From microscopic price dynamics to multidimensional rough volatility models
aa r X i v : . [ q -f i n . M F ] O c t From microscopic price dynamics tomultidimensional rough volatility models *Mathieu Rosenbaum † Mehdi Tomas ‡ October 31, 2019
Abstract
Rough volatility is a well-established statistical stylised fact of financial assets. This property has lead tothe design and analysis of various new rough stochastic volatility models. However, most of these devel-opments have been carried out in the mono-asset case. In this work, we show that some specific multi-variate rough volatility models arise naturally from microstructural properties of the joint dynamics of assetprices. To do so, we use Hawkes processes to build microscopic models that reproduce accurately high fre-quency cross-asset interactions and investigate their long term scaling limits. We emphasize the relevanceof our approach by providing insights on the role of microscopic features such as momentum and mean-reversion on the multidimensional price formation process. We in particular recover classical properties ofhigh-dimensional stock correlation matrices.
Keywords:
Rough volatility, multidimensional processes, microstructure, Hawkes processes, limit theorems,high-dimensional correlation matrices.
AMS 2000 subject classifications:
It is now widely accepted that volatility is rough (see [11] and among others [6, 24]): the log-volatility process iswell-approximated by a fractional Brownian motion with small Hurst parameter H ≈ H − ǫ , ǫ >
0. Furthermore, rough volatility models capture key features of the im-plied volatility surface and its dynamics (see [3, 10, 17]).The macroscopic phenomenon of rough volatility is seemingly universal: it is observed for a large class offinancial assets and across time periods. This universality may stem from fundamental properties such asmarket microstructure or no arbitrage. This raised interest in building microscopic models for market dynam-ics which reproduce rough volatility at a macroscopic scale. For us, the microscopic time scale is of the order * Mathieu Rosenbaum and Mehdi Tomas gratefully acknowledge financial support of the ERC 679836 Staqamof and the Chair Analyticsand Models for Regulation. Mehdi Tomas gratefully acknowledges the support of the Chair Econophysics and Complex Systems. Theauthors thank Michael Benzaquen and Iacopo Mastromatteo for their helpful comments and are grateful to Eduardo Abi Jaber, Jean-
Philippe Bouchaud, Antoine Fosset and Paul Jusselin for very fruitful discussions and suggestions. † CMAP, École Polytechnique, [email protected] ‡ CMAP & LadHyx, École Polytechnique, [email protected]
We first introduce the asymptotic framework which models the high endogeneity of financial markets in themono-asset case (as in [1, 8, 20, 21]) for clarity purposes before moving to the multivariate setting of interest. Atthe high frequency scale, the price is a piecewise constant process with upward and downward jumps capturedby a bi-dimensional counting process N = ( N + , N − ), with N + counting the number of upward price movesand N − the number of downward price moves. Assuming that all jumps are of the same size, the microscopicprice of the asset is the difference of the number of upward and downward jumps (where the initial price is setto zero for simplicity) and therefore can be written P t = N + t − N − t .Our assumption is that N is a Hawkes process with intensity λ = ( λ + , λ − ) such that λ + t = µ + t + Z t φ + ,1 + ( t − s ) d N + s + Z t φ + ,1 − ( t − s ) d N − s λ − t = µ − t + Z t φ − ,1 + ( t − s ) d N + s + Z t φ − ,1 − ( t − s ) d N − s ,where the µ : R + →∈ R + is called the baseline and φ : R + → M ( R + ) is called the kernel , where we write vectorsand matrices in bold and M n , m ( X ) (resp. M n ( X )) for the set of X -valued n × m (resp. n × n ) matrices. From afinancial perspective, we can easily interpret the different terms above:• on the one hand, µ + (resp. µ − ) is an exogenous source of upward (resp. downward) price moves;• on the other hand, φ is an endogenous source of price moves. For example, φ + ,1 − increases the inten-sity of upward price jumps after a downward price jump, creating a mean-reversion effect (while φ + ,1 + creates a trending effect).To further encode the long-memory property of the order flow, [8, 20] consider heavy-tailed kernels where,writing ρ ( M ) for the spectral radius of a matrix M , for some c > α ∈ (1/2,1) we have ρ ¡ Z ∞ t φ ( s ) d s ¢ ∼ t →∞ ct − α .Such a model satisfies the stability property of Hawkes processes (see for example [20]) as long as ρ ( °° φ °° ) < k·k for the L norm). In fact, calibration of Hawkes processes on financial data suggests that thisstability condition is almost violated. To account for this effect, the authors of [8, 20] model the market up totime T with a Hawkes process N T of baseline µ T and kernel φ T . The microscopic price until time T is then P T ,1 t = N T ,1 + t − N T ,1 − t .In order to obtain macroscopic dynamics, the time horizon must be large, thus the sequence T n tends towardsinfinity (from now on, we write T for T n ). As T tends to infinity, φ T almost saturates the stability condition:lim n →∞ ρ ( °° φ T °° ) =
1. A macroscopic limit then requires scaling the processes appropriately to obtain a non-trivial limit. Details on the proper rescaling of the processes are given in Section 1.4.
Having described the asymptotic setting in the mono-asset case, we now model m different assets. The associ-ated counting process is now a 2 m -dimensional process N T = ( N T ,1 + , N T ,1 − , N T ,2 + ,... , N T , m − ) and its intensitysatisfies λ Tt = µ T + Z t φ ( t −−− s ) T d N Ts .The counting process N includes the upward and downward price jumps of m different assets and the micro-scopic price of Asset i , where 1 ≤ i ≤ m , is simply P T , it = N T , i + t − N T , i − t .This allows us to capture correlations between assets since, focusing for example on Asset 1, we have λ T ,1 + t = µ T ,1 + t + Z t φ T + ,2 + ( t − s ) d N T ,2 + s + Z t φ T − ,2 + ( t − s ) d N T ,2 + s + ··· .Therefore φ T + ,2 + increases the intensity of upward jumps on Asset 1 after an upward jump of Asset 2 while φ T − ,2 + increases the intensity of downward jumps, etc.We now need to adapt the nearly-unstable setting to the multidimensional case. Thus we have to find howto saturate the stability condition and to translate the long memory property of the order flow.In [8], φ T ( t ) is taken diagonalisable (in a basis independent of T and t ) with a maximum eigenvalue ξ T ( t )such that lim T →∞ °° ξ T °° =
1. However this structure leads to the same volatility for all assets and thus cannotbe a satisfying solution for realistic market dynamics. We take here a sequence of trigonalisable (in a basis O independent of T and t ) kernels φ T ( t ) with n c > φ T ( t ) = O Ã A T ( t ) 0 B T ( t ) C T ( t ) ! O − ,where A T : R + → M n c ( R ), B T : R + → M m − n c , n c ( R ) and C T : R + → M m − n c ( R ). Note that we will see that in thelimit, macroscopic volatilities and prices are independent of the chosen basis. We assume that the stabilitycondition is saturated at the speed T − α where α ∈ (1/2,1) is again related to the tail of the matrix kernel (seebelow). The saturation condition translates to T α ¡ I − Z ∞ A T ¢ → T →∞ K ,where K is an invertible matrix.We now need to encode the long memory property of the order flow. We can expect orders to be sent jointly ondifferent assets (this can be due, for example, to portfolio rebalancing, risk management or optimal trading)and split under different time scales depending on idiosyncratic components (such as daily traded volume orvolatility). Empirically the approximation that despite idiosyncrasies a common time scale for order splittingexists is partially justified: for example [4] shows that market impact, which is directly related to the order flow,is well-approximated by a single time scale for many stocks. Finally, this property is encoded by imposing aheavy-tail condition for A : = lim T →∞ A T with the previous exponent α : α x α Z ∞ x A ( s ) d s → x →∞ M ,with M an invertible matrix. In the framework described above, we show that the macroscopic limit of prices is a multivariate version ofthe rough Heston model introduced in [9, 10], where the volatility process is a solution of a multivariate roughstochastic Volterra equation. Thus we derive a natural multivariate setting for rough volatility using nearly-unstable Hawkes processes.More precisely, define the rescaled processes (see [20] for details), for t ∈ [0,1]: X Tt : = T α N TtT (1) Y Tt : = T α Z tT λ s d s (2) Z Tt : = T α ( X Tt − Y Tt ) = T α M TtT (3) P Tt = T α ( N T ,1 + tT − N T ,1 − tT , ··· , N T , m + tT − N T , m − tT ). (4)We refer to P T as the (rescaled) microscopic price process. Under some additional technical and no statisticalarbitrage assumptions, there exists an n c dimensional process ˜ V , matrices Θ ∈ M n c ( R ), Θ ∈ M n − n c ( R ), Λ ∈ M n c ( R ), Λ ∈ M n c ( R ), Λ ∈ M n c , n − n c ( R ), θ ∈ R n c and a Brownian motion B such that• Any macroscopic limit point P of the sequence P T satisfies P t = ( I + ∆ ) † Q Z t diag( p V s ) d B s ,where Q : = ( e − e | ··· | e m − − e m ), writing † Q for the transpose of Q , ( e i ) ≤ i ≤ m for the canonicalbasis of R m and ∆ = ( ∆ i j ) ≤ i , j ≤ m ∈ M m ( R ) is defined in Section 3 while V is defined below.• Θ ˜ V = ( V , ··· , V n c ) and Θ ˜ V = ( V n c + , ··· , V n ).• ˜ V has Hölder regularity α − − ǫ for any ǫ > t in [0,1], ˜ V satisfies ˜ V t = Z t ( t − s ) α − ( θ − Λ ˜ V s ) d s + Z t ( t − s ) α − Λ diag( q Θ ˜ V s ) d W s + Z t ( t − s ) α − Λ diag( q Θ ˜ V s ) d Z s ,where W : = ( B , ··· , B n c ), Z : = ( B n c + , ··· , B n ) and we write p x for the component-wise square root ofvectors of non-negative entries.Thus the volatility process V is driven by ˜ V , which represents volatility factors, of which there are as many asthere are critical directions.We can use this result to provide microstructural foundations for some empirical properties of correlation ma-trices. Informally, considering that our assets have similar self-exciting features in their microscopic dynamics,we show that any macroscopic limit point P of the sequence P T satisfies PP t = Σ Z t diag( p V s ) d W s ,where W is a Brownian motion, V satisfies a stochastic Volterra equation and Σ has one very large eigenvaluefollowed by smaller eigenvalues that we can interpret as due to the presence of sectors and a bulk of eigenval-ues much smaller than the others. This is typical of actual stock correlation matrices (see for example [23] foran empirical study).The paper is organised as follows. Section 2 rigorously introduces the technical framework sketched in theintroduction. We present and discuss the main results in Section 3 which are then applied in examples de-veloped in Section 4. Proofs can be found in Section 5 while some technical results, including proofs of thevarious applications, are available in an appendix. Before presenting the main results, we make precise the framework sketched out in the introduction. Differentexamples of Hawkes processes satisfying our assumptions are given in Section 4.Consider a sequence of measurable functions φ T : R + → M m ( R + ) and µ T : R + → R m + , where the pair ( µ T , φ T )will be used to model the market dynamics until time T via a Hawkes process N T of baseline µ T and kernel φ T . Each kernel φ T is stable ( ρ ¡ °° φ T °° ¢ < Assumption 1.
There exists O an invertible matrix such that each φ T can be written as φ T = O Ã A T B T C T ! O − , where A T : R + → M n c ( R ) , B T : R + → M m − n c , n c ( R ) , C T : R + → M m − n c ( R ) . Furthermore, the sequence φ T con-verges towards φ : R + → M m ( R + ) as T tends to infinity and, writing A , B , C for the limits of A T , B T , C T as Ttends to infinity, ρ ( R ∞ C ) < .Additionally, there exists α ∈ (1/2,1) , K , M invertible matrices and µ : [0,1] → R + such thatT α ( I − °° A T °° ) → T →∞ K (5) α x α Z ∞ x A ( s ) d s → x →∞ M (6) T − α µ TtT → T →∞ µ t , (7) where K M − has strictly positive eigenvalues. Realistic market dynamics require enforcing no statistical arbitrage conditions on the kernels, as in the spirit of[20]. To determine which conditions need to be satisfied to prevent such arbitrage, we write the intensity of thecounting process λ T using the compensator process M Tt : = N Tt − λ Tt and ψ T = P k ≥ ( φ T ) ∗ k (see for exampleProposition 2.1 in [20]). We have λ Tt = µ T + Z t ψ T ( t −−− s ) µ Ts d s + Z t ψ T ( t −−− s ) d M Ts . (8)Thus, the expected intensities of upward and downward price jumps of Asset i are E [ λ T , i + t ] = µ T , i + t + X ≤ j ≤ m Z t ψ Ti + , j − ( t − s ) µ T , j − s d s + X ≤ j ≤ m Z t ψ Ti + , j + ( t − s ) µ T , j + s d s E [ λ T , i − t ] = µ T , i − t + X ≤ j ≤ m Z t ψ Ti − , j − ( t − s ) µ T , j − s d s + X ≤ j ≤ m Z t ψ Ti − , j + ( t − s ) µ T , j + s d s .The above leads us to the following assumption. Assumption 2.
For any ≤ i , j ≤ m:(i) ψ Ti + , j + + ψ Ti + , j − = ψ Ti − , j + + ψ Ti − , j − (no pair-trading arbitrage)(ii) lim T →∞ ³ R ∞ ψ Ti + , j − R ∞ ψ Ti + , j + ´ < ∞ (suitable asymptotic behaviour of the intensities) Under the above conditions and if µ T , i + = µ T , i − for all 1 ≤ i ≤ m , then E [ λ T , i + t ] = E [ λ T , i − t ] and there are on aver-age as many upward than downward jumps, which we interpret as a no statistical arbitrage property.Define, for any 1 ≤ i , j ≤ m , δ Ti j : = ψ Tj + , i + − ψ Tj − , i + (9) ∆ i j : = lim T →∞ °°° ψ Tj + , i + °°° − °°° ψ Tj − , i + °°° . (10)We can make the following remark. Remark 1.
Note that for any ≤ k ≤ m, defining v k : = e k + − e k − and using (i) of Assumption 2, we have † ψ T v k = † ψ T ( e k + − e k − ) = ( ψ Tk + ,1 + − ψ Tk − ,1 + ) e + + ( ψ Tk + ,1 − − ψ Tk − ,1 − ) e − + ··· + ( ψ Tm + ,1 + − ψ Tm − ,1 + ) e m − = ( ψ Tk + ,1 + − ψ Tk − ,1 + ) e + − ( ψ Tk + ,1 + − ψ Tk − ,1 + ) e − + ··· + ( ψ Tm + ,1 + − ψ Tm − ,1 + ) e m − = ( ψ Tk + ,1 + − ψ Tk − ,1 + ) v + ··· + ( ψ Tk + , m + − ψ Tk − , m + ) v m = δ Tk v + ··· + δ Tkn v m . A sufficient condition for the no pair-trading arbitrage Equation (i) of Assumption 2 to hold is that, for all ≤ i ≤ m, † φ T v i = X ≤ j ≤ m ( † φ T v i · v j ) v j , since then we have, for any ≤ k ≤ m, X ≤ l ≤ m ( ψ Tk + , l + − ψ Tk − , l + ) e l + − ( ψ Tk + , l + − ψ Tk − , l + ) e l − = X ≤ l ≤ m ( ψ Tk + , l + − ψ Tk − , l + ) e l + − ( ψ Tk + , l − − ψ Tk − , l − ) e l − . In our applications in Section 4 we will use this condition as it is easier to check assumptions on φ than on ψ . We are now in the position to rigorously state the main results of this paper. We use the processes X T , Y T and Z T defined in the introduction (see Equations (1), (2) (3)) and write O − = Ã O ( − O ( − O ( − O ( − ! , O = Ã O O O O ! .We set Θ : = ¡ O + O ( I − Z ∞ C ) − Z ∞ B ¢ K − Θ : = ¡ O + O ( I − Z ∞ C ) − Z ∞ B ¢ K − θ : = Ã O ( − O ( − ! µ Λ : = α Γ (1 − α ) K M − .We have the following theorem. Theorem 1.
The sequence ( X T , Y T , Z T ) is C -tight for the Skorokhod topology. Furthermore, for every limit point ( X , Y , Z ) of the sequence, there exists a positive process V and an m-dimensional Brownian motion B such that(i) X t = R t V s d s, Z t = R t diag( p V s ) d B s .(ii) There exists ˜ V a process of Hölder regularity α − − ε for any ε > such that Θ ˜ V = ( V , ··· , V n c ) , Θ ˜ V = ( V n c + , ··· , V m ) and ˜ V is solution of the following stochastic Volterra equation: ∀ t ∈ [0,1], ˜ V t = Γ ( α ) Λ Z t ( t − s ) α − ( θ − ˜ V s ) d s + Γ ( α ) Λ Z t ( t − s ) α − O ( − diag( q Θ ˜ V s ) d W s + Γ ( α ) Λ Z t ( t − s ) α − O ( − diag( q Θ ˜ V s ) d W s , (11) where W : = ( B , ··· , B n c ) , W : = ( B n c + , ··· , B m ) , Θ , Θ , O ( − , O ( − , θ do not depend on the chosenbasis.Finally, any limit point P of the rescaled price processes P T satisfies P t = ( I + ∆ ) † Q ( Z t diag( p V s ) d B s + Z t µ s d s ), where ∆ is defined in Equation (10) . Theorem 1 links multivariate nearly unstable Hawkes processes and multivariate rough volatility. We note that:• The resulting stochastic Volterra equation has non-trivial solutions, as the examples in Section 4 willshow.• From a financial perspective, Theorem 1 shows that the limiting volatility process for a given asset is asum of different factors. The matrix ∆ mixes them and is therefore responsible for correlations betweenasset prices. Remarks and comments on I + ∆ are developed in Section 4.• The theorem implies that adding/removing an asset to/from a market has an impact on the individualvolatility of other assets. We can estimate the magnitude of such volatility modifications by calibratingHawkes processes on price changes.• Since there is a one to one correspondence between the Hurst exponent H and the long memory pa-rameter of the order flow α , our model yields the same roughness for all assets. Extensions to allowfor different exponents to coexist, for example by introducing an asset-dependent scaling through D = ( α , ··· , α m ) and studying T − D λ TtT , are more intricate. In particular, one needs to use a special functionextending the Mittag-Leffler matrix function such that its Laplace transform is of the form ( I + Λ t D ) − . In this section, we give examples of processes obtained through Theorem 1 under different assumptions onthe microscopic parameters. The first example highlights the flexibility of our framework and shows that theobtained limit in Theorem 1 is non-trivial. We then study the influence of microscopic parameters on thelimiting price and volatility processes when modeling two assets. Finally, we model many different assets toreproduce realistic high-dimensional correlation matrices.
Before presenting some truly relevant results for finance, we develop an example demonstrating that the so-lutions to the Volterra equations of the form of Equation (11) are non-trivial. The structure of our Volterraequations is close to those studied in [19], which proves existence and uniqueness of affine Volterra equations.In particular, this paper covers Volterra equations of the following type, for α ∈ (1/2,1): X t = X + Z t ( t − s ) α − b ( X s ) d s + Z t ( t − s ) α − σ ( X s ) d B s ,where b : R → R n and σ : R → M n ( R ) are continuous functions. A key condition required for existence anduniqueness is sublinear growth condition on b and σ , that is k b ( x ) k ∨ k σ ( x ) k ≤ c (1 + k x k ), (12)for some constant c > k·k is the usual Euclidian norm for vectors and matrices. Thus, this settingcovers equations of the type X t = X + Z t ( t − s ) α − b ( X s ) d s + Z t ( t − s ) α − diag( p X s ) d B s ,0which are a particular case of Theorem 1. However, note that Condition 12 fails when σ ( x ) = Σ diag( p x ) forsome non-diagonal matrix Σ . Interestingly, this setting is covered in our approach as illustrated by the follow-ing corollary. Corollary 1.
We can find a microscopic process satisfying the assumptions of Theorem 1 such that V is a non-negative process which satisfies, for any t in [0,1], V t = Z t ( t − s ) α − ( θ − GV s ) d s + Z t ( t − s ) α − Σ diag( p V s ) d B s , where θ is a -dimensional vector, G , Σ are × non-diagonal matrices and B is a -dimensional Brownianmotion. Thus, our framework yields non-trivial solutions and leads to interesting new examples of processes. We nowfocus on building realistic models to discuss the correspondence between the microscopic parameters of theHawkes kernel and macroscopic quantities such as correlations and volatility.
Our first model to understand the price formation process focuses on two assets. Let µ , µ > α ∈ (1/2,1), γ , γ in [0,1], H c , H a , H c , H a in [0,1] such that (here p· is the principal square root, so that if x < p x = i p− x ):0 ≤ ( H c + H a )( H c + H a ) < ≤| − ( γ + γ ) − q ( H c − H a )( H c − H a ) + ( γ − γ ) |< ≤| − ( γ + γ ) + q ( H c − H a )( H c − H a ) + ( γ − γ ) |< K and M . For simplicity we choose the kernel such that M = α I . This leads us to define, for t ≥ φ T ( t ) = (1 − γ ) α (1 − T − α ) t ≥ t − ( α + φ T , c ( t ) = α T − α H c t ≥ t − ( α + φ T ( t ) = γ α (1 − T − α ) t ≥ t − ( α + φ T , a ( t ) = α T − α H a t ≥ t − ( α + ˜ φ T ( t ) = (1 − γ ) α (1 − T − α ) t ≥ t − ( α + φ T , c ( t ) = α T − α H c t ≥ t − ( α + ˜ φ T ( t ) = γ α (1 − T − α ) t ≥ t − ( α + φ T , a ( t ) = α T − α H a t ≥ t − ( α + .For a realistic model, we impose the exogenous source of upward and downward price moves to be equal: µ + = µ − and µ + = µ − . Thus, the sequence of baselines and kernels are chosen as µ T = T α − µ µ µ µ , φ T = φ T φ T φ T , c φ T , a φ T φ T φ T , a φ T , c φ T , c φ T , a ˜ φ T ˜ φ T φ T , a φ T , c ˜ φ T ˜ φ T . The superscripts c (resp. a ) stand for continuation (resp. alternation) to describe that after a price move in a given direction, H c (resp. H a ) encodes the tendency to trigger other price moves in the same (resp. opposite) direction will follow. Corollary 2.
Consider any limit point P of P T . Under the above assumptions, it satisfies P t = p γ γ − ( H c − H a )( H c − H a ) à γ H c − H a H c − H a γ !Z t q V s dW s q V s dW s , (13) with à V t V t ! = α Γ ( α ) Γ (1 − α ) Z t ( t − s ) α − Ãà µ µ ! − − ( H c + H a )( H c + H a ) à H c + H a H c + H a ! à V s V s !! d s + p α Γ ( α ) Γ (1 − α ) Z t ( t − s ) α − q V s d Z s q V s d Z s , (14)where W and Z are bi-dimensional independent Brownian motions. This model helps us understand howmicroscopic parameters drive the price formation process to generate a macroscopic price and volatility. Webegin our remarks with some definitions.We call momentum the trend (i.e., the imbalance between the number of upward and downward jumps) cre-ated by jumps of one asset on itself . The opposite effect is referred to as mean-reversion . For example, theparameter γ controls the intensity of self-induced bid-ask bounce on Asset 1: when γ close to zero corre-sponds to a strong momentum while γ close to one corresponds to a strong mean-reversion.We call cross-asset momentum the trend created by jumps of one asset on another. For example, cross-assetmomentum from Asset 2 to Asset 1 (resp. Asset 1 to Asset 2) appears via H c − H a (resp. H c − H a ): when both H c − H a and H c − H a are nill, the prices of Asset 1 and Asset 2 are uncorrelated. We now turn to commentson the volatility process.Because of its role in the single-asset case, we refer to V as the fundamental variance : for example V is thefundamental variance of Asset 1. The equation satisfied by V only depends on the sum of the feedback effectsbetween each asset through H c + H a : from a volatility viewpoint, upward and downward jumps have thesame impact. Furthermore, we can compute the expected fundamental variance using Mittag-Leffler func-tions (see Section 5).Mean-reversion drives down volatility while cross-asset momentum increases it. Indeed, computing E [( P t ) ]for example we get: E [( P t ) ] = γ R t E [ V s ] d s + ( H c − H a )( H c − H a ) R t E [ V s ] d s [4 γ γ − ( H c − H a )( H c − H a )] .In particular, increasing γ or γ does not change V but reduces E [( P t ) ]. This example may be particularlyrelevant to understand the contribution of Asset 2 to the volatility of Asset 1 through calibration to marketdata since if Asset 2 were removed from the market, we would have E [( P t ) ] = γ . Focusing now on the priceformation process, we see that it results from a combination of momentum, mean-reversion and cross-asset2momentum. We illustrate this in two extreme cases: when there is no cross-asset momentum and when cross-asset momentum is strong.• When there is no cross-asset momentum (i.e. H c = H a and H c = H a ) at the microscopic scale a pricemove on Asset 2 has the same impact on the intensity of upward and downward price moves of Asset1. Thus the difference between the expected number of upward and downward jumps does not changeafter a price move on Asset 2: the expected microscopic price of Asset 1 is unaffected and price moves ofAsset 2 generate no trend on Asset 1. This results in macroscopic prices being uncorrelated (see Equation(13)).• On the other hand, when cross-asset momentum is strong (i.e. ( H c − H a )( H c − H a ) ≈ γ γ , for ex-ample if H c − H a = γ p − ǫ , H c − H a = γ p − ǫ for some small ǫ > ∆ +++ I = γ γ ǫ à γ γ p − ǫγ p − ǫ γ ! .Using Equation (13) we can check that E [ P t P t ] q E [( P t ) ] E [( P t ) ] → ǫ → It is well-known that the correlation matrix of stocks has few large eigenvalues outside of a "bulk" of eigen-values attributable to noise (see for example [23]). The largest eigenvalue is referred to as the market mode(because the associated eigenvector places a roughly equal weight on each asset) and is much larger thanother eigenvalues. Other significant eigenvalues can be related to the presence of sectors: groups of compa-nies with similar characteristics.How can we provide microstructural foundations for this stylised fact? The large eigenvalue associated to themarket mode implies that, in a first approximation, stock prices move together: a price increase on one assetis likely followed by a price increase on all other assets. Translating this in our framework, an upward (resp.downward) jump on a given asset increases the probability of an upward (resp. downward) jump on all otherassets. We further expect that an upward price move on an asset increases this probability much more on anasset from the same sector than on an unrelated one.The above remarks lead us to consider a model where:• All stocks share some fundamental high-frequency properties by having similar self-excitement param-eters in the kernel.• Stocks have a stronger influence on price changes of stocks within the same sector.3• Within the same sector, all stocks have the same microscopic parameters.The technical details of our setting are presented in Appendix A.5 and we only provide here essential elementsto understand the framework. Let µ ,... , µ m > γ in [0,1], α in (1/2,1) and H c , H a >
0. We consider R > r having m r stocks. For a pair of stocks which we dub 1,2 to make an analogy with the previous example, we have that:• The self excitement parameters are equal: γ = γ = γ where γ is the same for all stocks.• If Stock 1 and Stock 2 do not belong to the same sector, H c = H c = H c , H a = H a = H a where H c , H a are the same for all stocks.• If Stock 1 and Stock 2 belong to the same sector r , H c = H c = H c + H cr , H a = H a = H a + H ar where H cr , H ar are the same for all stocks belonging to sector r .The asymptotic framework is built as in the previous example, with the details given in the proof of Corollary 3in Appendix A.5. We write i r : = m + m + ··· + m r − for 1 ≤ r ≤ R (with convention m =
1) so that stocks fromsector r are indexed between i r and i r + excluded and define the following vectors w : = p m ( e + ··· + e m ) w r : = p m r X i r ≤ i < i r + e i θ : = X ≤ i ≤ m µ i e i .We consider an asymptotic framework where the number of assets will eventually grow to infinity. As willbecome clear in the proof, the only non-trivial regime appears when H c , H a , H cr , H ar = m →∞ O ( m − ). Thus weassume that mH c , mH a , mH cr , mH ar converge to ¯ H c , ¯ H a , ¯ H cr , ¯ H ar as m tends to infinity. We also assume thatthe proportion of stocks in a given sector relative to the total number of stocks does not vanish: for each 1 ≤ r ≤ R , m r m → m →∞ η r >
0. Define the following constants which will appear in the price and volatility processes: λ + : = ¯ H c + ¯ H a , λ + r : = ¯ H cr + ¯ H cr , λ − : = ¯ H c − ¯ H a , λ − r : = ¯ H cr − ¯ H ar . Applying Theorem 1 yields the following result. Corollary 3.
Consider any limit point P of P T . Under the above assumptions, it satisfies: P t = p Σ ε Z t diag( p V s ) d W s , where W is a Brownian motion, Σ ε : = (2 γ I − λ − v † v − P ≤ r ≤ R η r λ − r v r † v r + ε ) − with ǫ a deterministic m × mmatrix such that ρ ( ǫ ) = m →∞ o ( m − ) and V satisfies the stochastic Volterra equation V t = α Γ ( α ) Γ (1 − α ) Z t ( t − s ) α − ( θ − V ǫ V s ) d s + p α Γ ( α ) Γ (1 − α ) Z t ( t − s ) α − diag( p V s ) d Z s , with Z a Brownian motion independent from W and V ǫ : = ¡ I − λ + v † v − P ≤ r ≤ R η r λ + r v r † v r + ǫ ¢ − where ε is adeterministic m × m matrix such that ρ ( ε ) = m →∞ o ( m − ) . Under the previous corollary, writing ∝ for equality up to a multiplicative constant, the expected fundamental4variance can be written using the cumulative Mittag-Leffler function E [ V t ] ∝ F α , V ǫ ( t ) θ .Since ρ ( ǫ ) = m →∞ o ( m − ), we neglect it in further comments and use the approximation V ǫ ≈ V . Writing ξ forthe largest eigenvalue of V and neglecting other eigenvalues (which is reasonable if λ + + P ≤ r ≤ R η r λ + r ≈ z for the associated eigenvector, using the definition of the Mittag-Leffler function (see Definition 4 inAppendix A.2), we have E [ V t ] ∝ F α , ξ ( t )( † θ z ) z .In the further approximation that η r λ + r is independent r , we have z ∝ (1, ··· ,1) and E [ P t † P t ] ∝ Σ ε diag( E [ V t ]) † Σ ε ∝ Σ ε diag( z ) † Σ ε ∝ Σ ε † Σ ε ∝ Σ ε .Therefore the eigenvectors of E [ P t † P t ] are those of Σ ε . As ρ ( ε ) = m →∞ o ( m − ), we neglect it in further commentsand use the approximation Σ ε ≈ Σ . When λ − + P ≤ r ≤ R η r λ − r ≈ γ , Σ has one large eigenvalue followed by R − λ − + P ≤ r ≤ R η r λ − r ≈ λ + + P ≤ r ≤ R η r λ + r ≈ λ − + P ≤ r ≤ R η r λ − r ≈ γ or λ − + P ≤ r ≤ R η r λ − r ≈
1, the spectralradius of R ∞ C is equal to one and we cannot split the kernel into a critical and a non-critical component.It would be interesting to study other implications of this model. In particular, we believe that encoding anegative price/volatility correlation into the microscopic parameters could explain the so-called index lever-age effect (see [25] for a definition and empirical analysis of this stylised fact). We split the proof into four steps. Our approach is inspired by [8, 20, 21]. First, we show that the sequence( X T , Y T , Z T ) is C -tight. Second, we use tightness and representation theorems to find equations satisfied byany limit point ( X , Y , Z ) of ( X T , Y T , Z T ). Third, properties of the Mittag-Leffler function enable us to proveEquation (11). Fourth and finally, we derive the equation satisfied by any limit point P of P T . Preliminary lemmas
We start with lemmas that will be useful in the proofs. Lemma A.1 from [8] yields1 T α λ TtT = µ TtT T α + T α Z tT ψ T ( tT −−− s ) µ Ts d s + T α Z tT ψ T ( tT −−− s ) d M Ts . (15)5Thus to investigate the limit of 1 T α λ T · T we need to study 1 T α ψ T ( T ··· ) , which we will do through its Laplacetransform. Given a L ( R + ) function f , we write its Laplace transform ˆ f ( t ) : = R ∞ f ( x ) e − tx d x , for t ≥ F = ( F i j ) where each F i j ∈ L ( R + )). Remark that d f ∗ k = ˆ f k , where ∗ k isthe convolution product iterated k times. The following lemma holds. Lemma 1.
We have the following convergence for any t ≥ :T − α á ψ T ( T ··· )( t ) → T →∞ O · Γ (1 − α ) α t α M + K ¸ − I − R ∞ C ) − R ∞ B · Γ (1 − α ) α t α M + K ¸ − O − , (16) where K and M are defined in Equation (5) and (6) .Proof. Define ϕ T : = O − ˆ φ T O . Then ˆ ψ T ( t ) = X k ≥ ˆ φ T , ∗ k = O ( I − ˆ ϕ T ) − ˆ ϕ T O − .We can use the shape of ϕ T and matrix block inversion to rewrite this expression. Doing so, we find ˆ ψ T ( t ) = O Ã ( I − ˆ A T ( t )) − ˆ A T ( t ) I − ˆ C T ( t ) ) − ˆ B T ( t ) ( I − ˆ A T ( t ) ) − ˆ A T ( t ) − ( I − ˆ C T ( t ) ) − ˆ B T ( t ) ( I − ˆ C T ( t ) ) − ˆ C T ( t ) ! O − .To derive the limiting process, we use Equations (5) and (6). Using integration by parts and a Tauberian theo-rem as in [8, 21], we have ZZZ ∞ A T − ˆ A T ( t / T ) = T →∞ Γ (1 − α ) α t α M T − α + o ( T − α ) I − Z ∞ A T = T →∞ K T − α + o ( T − α ).Therefore T ( I − ˆ A T ( t / T ) ) = T ( Z ∞ A T − ˆ A T ( t / T )) + T ( I − Z ∞ A T ) = T →∞ · Γ (1 − α ) α t α M + K ¸ T − α + o ( T − α ).Consequently T α − T ( I − ˆ A T ( t / T ) ) = T →∞ Γ (1 − α ) α t α M + K + o (1).By Assumption 1 M is invertible and K M − has strictly positive eigenvalues. Thus M t + K = ( K M − + t I ) M is invertible for any t ≥
0. The Laplace transform of T − α ψ T ( T ··· ) being T − α b ψ T ( ··· / T ) , we have proved for any6 t ≥ á T − α ψ T ( T ··· ) ( t ) → T →∞ O · Γ (1 − α ) α t α M + K ¸ − I − R ∞ C ) − R ∞ B · Γ (1 − α ) α t α M + K ¸ − O − .We show in the technical appendix that the inverse Laplace transform of Λ ( t α I + Λ ) − , where Λ ∈ M n ( R ) haspositive eigenvalues, is a simple extension of the Mittag-Leffler density function to matrices (see Definition 4in the appendix) denoted by f α , Λ . Thus we define for any t ∈ [0,1] f ( t ) : = O K − f α , α Γ (1 − α ) K M − ( I − R ∞ C ) − R ∞ BK − f α , α Γ (1 − α ) K M − O − . (17)The following lemma shows the weak convergence of ψ T towards f . Lemma 2.
For any bounded measurable function g and ≤ i , j ≤ n Z [0,1] g ( x ) T − α ψ Ti j ( T x ) d x → T →∞ Z [0,1] g ( x ) f i j ( x ) d x . Proof.
First note that when °° f i j °° = f i j = t = °°° T − α ψ Ti j °°° → T →∞ °° f i j °° = − α ≥ °°° ψ Ti j °°° → T →∞ ψ Ti j ≥
0, for any bounded measurable function g ¯¯¯ Z [0,1] g ( x ) T − α ψ Ti j ( T x ) d x ¯¯¯ ≤ c Z [0,1] T − α ψ Ti j ( T x ) d x ≤ c °°° T − α ψ Ti j °°° ,and the result holds. Assume now that °° f i j °° >
0. It will be convenient for us to proceed with random variables,so define ρ Ti j : = T − α ψ Ti j ( T · ) °°° T − α ψ Ti j °°° .We can view ρ Ti j as the density of a random variable taking values in [0,1], say S . Lemma 1 gives the conver-gence of the characteristic functions of S towardsˆ ρ i j : = ˆ f i j °° f i j °° .Since ρ i j is continuous (as ψ Ti j is continuous), Levy’s continuity theorem guarantees that ρ Ti j converges weakly7towards ρ i j . Therefore for any bounded measurable function g Z [0,1] g ( x ) ρ Ti j ( x ) d x → T →∞ Z [0,1] g ( x ) ρ i j ( x ) d x Z [0,1] g ( x ) T − α ψ Ti j ( T x ) °°° T − α ψ Ti j °°° d x → T →∞ Z [0,1] g ( x ) f i j ( x ) °° f i j °° d x .Equation (16) implies °°° T − α ψ Ti j °°° → T →∞ °° f i j °° , so that together with the above we have Z [0,1] g ( x ) T − α ψ Ti j ( T x ) d x → T →∞ Z [0,1] g ( x ) f i j ( x ) d x .We introduce the cumulative functions F T ( t ) = Z t T − α ψ T ( T s ) d s F ( t ) = Z t f ( s ) d s .We have just shown in particular that F T converges pointwise towards F and therefore, by Dini’s theorem,converges uniformly towards F . C -tightness of ( X T , Y T , Z T ) Recall the definition of the rescaled processes: X Tt : = T α N TtT Y Tt : = T α Z tT λ s d s Z Tt : = T α ( X Tt − Y Tt ) = T α M TtT .As in [8] and [21] we show that the limiting processes of X T and Y T are the same and that the limiting processof Z T is the quadratic variation of the limiting process of X T . We have the following proposition: Proposition 1 (C-tightness of ( X T , Y T , Z T )) . The sequence ( X T , Y T , Z T ) is C-tight and if ( X , Z ) is a possiblelimit point of ( X T , Z T ) , then Z is a continuous martingale with [ Z , Z ] = diag( X ) . Furthermore, we have theconvergence in probability sup t ∈ [0,1] °° Y Tt − X Tt °° P → T →∞ Proof.
The proof is esentially the same as in [8], adapting for our structure of Hawkes processes. We have λ Tt = µ Tt + Z t ψ T ( t −−− s ) µ Ts d s + Z t ψ T ( t −−− s ) d M Ts ,8and therefore E [ N TT ] = E [ Z T λ Ts d s ] = Z T µ Tt d t + Z T Z t ψ T ( t −−− s ) µ Ts d sd t ≤ cT α °° µ °° ∞ ,where we used the convergence of T − α µ TT · (see Equation (7)) together with the weak convergence of T − α ψ T ( T · )(see Lemma 2). It follows then that E [ X T ] = E [ Y T ] ≤ c ,and since the processes are increasing, X T and Y T are tight. As the maximum jump size of X T and Y T tendsto 0, we have the C -tightness of ( X T , Y T ). Since N T is the quadratic variation of M T , ( M T , i ) − N T , i is an L martingale starting at 0 and Doob’s inequality yields X ≤ i ≤ n E [ sup t ∈ [0,1] ( X T , it − Y T , it ) ] ≤ X ≤ i ≤ n E [( X T , i − Y T , i ) ] ≤ T − α X ≤ i ≤ n E [( M T , iT ) ] ≤ T − α X ≤ i ≤ n E [ N T , iT ] ≤ cT − α .Using the same approach as in [8] we conclude that Z is a continuous martingale and [ Z , Z ] is the limit of[ Z T , Z T ]. ( X T , Y T , Z T ) By Proposition 1, for any limit point ( X , Y ) of ( X T , Y T ), we have X = Y almost surely. We use Y T to derive anequation for Y = X . As Y T = T α R tT λ Ts d s , we first study λ TsT . Using Equation (15) we get Z t λ Ts d s = Z t µ Ts d s + Z t Z u ψ T ( s −−− u ) µ Tu dud s + Z t ψ T ( t −−− s ) M Ts d s = Z t µ Ts d s + Z t ψ T ( t −−− s ) Z s µ Tu dud s + Z t ψ T ( t −−− s ) M Ts d s .A change variables of leads to Z tT λ Ts d s = Z tT µ Ts d s + Z tT ψ T ( tT −−− s ) Z s µ Tu dud s + Z tT ψ T ( tT −−− s ) M Ts d s = Z tT µ Ts d s + T Z t ψ T ( tT −−− sT ) Z sT µ Tu dud s + Z t ψ T ( tT −−− sT ) M TsT
T d s = T Z t µ TsT d s + T Z t ψ T ( T ( t −−− s )) Z sT µ Tu dud s + T Z t ψ T ( T ( t −−− s )) M TsT d s .9Therefore T α Y Tt = T Z t µ TsT d s + T Z t ψ T ( T ( t −−− s )) Z sT µ Tu dud s + T Z t ψ T ( T ( t −−− s )) M TsT d s (18) = : T α ( Y T ,1 t + Y T ,2 t + Y T ,3 t ), (19)with obvious notations. Thus, to obtain our limit we use the convergence properties of F T which we derivedpreviously. We have the following proposition. Proposition 2.
Consider ( X , Z ) a limit point of ( X T , Z T ). Then, X t = Z t F ( t −−− s ) µ s d s + Z t F ( t −−− s ) d Z s . Proof.
Let ( X , Y , Z ) be a limit point of ( X T , Y T , Z T ). First, since T − α µ TtT → T →∞ µ t (see Equation (7)), Y T ,1 t converges to 0 as T tends to infinity. Moving on to Y T ,2 , by integration by parts we have Y T ,2 t = Z t T − α ψ T ( T ( t −−− s )) T − α Z sT µ Tu dud s = · F T ( t −−− s ) T − α Z sT µ T u du ¸ t + Z t F T ( t −−− s ) T − α µ T sT d s = Z t F T ( t −−− s ) T − α µ T sT d s .Using Equation (7) again together with the uniform convergence of F T (see Lemma 2) we have the convergence Y T ,2 t → T →∞ Z t F ( t −−− s ) µ s d s .Finally, Y T ,3 t can be written as Y T ,3 t = T − α Z t ψ T ( T ( t −−− s )) M TsT d s = Z t F T ( t −−− s ) d Z Ts = Z t F ( t −−− s ) d Z s + Z t F ( t −−− s ) ( d Z Ts − d Z s ) + Z t ( F T ( t −−− s ) − F ( t −−− s ) ) d Z Ts .The Skorokhod representation theorem applied to ( Z T , Z ) yields the existence of copies in law ( ˜ Z T , ˜ Z ), ˜ Z T converging almost surely to ˜ Z . We proceed with ( ˜ Z T , ˜ Z ) and keep previous notations. The stochastic Fubinitheorem [27] gives, almost surely Z t F ( t −−− s ) ( d Z Ts − d Z s ) = Z t f ( s ) ( Z Tt − s − Z t − s ) d s .From the dominated convergence theorem we obtain the almost sure convergence Z t f ( s ) ( Z Tt − s − Z t − s ) d s → T →∞ Z T , Z T ] = diag( X T ) we have X ≤ i ≤ n E "µZ t ( F T ( t −−− s ) − F ( t −−− s ) ) d Z Ts ¶ i ≤ X ≤ i , j ≤ n Z t ( F Ti j ( t − s ) − F i j ( t − s )) T − α E [ λ T , jsT ] d s .Using Equation (15) together with Lemma 1 we can bound E [ λ T , jsT ] independently of T and X ≤ i ≤ n E "µZ t ( F T ( t −−− s ) − F ( t −−− s ) ) d Z Ts ¶ i ≤ c X ≤ i , j ≤ n Z t ( F Ti j ( t − s ) − F i j ( t − s )) d s .The right hand side converges to 0 by the dominated convergence theorem together with the uniform con-vergence of F T towards F (see Lemma 2). From Proposition 1 we know that Y = X almost surely. Puttingeverything together, almost surely, X t = Z t F ( t −−− s ) µ s d s + Z t F ( t −−− s ) d Z s .This is valid for any limit point ( X , Z ) of ( X T , Z T ), which concludes the proof.The previous proposition gives suitable martingale properties of limit points of Z T to apply the martingalerepresentation theorem, which is the topic of the following proposition. Proposition 3.
Let ( X , Z ) be a limit point of ( X T , Z T ) . There exists, up to an extension of the original probabilityspace, an n-dimensional Brownian motion B and a non-negative process V such that X t = Z t V s d s Z t = Z t diag( p V s ) d B s V t = Z t f ( t −−− s ) µ s d s + Z t f ( t −−− s ) diag( p V s ) d B s . Proof.
This proof relies on the martingale representation theorem applied to Z . Consider ( X , Z ) a limit pointof ( X T , Z T ). Following the proof of Theorem 3.2 in [21], X can be written as the integral of a process VX t = Z t V s d s ,with V satisfying the equation V t = Z t f ( t −−− s ) µ s d s + Z t f ( t −−− s ) d Z s .Therefore, as [ Z , Z ] t = diag( X t ) = diag( R t V s d s ) and Z is a continuous martingale, by the martingale represen-tation theorem (see for example Theorem 3.9 from [26]), there exists (up to an enlargement of the probabilityspace) a multivariate Brownian motion B and a predictable square integrable process H such that Z t = Z t H s d B s .1Furthermore, note that as V is a non-negative process as X is a non-decreasing process and we have Z t = Z t diag( p V s )diag( p V s ) − H s d B s .A simple computation shows that, since [ Z , Z ] t = R t H s † H s d s = X t = R t V s d s , the process ˜ B t : = R t diag( p V s ) − H s d B s is a Brownian motion. Finally, V t = Z t f ( t −−− s ) µ s d s + Z t f ( t −−− s ) diag( p V s ) d ˜ B s .A straightforward application of Lemma 4.4 and Lemma 4.5 in [21] yields the following lemma. Lemma 3.
Consider a (weak) non-negative solution V of the stochastic Volterra equation V t = Z t f ( t −−− s ) µ s d s + Z t f ( t −−− s ) diag( p V s ) d B s , where B is a Brownian motion. Then every component of V has pathwise Hölder regularity α − − ǫ for any ǫ > . (11) Properties of the Mittag-Leffler function (as in [8]) enable us to rewrite the previous stochastic differentialequation using power-law kernels, which is the subject of the next proposition. Let Θ : = ( O + O ( I − R ∞ C ) − R ∞ B ) K − , Θ : = ( O + O ( I − R ∞ C ) − R ∞ B ) K − and Λ : = α Γ (1 − α ) K M − . Proposition 4.
Given an m-dimensional Brownian motion B , a non-negative process V is solution of the fol-lowing stochastic differential equation V t = Z t f ( t −−− s ) µ s d s + Z t f ( t −−− s ) diag( p V s ) d B s , if and only if there exists a process ˜ V of Hölder regularity α − − ǫ for any ǫ > such that Θ ˜ V t = ( V , ··· , V n c ) and Θ ˜ V t = ( V n c + , ··· , V m ) are non-negative processes and ˜ V is solution of the following stochastic Volterraequation ˜ V t = Γ ( α ) Λ Z t ( t − s ) α − ( O ( − µ + O ( − µ − ˜ V s ) d s + Γ ( α ) Λ Z t ( t − s ) α − O ( − diag( q Θ ˜ V s ) d W s + Γ ( α ) Λ Z t ( t − s ) α − O ( − diag( q Θ ˜ V s ) d W s , where W : = ( B , ··· , B n c ) and W : = ( B n c + , ··· , B m ) .Proof. We begin by showing the first implication. Starting from Proposition 3 we have V t = Z t f ( t −−− s ) µ s d s + Z t f ( t −−− s ) diag( p V s ) d B s .2Developing from the definition of f in Equation (17), for any t ∈ [0,1], f can be written f ( t ) = Ã ( O + O ( I − R ∞ C ) − R ∞ B ) K − f α , Λ ( t ) 0 ( O + O ( I − R ∞ C ) − R ∞ B ) K − f α , Λ ( t ) 0 ! Ã O ( − O ( − O ( − O ( − ! .Defining V : = ( V , ··· , V n c ) and V : = ( V n c + , ··· , V m ), we have V t = Θ Z t f α , Λ ( t −−− s ) O ( − µ s d s + Θ Z t f α , Λ ( t −−− s ) O ( − µ s d s + Θ Z t f α , Λ ( t − s ) O ( − diag( q V s ) d W s + Θ Z t f α , Λ ( t −−− s ) O ( − diag( q V s ) d W s .If Θ were non-singular, we could express V with power-law kernels thanks to the same approach as in [8]. Ingeneral we define˜ V t : = Z t f α , Λ ( t −−− s ) ( O ( − µ s + O ( − µ s ) d s + Z t f α , Λ ( t − s ) O ( − diag( q V s ) d W s + Z t f α , Λ ( t − s ) O ( − diag( q V s ) d W s .From the same arguments as in Lemma 3, Hölder regularity of V carries to ˜ V , and the components of ˜ V areof Hölder regularity α − − ǫ for any ǫ >
0, hence Lemma 3 shows K : = I − α ˜ V is well-defined, where I − α isthe fractional integration operator of order 1 − α (see Definition 1 in Appendix A.2). Note that for any t in [0,1],using Lemma 4 of Appendix A.2, we have K t = Z t Λ ( I − F α , Λ ( t −−− s ) )( O ( − µ s + O ( − µ s ) d s + Z t Λ ( I − F α , Λ ( t − s )) O ( − diag( q V s ) d W s + Z t Λ ( I − F α , Λ ( t − s )) O ( − diag( q V s ) d W s = Λ Z t ( O ( − µ s + O ( − µ s ) d s + Z t Λ O diag( q V s ) d W s + Z t Λ O ( − diag( q V s ) d W s − Λ Z t · F α , Λ ( t −−− s ) O ( − µ s + Z s f α , Λ ( s −−− u ) O ( − diag( q V u ) d W u ¸ d s − Λ Z t · F α , Λ ( t −−− s ) O ( − µ s + Z s f α , Λ ( s −−− u ) O ( − diag( q V u ) d W u ¸ d s .The last two terms can be rewritten using the definition of ˜ V , so that K t = Λ Z t ( O ( − µ s + O ( − µ s − ˜ V s ) d s + Λ Z t O ( − diag( q Θ ˜ V s ) d W s + Λ Z t O ( − diag( q Θ ˜ V s ) d W s .Thanks to the Hölder regularity of ˜ V , we can now apply the fractional differentiation operator of order 1 − α (see Definition 1 in Appendix A.2) together with the stochastic Fubini Theorem to deduce˜ V t = Γ ( α ) Λ Z t ( t − s ) α − ( O ( − µ s + O ( − µ s − ˜ V s ) d s + Γ ( α ) Λ Z t ( t − s ) α − O ( − diag( q Θ ˜ V s ) d W s + Γ ( α ) Λ Z t ( t − s ) α − O ( − diag( q Θ ˜ V s ) d W s .This concludes the proof of the first implication. We now show the second implication. Suppose there exists3˜ V of Hölder regularity α − − ǫ for any ǫ > Θ ˜ V and Θ ˜ V are positive, solution of the followingstochastic Volterra equation:˜ V t = Γ ( α ) Λ Z t ( t − s ) α − ( O ( − µ s + O ( − µ s − ˜ V s ) d s + Γ ( α ) Λ Z t ( t − s ) α − O ( − diag( q Θ ˜ V s ) d W s + Γ ( α ) Λ Z t ( t − s ) α − O ( − diag( q Θ ˜ V s ) d W s .Let us write for this proof θ : = Λ O ( − µ + Λ O ( − µ , Λ : = Λ O ( − , Λ : = Λ O ( − so that, for any t in [0,1],˜ V t = Γ ( α ) Z t ( t − s ) α − ( θ s − Λ ˜ V s ) d s + Γ ( α ) Z t ( t − s ) α − Λ diag( q Θ ˜ V s ) d W s + Γ ( α ) Z t ( t − s ) α − Λ diag( q Θ ˜ V s ) d W s .Remark that the above can be written˜ V t = I α ( θ − Λ ˜ V ) t + I α B ( Λ diag( p Θ ˜ V )) t + I α B ( Λ diag( p Θ ˜ V )) t ,where I α B is the fractional integration operator with respect to B (see Definition 2 in Appendix A.2). Iteratingthe application of I α we find that, for any N ≥
1, ˜ V satisfies˜ V = X ≤ k ≤ N Λ k − ( − k − I ( k − α [ I α θ + I α B ( Λ diag( p Θ ˜ V )) + I α B ( Λ diag( p Θ ˜ V ))] + Λ N ( − N I ( N + α ˜ V .Now, note that θ , diag( p Θ ˜ V ), diag( p Θ ˜ V ) and ˜ V are square-integrable processes and Lemma 8 in AppendixA.2 shows that the sum converges almost surely to the series while Λ N ( − N I ( N + α ˜ V converges almost surelyto zero as N tends to infinity. Thus we have˜ V = X k ≥ Λ k ( − k I k α [ I α θ + I α B ( Λ diag( p Θ ˜ V )) + I α B ( Λ diag( p Θ ˜ V ))] = X k ≥ Λ k ( − k I k α I α θ + X k ≥ Λ k ( − k I k α I α B ( Λ diag( p Θ ˜ V )) + I α B ( Λ diag( p Θ ˜ V ))] = Λ − X k ≥ Λ k + ( − k I ( k + α θ + X k ≥ Λ k ( − k I k α I α B ( Λ diag( p Θ ˜ V )) + I α B ( Λ diag( p Θ ˜ V ))].Lemmas 5 and 7 shown in Appendix A.2 enable us to rewrite the above using the matrix Mittag-Leffler function.This yields, for any t in [0,1] and almost surely,˜ V t = Λ − Z t f α , Λ ( t −−− s ) θ s d s + Λ − Z t f α , Λ ( t −−− s ) Λ diag( q Θ ˜ V s ) d W s + Λ − Z t f α , Λ ( t −−− s ) Λ diag( q Θ ˜ V s ) d W s .Replacing θ , Λ , Λ by their expressions, almost surely and for any t in [0,1],˜ V t = Z t f α , Λ ( t −−− s ) ( O ( − µ s + Λ O ( − µ s ) d s + Z t f α , Λ ( t −−− s ) O ( − diag( q Θ ˜ V s ) d B s + Z t f α , Λ ( t −−− s ) O ( − diag( q Θ ˜ V s ) d B s .This concludes the second implication and the proof.4 The previous results on the convergence of the intensity process enable us to now turn to the question of thelimiting price dynamics. Recall that the sequence of rescaled price processes P T is defined as P T : = † QX T ,where Q = ³ e − e | ··· | e m − − e m ´ . We have the following result. Proposition 5.
Let ( X , Z ) be a limit point of ( X T , Z T ) and P = † QX . Then P t = ( I + ∆ ) † Q ( Z t + Z t µ s d s ). where ∆ = ( R ∞ δ Ti j ) ≤ i , j ≤ m .Proof. Let ( X , Z ) be a limit point from ( X T , Z T ). For any 1 ≤ i ≤ m we can compute the difference betweenupward and downard jumps on Asset i v i · N T t = v i · M Tt + v i · Z t λ s d s ,with the following expression for the integrated intensity: Z tT λ T s d s = T Z t µ TsT d s + T Z t Z T ( t − s )0 ψ T ( u ) du µ TTs d s + °° ψ T °° M TtT − Z tT Z ∞ tT − s ψ T ( u ) dud M Ts .Thus the microscopic price for the Asset i satisfies T − α v i · N T tT = T − α Z t v i · µ TsT d s + T − α † °° ψ T °° v i · Z t µ TTs d s + v i · Z Tt + † °° ψ T °° v i · Z Tt − T − α Z t Z ∞ T ( t − s ) † ψ T ( u ) v i · µ TTs dud s − T − α Z tT Z ∞ tT − s ψ T ( u ) dud M Ts = X ≤ k ≤ m ( ik + Z ∞ δ Tik ), v k · Z Tt + X ≤ k ≤ m ( ik + Z ∞ δ Tik ) T − α Z t v k · µ TsT d s − Z t Z ∞ tT − s † ψ T ( u ) v i du · d Z Ts − T − α Z t Z ∞ T ( t − s ) † ψ T ( u ) v i · µ TTs dud s .It is straightforward to show that the last two terms converge to zero and thus, any limit point P of P T = † QX T is such that P t = ( I + ∆ ) † Q ( Z t + Z t µ s d s ).Replacing Z by the expression obtained in Proposition 3 concludes the proof of Theorem 1 since P t = ( I + ∆ ) † Q ¡Z t diag( p V s ) d B s + Z t µ s d s ¢ .5 A Technical appendix
A.1 Independence of Equation (11) from chosen basis
We consider two representations which satisfy Assumption 1. Let P , ˜ P be invertible matrices, 0 ≤ n c , n c ′ ≤ n and A T ∈ F ( M n c ( R )), C T ∈ F ( M n − n c ( R )), B T ∈ F ( M n − n c , n c ( R )) and ˜ A T ∈ F ( M n c ′ ( R )), ˜ C T ∈ F ( M n − n c ′ ( R )), ˜ B T ∈ F ( M n − n c ′ , n c ′ ( R )) such that φ T = P Ã A T B T C T ! P − = ˜ P Ã ˜ A T B T ˜ C T ! ˜ P − .We write A for the limit of A T (and similarly for B T , C T , etc.). First, remark that we must have n c = n c ′ . Indeed,since ρ ( R ∞ C ) < ρ ( R ∞ ˜ C ) <
1, 1 is neither an eigenvalue of R ∞ C nor of R ∞ ˜ C . Yet, since A = I and ˜ A = I ,1 is an eigenvalue of φ with multiplicity n c and n c ′ . Therefore n c = n c ′ .We have, writing L = P − ˜ P , Ã A B C ! = L Ã ˜ A B ˜ C ! L − .Since A = ˜ A = I because of Equation (5), developing and using the assumption that I − C is invertible, we get L = ( I − C ) L = BL − L ˜ BC L = L ˜ C .Since LL − = I , L = I , L = I , L = − L ( − , we deduce L = I , L = I , L = , ( I − C ) L = B − ˜ B , C = ˜ C .As L = P − ˜ P , we have P − = Ã I ( I − C ) − ( B − ˜ B ) I ! ˜ P − = Ã ˜ P ( − ˜ P ( − ( I − C ) − ( B − ˜ B ) ˜ P ( − + ˜ P ( − ( I − C ) − ( B − ˜ B ) ˜ P ( − + ˜ P ( − ! .Developing ˜ P = PL together with the above, we find˜ P ( − = P ( − , ˜ P ( − = P ( − , ˜ P = P , ˜ P = P ˜ P = P + P ( I − C ) − ( B − ˜ B )˜ P = P + P ( I − C ) − ( B − ˜ B ).6Thus ˜ P ( − = P ( − , ˜ P ( − = P ( − ˜ P + ˜ P ( I − C ) − ˜ B = P + P ( I − C ) − B ˜ P + ˜ P ( I − C ) − ˜ B = P + P ( I − C ) − B .Therefore regardless of the chosen basis, Equation (11) is the same, which concludes the proof. A.2 Fractional operators
This section is a brief reminder on fractional operators which are used in proofs. We also introduce the matrixextended Mittag-Leffler function.
Definition 1 (Fractional differentiation and integration operators) . For α ∈ (0,1) , the fractional differentiation(resp. integration) operator denoted by D α is defined asD α f ( t ) : = Γ (1 − α ) dd t Z t ( t − s ) − α f ( s ) d s , where f is a measurable, Hölder continuous function of order strictly greater than α . The fractional integrationoperator denoted by I α is defined as I α f ( t ) : = Γ ( α ) Z t ( t − s ) α − f ( s ) d s . where f is a measurable function. It will be convenient for us to define fractional integration with respect to a Brownian motion.
Definition 2 (Fractional integration operator with respect to a Brownian motion) . Given a Brownian motion Band α ∈ (1/2,1) , the fractional integration operator with respect to B, denoted by I α B , is defined asI α B f ( t ) = Γ ( α ) Z t ( t − s ) − α f ( s ) dB s . for f a measurable, square integrable stochastic process. Remark 2.
The fractional integration of a matrix-valued stochastic process f with respect to a multivariateBrownian motion B is: I α B f ( t ) = Γ ( α ) Z t ( t − s ) − α f ( s ) d B s .We now extend the Mittag-Leffler function to matrices (for a theory of matrix-valued functions, see for example[16]). We have the following definition. Definition 3 (Matrix-extended Mittag-Leffler function) . Let α , β ∈ C such that Re( α ),Re( β ) > , Λ ∈ M n ( R ) .Then the matrix Mittag-Leffler function is defined as E α , β ( Λ ) : = X n ≥ Λ n Γ ( α n + β ) .7We also extend the Mittag-Leffler density function for matrices. Definition 4 (Mittag-Leffler density for matrices) . Let α ∈ C such that Re( α ) > , Λ ∈ M n ( R ) . Then, the matrixMittag-Leffler density function f α , Λ is defined as f α , Λ ( t ) : = Λ t α − E α , α ( −−− Λ t α ) We write F α , Λ for the cumulative matrix Mittag-Leffler density function F α , Λ ( t ) : = Z t f α , Λ ( s ) d s Using Definition 3, it is easy to show the following lemma.
Lemma 4.
Let α ∈ C such that Re( α ) > , Λ ∈ M n ( R ) . Then,I − α f α , Λ = Λ ( I − F α , Λ ) . Furthermore, if α ∈ (1/2,1) (cid:129) f α , Λ ( z ) = Λ ( I z α + Λ ) − .We need another important property relating Mittag-Leffler functions with fractional integration operators. Lemma 5.
Let α > and Λ ∈ M m ( R ) . ThenI f α , Λ = X n ≥ ( − n − Λ n I n α (1) Proof.
Using Lemma 4 and repeated applications of I α , for all N ≥ I f α , Λ = X ≤ n ≤ N ( − n − Λ n I n α (1) + ( − N − Λ N I N α I f α , Λ .Therefore, if we show that ( − N − Λ N I N α I f α , Λ → N →∞ I N α f α , Λ to deduce bounds whichwill converge to zero. Writing C a constant independent of t and N which may change from line to line, N α =⌊ α ⌋ and k·k op for the operator norm, we have °° Λ N f α , Λ ( t ) °° op = °°°° Λ N + X n ≥ ( − n t ( n + α − Γ (( n + α ) °°°° op ≤ °°°°° Λ N + X ≤ n ≤ N α ( − n t ( n + α − Γ (( n + α ) + Λ N + C °°°°° op .8Therefore, when applying the fractional integration operator of order N α we have, writing g n : t t ( n + α − I N α °° Λ N f α , Λ ( t ) °° op ≤ °°°°° Λ N + I N α ( X ≤ n ≤ N α ( − n g n Γ (( n + α ) ) + Λ N + I N α ( C ) °°°°° op ≤ X ≤ n ≤ N α Γ (( n + α ) °° Λ N + I N α ( g n ) °° op + °° Λ N + I N α ( C ) °° op .An explicit computation of I N α ( g n ) shows the convergence to zero of the right hand side as N tends to infinity,which concludes the proof.Finally, we need to combine fractional integration I α with I α B . We have the following lemma. Lemma 6.
Let m ≥ , B an m-dimensional Brownian motion, X a m × m matrix valued adapted square-integrable stochastic process and α , β > . Then we have:I α I β B ( X ) = I α + β B ( X ). Proof.
The proof is a straightforward application of the definition of the operators together with the stochasticFubini theorem.The next lemma is useful to transform stochastic convolutions of stochastic processes with the Mittag-Lefflerdensity function into series of repeated applications of I α B . Lemma 7.
Let m ≥ , B an m-dimensional Brownian motion, X a m × m matrix valued adapted and square-integrable stochastic process, α > and Λ ∈ M m ( R ) . Then, for all t ≥ and almost surely Z t f α , Λ ( t −−− s ) X s d B s = X n ≥ ( − n − Λ n I n α B ( X ), where the series converges almost surely.Proof. Using Lemma 5, we can write the integral using a series of fractional integration operators and applythe stochastic Fubini theorem (as X is square-integrable) to obtain Z t f α , Λ ( t −−− s ) X s d B s = Z t X n ≥ ( − n − Λ n I n α − (1) t − s X s d B s = X n ≥ Z t ( − n − Λ n I n α − (1) t − s X s d B s = X n ≥ ( − n − Λ n Z t I n α − (1) t − s X s d B s = X n ≥ ( − n − Γ ( n α − Λ n Z t Z t − s ( t − s − τ ) n α − d τ X s d B s .After a change of variables and using the stochastic Fubini theorem (see for example [27]), we deduce thesimpler expression Z t f α , Λ ( t −−− s ) X s d B s = X n ≥ ( − n − Γ ( n α − Λ n Z t ( t − τ ) n α − Z τ X s d B s d τ .9Integrating by parts, we finally obtain the result: Z t f α , Λ ( t −−− s ) X s d B s = X n ≥ ( − n − Γ ( n α − n α − Λ n Z t ( t − τ ) n α − X τ d B τ , = X n ≥ ( − n − Γ ( n α ) Λ n Z t ( t − τ ) n α − X τ d B τ , = X n ≥ ( − n − Λ n I n α B ( X ).The last lemma gives convergence for terms of a series of repeated iterations of I α . Lemma 8.
Let α > , Λ ∈ M m ( R ) , B an m-dimensional Brownian motion and X a m-dimensional vector valuedsquare-integrable stochastic process. Then, almost surely and for all t ∈ [0,1]( − N − Λ N I N α ( X ) t → N →∞ X n ≥ N ( − n − Λ n I n α B (diag( X )) t → N →∞ Proof.
Let N ∗ > N α : = ⌊ α ⌋ . Since X is square-integrable, we have E h°°° X N > N ∗ Λ N I ( N + α B (diag( X )) t °°° i ≤ X N , N > N ∗ E [ † ( Λ N I ( N + α B (diag( X )) t )( Λ N I ( N + α B (diag( X )) t )].Using the Cauchy-Schwartz inequality and writing k·k op for the operator norm associated to the Euclidiannorm, we find E h°°° X N > N ∗ Λ N I ( N + α B (diag( X )) t °°° i ≤ X N , N > N ∗ k Λ k N + N op X ≤ k , l ≤ m E [ I ( N + α B k ( X k ) t I ( N + α B l ( X l ) t ] ≤ X N , N > N ∗ k Λ k N + N op Γ (( N + α ) Γ (( N + α ) X ≤ i ≤ m Z t ( t − s ) ( N + N ) α − E [( X is ) ] d s ≤ c X N , N > N ∗ k Λ k N + N op Γ (( N + α ) Γ (( N + α ) ≤ c ³ X N > N ∗ k Λ k N op Γ (( N + α ) ´ .Thus by comparison of functions (for example by application of Stirling’s formula), for all ǫ > X N > N α P ³°°° X N > N ∗ Λ N I ( N + α B (diag( X )) t °°° > ǫ ≤ ǫ X N ∗ ≥ N α E h°°° X N > N ∗ Λ N I ( N + α B (diag( X )) t °°° i < ∞ .The Borel-Cantelli lemma yields the almost sure convergence to zero of Λ N I ( N + α B (diag( X )) as N → ∞ . Thesame approach yields the almost sure convergence to zero of ( − N − Λ N I N α ( X ) as N → ∞ .0 A.3 Proof of Corollary 1
Take µ , µ > α ∈ (1/2,1), κ ∈ [0,1], H b , H s , H b , H s ∈ [0,1] such that (here p· is the principal square root, sothat if x < p x = i p− x ): 0 ≤ ( H b + H s )( H b + H s ) < ≤| κ − q ( H b − H s )( H b − H s ) |< ≤| κ + q ( H b − H s )( H b − H s ) |< t ≥
0, which will appear in the structure of the kernel: φ T ( t ) : = α (1 + κ /2) t ≥ t − ( α + φ b , T ( t ) = α T − α H b t ≥ t − ( α + φ T ( t ) : = α (1 + κ /2) t ≥ t − ( α + φ s , T ( t ) = α T − α H s t ≥ t − ( α + λ T ( t ) : = α ( κ − κ T − α ) t ≥ t − ( α + φ b , T ( t ) : = α T − α H b t ≥ t − ( α + ˜ λ T ( t ) : = α ( κ − κ T − α ) t ≥ t − ( α + φ s , T ( t ) = α T − α H s t ≥ t − ( α + .The sequence of baselines and kernels are chosen as: µ T = T α − µ µ µ µ , φ T = φ T φ T − λ T φ T , b φ T , s φ T − λ T φ T φ T , s φ T , b φ T , b φ T , s φ T φ T − ˜ λ T φ T , s φ T , b φ T − ˜ λ T φ T .The above sequence naturally satisfies the different assumptions outlined in Section 2. Indeed, using the fol-lowing change of basis O = − − ,we have, with notations from Section 2, A = à φ + φ − λ φ b + φ s φ b + φ s φ + φ − ˜ λ ! B = ( φ − φ ) I C = à λ φ b − φ s φ b − φ s ˜ λ ! M = α IK = à κ H b + H s H b + H s κ ! .1Furthermore, we can check that the assumptions of Section 2 are satisfied if0 ≤ H H < ≤| κ − q ( H b − H s )( H b − H s ) |< ≤| κ + q ( H b − H s )( H b − H s ) |< à x yz w ! : = ³ I − Z ∞ C ´ − Z ∞ B .Then, straightforward linear algebra yields O + O ³ I − Z ∞ C ´ − Z ∞ B = à + x y − x − y !³ O + O ³ I − Z ∞ C ´ − Z ∞ B ´ O ( − = à + x + x − x − x !³ O + O ³ I − Z ∞ C ´ − Z ∞ B ´ O ( − = à y y − y − y ! ,so that, using the notations of Theorem 1 for the Brownian motion B , W and W , we have ³ O + O ³ I − Z ∞ C ´ − Z ∞ B ´ O ( − Z t diag ³q Θ ˜ V s ´ d W s = Z t (1 + x ) ¡q V t dB s + q V t dB s ¢ (1 − x ) ¡q V t dB s + q V t dB s ¢³ O + O ³ I − Z ∞ C ´ − Z ∞ B ´ O ( − diag ³q Θ ˜ V s ´ d W t = Z t y ¡q V t dB t + q V t dB t ¢ − y ¡q V t dB t + q V t dB t ¢ .Therefore, writing Σ : = à + x + x y y − x − x − y − y ! ,we have the following equation for the fundamental variance of Asset 1 Γ (1 − α ) Γ ( α ) α à V t V t ! = Z t ( t − s ) α − hà + x y − x − y ! à µ µ ! − à + x y − x − y ! K − à + x y − x − y ! − à V s V s ! i d s + Z t ( t − s ) α − Σ diag( p V s ) dB s .2By symmetry, we can find the analogue to the above on the second asset. Using the following notations Σ : = α Γ (1 − α ) Γ ( α ) 12 + x + x y y − x − x − y − yz z + w + w − z − z − w − w , D : = α Γ (1 − α ) Γ ( α ) + x y − x − y + w z − w − z , G : = α Γ (1 − α ) Γ ( α ) à + x y − x − y ! K − à + x y − x − y ! − à z + w − z − w ! K − à z + w − z − w ! − ,where we have written for convenience à x yz w ! : = R ∞ φ − R ∞ φ R ∞ λ R ∞ ˜ λ − ( R ∞ φ b − R ∞ φ s )( R ∞ φ b − R ∞ φ s ) à R ∞ ˜ λ − ( R ∞ φ b − R ∞ φ s ) − ( R ∞ φ b − R ∞ φ s ) R ∞ λ ! .Therefore V satisfies the following stochastic Volterra equation Γ (1 − α ) Γ ( α ) α V t = Z t ( t − s ) α − h D à µ µ ! − GV s i d s + Z t ( t − s ) α − Σ diag( p V s ) d B s .This concludes the proof of Corollary 1. A.4 Proof of Corollary 2
We split the proof into two steps. First, we show that the structure of the kernel satisfies the assumptions ofSection 2. Then we compute the equations satisfied by variance and prices.
Checking for the assumptions of Theorem 1
We write O : = O : = O : = − O : = − .Then, setting O : = ³ O | O | O | O ´ , we have φ T = O φ T + φ T φ T , c + φ T , a φ b + φ s ˜ φ T + ˜ φ T φ T − φ T φ T , c − φ T , a φ b − φ s ˜ φ T − ˜ φ T O − .3It is straightforward to check that the assumptions are satisfied if0 ≤ ( H c + H a )( H c + H a ) < ≤| − ( γ + γ ) − q ( H c − H a )( H c − H a ) + ( γ − γ ) |< ≤| − ( γ + γ ) + q ( H c − H a )( H c − H a ) + ( γ − γ ) |< K = I − H has positive eigenvalues and therefore K M − = α K has positive eigenvalues.Therefore all the assumptions of Theorem 1 are satisfied. Limiting variance process
Since we can apply Theorem 1, we now compute the relevant quantities. As B = , writing H : = H a + H c and H : = H a + H c , we have O − = − − K − = − H H à H H ! Θ = − H H à H H ! Θ = − H H à H H ! .One can check that the equations satisfied by Θ ˜ V and Θ ˜ V are, where B is a Brownian motion, Θ ˜ V t = α Γ ( α ) Γ (1 − α ) Z t ( t − s ) α − "à µ µ ! − à ˜ V s ˜ V s ! d s + α Γ ( α ) Γ (1 − α ) Z t ( t − s ) α − q ˜ V s + H ˜ V s à dB s + dB s dB s + dB s ! Θ ˜ V t = α Γ ( α ) Γ (1 − α ) Z t ( t − s ) α − "à µ µ ! − à ˜ V s ˜ V s ! d s + α Γ ( α ) Γ (1 − α ) Z t ( t − s ) α − q ˜ V s + H ˜ V s à dB s + dB s dB s + dB s ! .Note that the above implies that V + = V − and V + = V − . This property is due to the the symmetric structureof the baselines and kernels. Therefore, the joint dynamics can be fully captured by considering the jointdynamics of ( V + , V + ). Thus, writing V : = V + = V − and V : = V + = V − , we have Γ ( α ) Γ (1 − α ) α V t = Z t ( t − s ) α − ( µ − ˜ V s ) d s + Z t q V t ( dB s + dB s ) Γ ( α ) Γ (1 − α ) α V t = Z t ( t − s ) α − ( µ − ˜ V s ) d s + Z t q V t ( dB s + dB s ).4We can write the above without ˜ V as Γ ( α ) Γ (1 − α ) α à V t V t ! = Z t ( t − s ) α − ¡ à µ µ ! − K − à V s V s ! ¢ d s + Z t ( t − s ) α − q V s ( dB s + dB s ) q V s ( dB s + dB s ) . Limiting price process
Turning now to the price process, it remains to compute ∆ (see Equation (10)) using the definition. We have † °° ψ °° O = X k ≥ °° φ °° k O = O X k ≥ £¡ Z ∞ C ¢ k e + ¡ Z ∞ C ¢ k e ¤ = X k ≥ £¡ Z ∞ C ¢ k O + ¡ Z ∞ C ¢ k O ¤ = [( I − Z ∞ C ) − − I ] O + [( I − Z ∞ C ) − − I ] O ,which, by definition of ∆ , yields ∆ = £¡ I − Z ∞ C ¢ − − I ¤ = γ γ γ − ( H c − H a )( H c − H a ) − ∆ = £¡ I − Z ∞ C ¢ − − I ¤ = H c − H a γ γ − ( H c − H a )( H c − H a ) .Therefore, ∆ = γ γ − ( H c − H a )( H c − H a ) à γ H c − H a H c − H a γ ! − I .Finally, any limit point P of the sequence of microscopic price processes satisfies the following equation P t = γ γ − ( H c − H a )( H c − H a ) à γ H c − H a H c − H a γ ! à − − ! Z t q V s dB s q V s dB s q V s dB s q V s dB s = γ γ − ( H c − H a )( H c − H a ) à γ H c − H a H c − H a γ ! Z t q V s ( dB s − dB s ) q V s ( dB s − dB s ) .This concludes the proof of Corollary 2.5 A.5 Proof of Corollary 3
We define the interaction kernel between Asset i and Asset j , for 1 ≤ i , j ≤ m , define φ Ti j ( t ) : = α (1 − T − α ) t ≥ t − ( α + Ã (1 − γ ) γγ (1 − γ ) ! if i = j , α T − α t ≥ t − ( α + Ã H c H a H a H c ! if Asset i and Asset j belong to the same sector, α T − α t ≥ t − ( α + Ã H c + H cr H a + H ar H a + H ar H c + H cr ! otherwise.Finally, the complete Hawkes baseline and kernel structure is µ T = T α − µ µ ... µ m µ m , φ T = φ T φ T ... φ T m φ T φ T ... φ T m ... ... ... ... φ Tm ... ... φ Tmm .As in the previous example, the proof is split into three steps. First, we show that the structure of the kernel sat-isfies the assumptions required to apply Theorem 1. Then, we compute the equation satisfied by the varianceand finally the limiting price process.
Checking assumptions of Theorem 1
We can examine the structure of the kernel as in the two-asset example. Define the following basis: O i : = ( e i + e i + if 1 ≤ i ≤ m , e i − e i if m + ≤ i ≤ m .Using the notations of Section 2, straightforward computations allow us to write φ T = O Ã A T B T C T ! O − = O Ã A T C T ! O − ,where we can compute A T and C T . Checking the assumptions is done as in the two-asset case, though theconditions have changed here due to the new structure of the kernel. For example, sincelim T →∞ Z ∞ φ T O m + i = (1 − γ ) O n + i + ( H c − H a ) X ≤ j i ≤ m O m + j + X ≤ j i ≤ m X ≤ r ≤ R ( H cr − H ar ) O m + j ,we have, writing J : = e † e + ··· + e m † e m and for any 1 ≤ r ≤ R , J r : = e i r † e i r + ··· + e i r + m r † e i r + m r , Z ∞ C = (1 − γ ) I + ( H c − H a ) J + X ≤ r ≤ R ( H cr − H ar ) J r .6Therefore, as the eigenvalues of R ∞ C can be made explicit, if | λ − + X ≤ r ≤ R λ − r |< γ ,then ρ ( R ∞ C T ) < ρ ( R ∞ C ) <
1. Similarly, we can easily check that a necessary condition for ρ ( R ∞ A T ) < T large enough is | H c + H a + X ≤ r ≤ R m r − m − H cr + H ar ) |< m − | λ − + X ≤ r ≤ R η r λ − r | < γ | λ + + X ≤ r ≤ R η r λ + r | < K and Λ : = K M − . As in the two-asset example, we have here M = α I . Since K = I − ( H c + H a ) J − P ≤ r ≤ R ( H cr + H ar ) J r , the eigenvalues of K (and therefore those of Λ ) are all strictly positive. Thus we havechecked all necessary conditions to apply Theorem 1. We can thus state the equation satisfied by the varianceprocess. Limiting variance process
As in the previous example, we have V i + = V i − . Thus, we write the underlying variance of asset i V i and usethe (slight) abuse of notation and define V : = ( V , V , ··· , V m ). Then V satisfies V t = α Γ ( α ) Γ (1 − α ) Z t ( t − s ) α − £ θ − K − V s ¤ d s + α p Γ ( α ) Γ (1 − α ) Z t ( t − s ) α − diag( p V s ) d B s ,where B is a Brownian motion. We can rewrite K − as K − = Ã I − ( H c + H a ) J − X ≤ r ≤ R ( H cr + H ar ) J r ! − , = Ã I − ( H c + H a )( m − w † w − X ≤ r ≤ R ( H cr + H ar )( m r − w r † w r − ǫ ! − ,with the small term ǫǫ : = ( H c + H a )( J − ( m − w † w ) + X ≤ r ≤ R ( H cr + H ar )( J r − ( m r − w r † w r ).It is easy to check that ρ ( ǫ ) = m →∞ o ( 1 m ), which concludes our study of the variance process. We now turn to theequation satisfied by the limiting price process.7 Limiting price process
Using the same approach as in the two-asset case, computing ∆ boils down to computing ( I − R ∞ C ) − . Usingthe expression for R ∞ C derived previously, we have( I − C ) − = γ ( I − H c − H a γ J − X ≤ r ≤ R H cr − H ar γ J r ) − .Therefore, repeating the same approach we used for K − yields( I − C ) − = (2 γ I − λ − w † w − X ≤ r ≤ R η r λ − r w r † w r − ǫ ) − ,with ρ ( ǫ ) = o ( 1 m ). Thus, we have the expression of ∆∆ = (2 γ I − λ − w † w − X ≤ r ≤ R η r λ − r w r † w r − ǫ ) − − I .Plugging this into Theorem 1, we have the equation satisfied by macroscopic prices, which concludes the proofof Corollary 3. References [1] Emmanuel Bacry, Sylvain Delattre, Marc Hoffmann, and Jean-FranÃ˘gois Muzy. Modelling microstructurenoise with mutually exciting point processes.
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