 3(b)
shows that this is true for almost all the stocks and reinforces our findings in Sections 3.1 and 3.2. Furthermore, as (3.4) suggests, we suppose a linear relationship between (C i,i k ) 1 k M and (F i,i k ) 1 k M ; thus for every 1 i d we fit the following regression:

(3.8) C i,i = + δt • Σ + α 2 • F i,i
where is the error term (assumed normal), the coefficient Σ is related to the "fundamental" covariance matrix of asset returns and the square root of the coefficient α 2 is homogeneous to the market impact factor (cf. (3.4)). In Table 1 we show estimates of α, Σ and the correlation between For each estimate the standard deviation (std.) is shown in parentheses and the p-value is given in the third row. Numbers in bold are significant at a level of at least 99%.

(C i,i k ) 1 k M and (F i,i k ) 1 k M (

3.3.2.

The Typical Intraday Pattern. Next, we are interested in the intraday evolution of the diagonal and off-diagonal terms of the covariance matrix of returns, and in the way this evolution is affected when the typical size of trades diminishes. For that purpose, we compute the intraday covariance matrix of returns for our pool of US stocks and we normalize each term (C i,j k ) 1 k M by its daily average, then we consider the median value of diagonal terms and off-diagonal terms as a way of characterizing the evolution of a typical diagonal term and a typical off-diagonal term respectively. The impact of the relative size of orders on the intraday patterns is assessed by conditioning our estimations.

More exactly, we start by defining the matrix of trade imbalances for each stock n in order to be able to compare the relative size of trades. Namely, for any n, k, l, we set:

w n k,l := ν n k,l mean 1 l N k |ν n k,l |
, where mean n∈A {x n } denotes the average of (x n ) as n varies in A. This mean is an estimate of the expectation of the sum of the absolute values of ν n k,l over a day; it can be seen as a renormalizing constant, enabling us to mix different stocks on Figure 4.

Next, we define the conditioned intraday covariance matrix C i,j k (λ) 1 i,j d

1 k M
for every λ 0 as follows:

(3.9)

C i,j k (λ) := 1 #E i,j k (λ) -1 l∈E i,j k (λ) δS i,k,l -δS i,j,k λ δS j,k,l -δS j,i,k λ
, where:

• the set E i,j k (λ) corresponds to a conditioning: it contains only days for which this 5 min bins (indexed by k) for this pair of stocks (indexed by (i, j), note that we can have i = j) is such that the renormalized net volumes are (in absolute value) below λ. It is strictly defined as follows:

E i,j k (λ) := 1 l N : |w i k,l | λ and |w j k,l | λ ;
• δS n,k,l is the price increment defined as for (3.1) and is computed from the historic stock prices; • δS i,j,k λ is the average price increment over selected days, given by: δS

i,j,k λ = l∈E i,j k (λ) δS i,k,l / #E i,j k (λ) ; • #E i,j
k (λ) denotes the number of elements of E i,j k (λ): the number of selected days. Note that the stricter the conditioning (i.e. the smaller λ), the less days in the selection, and hence the smaller #E i,j k (λ). Here C i,j k (λ) 1 i,j d

1 k M
represents the intraday covariance matrix of returns conditioned on trade imbalances between -λ and λ. In all our examples, the coefficient λ is chosen to have enough days in the selection (for obvious statistical significance reasons), i.e. so that #E i,j k (λ) 1 for any 1 i, j d and 1 k M.

Now we define the median diagonal pattern

C diag (λ) := C diag k (λ) 1 k M and the median off-diagonal pattern C off (λ) := C off k (λ) 1 k M as follows: C diag k (λ) := median 1 i d C i,i k (λ)/ mean 1 k M C i,i k (1)
and

C off k (λ) := median 1 i<j d C i,j k (λ)/ mean 1 k M C i,j k (1)
,

for any k = 1, ..., M and λ 0. Here the notation median n∈A {x n } denotes the median value of (x n ) as n varies in A. One should note that the choice of the normalization constant (i.e. the mean over bins of C i,j k (1)) will allow us to compare the different curves with respect to the reference case, i.e. without conditioning. In fact, it turns out that λ = 1 removes all conditionings: 1 is above the maximum value of our renormalized flows. Moreover, we set #E diag k (λ) := median We take medians instead of means to have robust estimates of the expectations. We do not want our estimates to be polluted by few days of potential erratic market data, that could for instance be due to trading interruptions.

Figures 4(a) and 4(b) displays representations of C diag (λ) and C off (λ) for various values of λ. Observe that C diag (1) and C off (1) exhibits a pattern that is very similar to our simulation in Figures 2(b) and 2(d), especially between the beginning of the trading period and 13 : 00. Indeed, the observed quantities are high at the beginning of the trading period, lower as the day progresses until it reaches a minimum around 13 : 00, followed by a slight increase until market close. The general shape of these curves (left-slanted smile) is well-known (see e.g. [START_REF] Cartea | Algorithmic and High-Frequency Trading[END_REF] and references therein).

Our core observation is that: given the absolute value of the net flows are small, this average curve flattens out, even at the beginning of the day. At our knowledge, it is the first time that this conditioning is mentioned, and it is perfectly in line with our simulated Figures 2(b) and 2(d). This suggests the slopes of the "averaged volatility curves" comes essentially from the days during which there is a large positive or negative imbalance of large orders, that are "optimally" executed. We believe that this analysis should be completed by using a larger data set.

3.3.3.

A Toy Model Calibration. Now, we use historical data to fit our MFG model to some examples of traded stocks. For that purpose, we use a very simple approach by reducing as much as possible the number of parameters: (S1) We suppose that E 0 is a centered Gaussian random vector with a covariance matrix Γ . Moreover, as suggested by (2.12) , we use Corr k ν i k , k ν j k as a proxy for Corr E i 0 , E j 0 , and which is in turn estimated from data by using the standard estimator :

1 N -1 N k ν i k,l - k ν i k,l k ν j k,l - k ν j k,l ,
where k ν i k,l = N C i,j k (1) ; (S3) Finally, we fix the penalization parameters A i = A = 10, and choose k i := V i /η i , γ, and Γ i,i by minimizing the L 2 -error between the simulated curves and real curves.

10:00:00 11:00:00 12:00:00 13:00:00 14:00:00 2.

Here Γ 1,2 , α 1 , α 2 , σ 1 , σ 2 , ρ 1,2 are estimated from data (cf. Table 1 and Figures 4(a)-4(b)), while Γ 1,1 , Γ 2,2 , γ, k 1 , k 2 are computed by minimizing the L 2 -error between the simulated curves and real curves. Following this approach, one requires 2d + 1 parameters to fit a portfolio of d stocks (i.e. d(d + 1)/2 curves).

Asset 1 (

 1 market trend ) Asset 2 ( market trend ) Asset 3 ( market trend ) Asset 1 ( an individual investor ) Asset 2 ( an individual investor ) Asset 3 ( an individual investor )

( d )

 d Intraday covariance for λ = 10 4 and λ = 6.10 3

FIGURE 2 .

 2 FIGURE 2. Simulated examples of intraday covariance and correlation matrices using (2.1).

FIGURE 3 .

 3 FIGURE 3. Dependence structure between(C i,i k ) 1 k M and (F i,i k ) 1 k M . Figure 3(a) displays the relationship between (C i,i k ) 1 k M and (F i,i k ) 1 k M for GOOG.Figure 3(b) exhibits the histogram of correlations denoted Corr (C, F).

Figure 3 (

 3 FIGURE 3. Dependence structure between(C i,i k ) 1 k M and (F i,i k ) 1 k M . Figure 3(a) displays the relationship between (C i,i k ) 1 k M and (F i,i k ) 1 k M for GOOG.Figure 3(b) exhibits the histogram of correlations denoted Corr (C, F).

Figure 3 (

 3 Figure 3(a) shows a strong positive correlation between(C i,i k ) 1 k M and (F i,i k ) 1 k M for GOOG. Figure 3(b)shows that this is true for almost all the stocks and reinforces our findings in Sections 3.1 and 3.2. Furthermore, as (3.4) suggests, we suppose a linear relationship between (C i,i k ) 1 k M and (F i,i k ) 1 k M ; thus for every 1 i d we fit the

  1 i d #E i,i k (λ) and #E off k (λ) := median 1 i<j d #E i,j k (λ) .

  The median diagonal pattern (squared volatility) The median off-diagonal pattern

FIGURE 4 .

 4 FIGURE 4. Plots of the median diagonal pattern C diag (λ) and the median off-diagonal pattern C off (λ) for diverse values of λ. The secondary axis corresponds to the number of observations for each 5 minutes bin after the conditioning.

  As suggested by Figures 4(a)-4(b) we chooseδtΣ i,j = 0.2 × mean 1 k M C i,j k (1) ,and we shift upward the simulated curves by δ = 0.3 × mean1 k M

FIGURE 5 .

 5 FIGURE 5. Comparison between the simulated curves and the real curves for two examples. Figure 5(a) corresponds to i ≡ j ≡ GOOG and Figure 5(b) corresponds to (i, j) ≡ (GOOG, AAPL).

Figures 5 (

 5 Figures 5(a)-5(b) show illustrative examples by considering the two-stocks portfolio: Asset 1 ≡ GOOG; Asset 2 ≡ AAPL. For that example, the parameters of our model are presented in Table2.Here Γ 1,2 , α 1 , α 2 , σ 1 , σ 2 , ρ 1,2 are estimated from data (cf. Table1and Figures 4(a)-4(b)), while Γ 1,1 , Γ 2,2 , γ, k 1 , k 2 are computed by minimizing the L 2 -error between the simulated curves and real curves. Following this approach, one requires 2d + 1 parameters to fit a portfolio of d stocks (i.e. d(d + 1)/2 curves).

Estimated using the regression ( 3 . 8 )Γ 1 , 1 = 3 . 6 × 3 ,k 1 = 2 × 10 7 , k 2 = 8 ×

 381136312728 of Section 3.3 and (S1)-(S2)Corr E 1 0 , E 2 0 = 20% , α 1 = 2.5 × 10 -4 , σ 1 = 1.55, ρ 1,2 = 0.5%, α2= 7.9 × 10 -5 , σ 2 = 0.43, Calibrated on curves of Figure 5(a) and 5(b) 10 9 , Γ 2,2 = 2.02 × 10 9 , γ = 10 -10 8 .

  where the order in the above inequality should be understood in the sense of positive symmetric matrices. Moreover, note that ΣV and VΣ are both diagonalizable with nonnegative eigenvalues. Thus by using the ODE satisfied by H a , we know that H a V and , where the symbol [B, A] denotes the Lie Bracket:[B, A] = BA -AB.Given H a , we aim to construct Ėa in L 1 m0 (D; C([0, T ])) by solving a fixed point relation, and then deduce E a . For that purpose, we start by deriving a fixed point relation for Ėa . By virtue of (2.[START_REF] Curato | Optimal execution with non-linear transient market impact[END_REF], observe that any solution E a to (2.16) fulfills (2.15) with (see e.g.[START_REF] Martin | On the exponential representation of solutions of linear differential equations[END_REF])

	VH a are both diagonalizable with a constant change of basis matrix. In particular, it holds
	that
		w	w
	(2.19)	H a (t)V,	H a (u)V du = VH a (t),	VH a (u) du = 0
		t	t
	for any 0 t, w T

. At first, note that the solution H a to the matrix Riccati equation in

(2.14)

exists on [0, T ], is unique, depends only on data, and satisfies (see e.g.

[START_REF] Kučera | A review of the matrix Riccati equation[END_REF]

)

(2.18)

-2A a -T γ a Σ H a 0,

TABLE 1 .

 1 denoted Corr (C, F)) for several examples. In particular, we obtain estimates for the permanent market impact α. Estimates

		AAPL	BMRN	GOOG	INTC
	α (bp)	0.8	8.43	2.5	0.01
	α 2	6.41 × 10 -11	7.11 × 10 -9	6.25 × 10 -8	1.79 × 10 -12
	std.	(4.15 × 10 -12 )	(3.98 × 10 -10 ) (3.17 × 10 -9 )	(1.58 × 10 -13 )
	p-value	0.01%	0.01%	0.01%	0.01%
	Σ	0.16	-0.01	0.15	5.5 × 10 -3
	std.	(0.05)	(0.05)	(0.49)	(2 × 10 -4 )
	p-value	0.01%	60%	75%	2%
	Corr (C, F)	90%	92%	94%	84%

for α, Σ and the realized correlation between (C i,i k ) 1 k M and (F i,i k ) 1 k M for Nasdaq stocks.

TABLE 2 .

 2 The MFG model parameters for the two-stocks portfolio: Asset 1 ≡ GOOG; Asset 2 ≡ AAPL.

Acknowledgement. C. Mouzouni was supported by LABEX MI-LYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR), and partially supported by project (ANR-16-CE40-0015-01) on Mean Field Games. Authors would like to thank Pierre Cardaliaguet for a careful reading of large parts of this paper, and Jean-Philippe Bouchaud to have insisted on the fact that results of [13] should be observable using intraday data.


Diya Moubdi

Bernard Yannou

François Cluzel

Asma Ghaffari

Caroline Vène-Rautureau

Pierre Jammes

Competency framework to support need seeker innovation training

Keywords: Design Learning, Innovation Learning, Innovation Skills, Innovation Strategy, Competency Framework Need Seeker Innovation, competency-based education

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Strategy positions a company for sustainable competitive advantage to achieve long term superior performance. After Porter, a strategy consists in "choosing a unique and valuable position rooted in systems of activities" [START_REF] Porter | What is strategy[END_REF]. Setting up a suitable innovation strategy is therefore, a way to differentiate a company from other firms in the same industry [START_REF] Tushman | Ambidextrous organizations: Managing evolutionary and revolutionary change[END_REF]. A statistical analysis of a representative sample of 2007 'global innovation 1000' study, profiles companies into three distinct categories of innovation strategies [START_REF] Jaruzelski | Proven paths to innovation success[END_REF]:  Techno driver strategy: drive innovation through leading edge new technology  Market reader strategy: drive innovation through market research and competitive intelligence  Need seeker strategy: drive innovation through superior end-user understanding Cited in more than 450 publications, this study is well recognised for its comprehensive assessment of the relationship between innovation strategy and corporate performance. The competency framework that we build in this study aims at supporting in particular, the need-seeker strategy. This strategy turns out to be the best performing strategy because generating superior profitability in the long term and for its potential to produce radical innovations (see [START_REF] Jaruzelski | Proven paths to innovation success[END_REF]). Interrogating end users for capturing their pains is not sufficient anymore. Indeed, imagining new usage situations in the future is needed; this is how need-seeker firms as Apple, Procter & Gamble, and Tesla proceed to get a deep end user understanding. This envisioning ability also gives the need-seeker firms the advantage of being the first mover to market. In addition to its propensity to generate radical innovations driven by usage, the need-seek strategy naturally conducts to highly differentiating value propositions. At the same time, the market reader strategy is most of the time limited to incremental innovation and the techno driver strategy is often more expensive in terms of R&D investments. However, choosing the need-seeker strategy puts firms in the face of two challenges. The first is to establish a 'cross-organisational alignment' between business strategy, innovation strategy and corporate culture. The second challenge is to develop specific skills for achieving a superior performance. After Leifer et al. "people who drive radical innovation have different characteristics from those in more traditional roles" [START_REF] Leifer | Implementing radical innovation in mature firms: The role of hubs[END_REF]. This brings a second challenge, to develop specific skills for achieving a superior performance [START_REF] Jaruzelski | Proven paths to innovation success[END_REF]. In this study, we propose, to our knowledge, the first competency framework supporting the need seeker strategy. Through each stage of the innovation process, we specify the individual and collective competencies as well as critical strategic leadership competencies to support the need seeker strategy. In the literature review section, we address the concept of competency, the possible methods for designing a competency framework, and the models of competency framework existing in literature that inspired us for our context of need-seeker innovation. In Section 4, we propose an original qualitative multi-method (Figure 1) combining extensive literature review, expert interviews, and empirical study on student project-based learning. We present in Section 5 the competency framework designed to support the need seeker strategy (Table 1 and Figure 2). Finally, we discuss the next validation steps we are considering, as well as the implementation of this framework in curriculum development process.

LITERATURE REVIEW

To build the competency framework supporting need seeker innovation training, we mainly addressed the following three questions: What are the specific concepts related to the notion of competency? What methods for designing a competency framework? What are the existing competency frameworks that may be of particular interest in the context of need-seeker innovation?

The notion of competency

We outline in this paragraph underlying concepts that are useful for understanding the notion of competency and so for designing a competency framework. Individual competency is defined as a combination of "visible 'competencies' of 'knowledge and skills' and 'underlying elements of competencies', as 'traits and motives'" [START_REF] Hartle | How to re-engineer your performance management process[END_REF]. A team's collective competence "consists of trust, commitment, communication and joint problem solving" [START_REF] Blomqvist | Collaboration capability-a focal concept in knowledge creation and collaborative innovation in networks[END_REF], and it appears in the literature under wordings such as 'collective mind ', 'shared leadership' or 'collective capacity building'. To highlight collective competencies in organisations, Prahalad and Hamel introduce the concept of 'core competence' as "the collective learning in the organisation" [START_REF] Prahalad | The core competence of the corporation[END_REF]. Given our context of need-seeker innovation, we introduce the concept of 'strategic leadership' reflecting the ability to share values and clear vision for a minimal controlled day-to-day decision-making, enabling immediate impact and preserving long-term goals, which is different from managerial and visionary leadership [START_REF] Rowe | Strategic leadership: short term stability and long-term viability[END_REF].

The design of competency framework

A competency framework "identifies the skills, knowledge, personal characteristics, and behaviours needed to effectively perform a role in the organisation and help the businesses meet its strategic objectives" [START_REF] Lucia | Art & science of competency models[END_REF]. Lévy-Leboyer sums up in his book 'La gestion des compétences' [START_REF] Lévy-Leboyer | La gestion des compétences : Une démarche essentielle pour la compétitivité des entreprises[END_REF], six methods for designing a competency framework. We did not identify any additional methods in the literature. In our study, observations, questionnaires and self-description of tasks are methods that we have realised by adopting the 'Jobs to be Done' mindset [START_REF] Christensen | Know your customers "jobs to be done[END_REF]. In addition, we have used the 'Behavioural Events Technique' (BET by Boyatzis in 1982) [START_REF] Boose | Personal construct theory and the transfer of human expertise[END_REF], which is a variant of the 'Critical Incident Technique' (CIT by Flanagan in 1954) [START_REF] Hughes | Critical incident technique[END_REF], to identify empirically the leadership skills in our context of need seeker innovation. Although the BET and CIT methods, lack a strong theoretical underpinning, they are particularly adapted to develop a conceptual frame or to follow a 'grounded' approach [START_REF] Hughes | Critical incident technique[END_REF]. To validate the resulting competency framework, we are considering as a next step the use of the Kelly Grid technique developed by Kelly from the 'construct theory' [START_REF] Boose | Personal construct theory and the transfer of human expertise[END_REF]. Although this method is demanding for the interviewed experts as for the competency framework designer, however, it has the advantage of being an effective technique to complement the methods we have combined.

Competency frameworks: existing models in the literature

The competency framework supporting the need seeker strategy -given our description of the need seeker strategy below -is at the intersection of two dimensions. First, the user-centricity of the new product development process, to reach deep end-user understanding. The second dimension, is the radical level of the innovation "for which the market is not clearly identified (…) riskier and (…) with the potential to move the organisation in new directions that provide rich platforms for growth" [START_REF] Mcdermott | Managing radical innovation: an overview of emergent strategy issues[END_REF]. An extensive literature review allowed us to identify 22 models of competency frameworks, not given here for reasons of brevity. Let us evoke two of them. First, the innovator's DNA model [START_REF] Dyer | The innovator's DNA: Mastering the five skills of disruptive innovators[END_REF], designed by Dyer, Gregersen and Christensen. This competency framework analyses: how known innovative entrepreneurs came up with and implement new ideas. The second example is a model designed by Veryzer and Borja de Mozota, studying how user oriented design skills contributes to the new product development, without positioning this analysis in the specific case of radical innovation [START_REF] Veryzer | The Impact of User-Oriented Design on New Product Development: An Examination of Fundamental Relationships[END_REF]. Several studies have focused on innovation competencies. However, specific skills supporting a need seeker innovation project derived by a superior end-user understanding, generating highly differentiating value propositions and giving the advantage of being the first mover to market, have not yet been clarified.

STRUCTURE OF THE COMPETENCY FRAMEWORK

The structure adopted for the competency framework has an important role in highlighting and organising the critical competencies over the various phases of the innovation process. To enable the implementation of the competency framework in curriculum development, we have defined the following main specifications:  1 st : Supporting different processes that can structure a need seeker strategy such as the 'Radical Innovation Design Methodology' [START_REF] Yannou | Supporting need seeker innovation: the Radical Innovation Design methodology[END_REF], 'Blue Ocean Strategy' [START_REF] Kim | Blue ocean strategy. If you read nothing else on strategy[END_REF] and 'Design Thinking process' [START_REF] Johansson-Sköldberg | Design thinking: past, present and possible futures[END_REF]. For this goal, we organise our competency framework by drawing on the pattern used by Howard et al. [START_REF] Howard | Describing the creative design process by the integration of engineering design and cognitive psychology literature[END_REF], to compare different creative innovation processes through the four following phases: Analysis phase, Generation phase, Evaluation phase and implementation phase.  2 nd : Including a New Business Development phase to allow "student design teams to assess the impacts of different design variables on the market success of a product". Besides, the "issue of whether or not a market exists for an innovation is an early, critical differentiator in the challenges that project teams face in managing this process" [START_REF] Veryzer | The Impact of User-Oriented Design on New Product Development: An Examination of Fundamental Relationships[END_REF].  3 rd : Simplifying the classification of competencies to operationalise the competency framework in curriculum development. Therefore, we use the following two dimensions: the innovative behaviour as addressed by Scott [START_REF] Scott | Determinants of Innovative Behavior: A Path Model of Individual Innovation in the Workplace[END_REF] and the dimension of ability to describe the ability to exercise' knowledge [START_REF] Fantl | Knowledge How[END_REF]. To classify collective and leadership competencies, we use the notions of collective competencies and strategic leadership as defined in Section 2.1. In the following section we describe the method that combined with the above criteria allowed us to design the competency framework supporting the need seeker strategy.

METHODOLOGY

To study the addressed research questions, we conducted a study combining concurrently: extensive literature review, empirical study and expert based knowledge. Firstly, we have identified by combination of keywords (e.g. need seeker innovation, human-centred innovation, radical innovation, competency framework) a selection of articles on the ASME database and using Google scholar and through it Wiley, JSTOR and ScienceDirect. A first screening of the articles was made according to the title and the number of quotations. Then, we made a second selection by reading the abstract, summary and figures titles'. Finally, we have identified 22 models belonging to one of the two dimensions described in Section 2.3. However, no framework was found in the literature addressing in a single competency framework, the competencies needed to support new product development, which is radically innovative and driven by deep end-user understanding. The empirical study consisted of a series of observations and interviews carried out during a project-based learning of radical innovation processes at CentraleSupélec engineering school. For four months, 29 students from engineering school, business school, and design school worked in a multidisciplinary team on six industrial projects, proposed by four major companies and two research institutes, on topics related to smart building, shared mobility, comfort in cars, and intelligent data management for car manufacturer. In these innovation contexts, students applied the Radical Innovation Design Methodology® [START_REF] Yannou | Supporting need seeker innovation: the Radical Innovation Design methodology[END_REF] to structure the innovation process in need seeker mode (what this methodology particularly claims). The observation consisted of describing and characterising the impact of project team members' behaviours and skills, on the progress of the innovation projects. So far, critical competencies impacting the progress of the project were experimentally reported. Furthermore, to identify the competencies of team leaders, we also used the appropriate behavioural events technique during the gates of the innovation process [START_REF] Boyatzis | The Competent Manager: A Model for Effective Performance[END_REF] (see also Section 2.3). The third approach for identifying the critical competencies was through eight semi-structured interviews of one to three hours plus four face-to-face exchanges, with industrials experts in the field of New Product Development and R&D strategies. As illustrated by Figure 1, for each skill identified -in a competency framework present in the literature -we observed its existence in the field (student projects), then we interviewed the experts on it through detailed interviews. If a skill was not identified through existing models but rather through field observations or expert interviews, we were then looking for whether this skill was individually present in the literature.

Figure 1. All the possible paths we took to identify a competency

Surprisingly, this search by individual innovation competency has led us to multidisciplinary fields of research, in particular: Management Science, Managerial Psychology, Change Management, Knowledge Management and Cognitive Science. For example, 'independent thinking' (illustrated in the result Table 1) is a competency that we have not identified in the 22 competency models; however it has been expressed by innovation experts during detailed interviews. We have also noted the importance of this competency through student projects and afterwards, we identified its presence individually in literature. By now, we present the results obtained through the process described below.

RESULTS

Following the process described in the methodology section and given the specifications that we have established in Section 3, we get the results illustrated in Table 1 and in Figure 2. Figure 2 highlights critical strategic leadership competencies in particular, during the gates of the need seeker innovation project. As specified in Section 3, we have identified the six strategic leadership competencies by combining the behavioural events technique (Section 2.2) and the methodology described in Figure 1. On the other hand, Table 1 illustrated below cluster individual competencies (behavioural competencies and abilities) as well as collective competencies of the members of the need seeker innovation project. We have obtained the results summarised in Table 1 through the method illustrated in the figure 1. In this paper we have presented to our knowledge the first competency framework supporting the need seeker innovation strategy. A novel approach has been proposed combining different design methods of competency framework and highlighting the work of other disciplines on innovation competencies. This study is a part of a broader design education research project aiming to design a competencybased-learning of innovation processes. The first validation method we are considering is to compare the performance and practices of learners before and after using this framework to guide the curriculum development process. The second method is by using the Kelly Grid technique, since it has the advantage of being an effective technique to complement the methods we have combined. The third validation method is to evaluate the competency framework, by innovation experts. In a second phase of this study, we consider future research using this competency framework as a basis of a diagnosis tool to build personalised training journey of innovation processes. It is our hope that these findings stimulate more research from multidisciplinary background, on the critical competencies supporting a need seeker innovation strategy. Specifying these competencies is a key factor to guide the implementation of a comprehensive educational curriculum about design and innovation processes.

Figure 2 .

 2 Figure 2. Strategic leadership competencies supporting need seeker innovation project. (See Legend of Figure 1 for the signification of R, O, I and Ri)

Table 1 .

 1 The competency framework supporting the need seeker strategy (See Legend of Figure1for the signification of R, O, I and Ri)

		ANALYSING PHASE (PROBLEM DISCOVERY PHASE)	IDEA PHASE) GENERATION PHASE (PROBLEM SYNTHESIS	IDEA TESTING PHASE) EVALUATION PHASE (SOLUTION FINDING AND	NEW BUSINESS DEVELOPMENT PHASE
		ability to	synthesising	ability to grasp	opportunity
		tackle ill	skills	aspects of	recognising skills
	ABILITY TO EXERCICE A KNOWLEDGE	structured problem (O, Ri) [24]	(R, O, I) knowledge management	intellectual property (R, I) experimenting	(I, O, Ri) [25] system thinking skills
		empathy	skills	and	(I, O, Ri)
		skills	(R, O, I)	prototyping
		(R, O)		skills
			project portfolio	(R, O, I)
		analytical	management	problem-solution
		skills	skills	pairing
		(R, O, I)	(I, Ri)	(R, O, I)

All the more since, the empowerment of these specific competencies becomes for several businesses a critical factor to their long term success.