Financial asset bubbles in banking networks
aa r X i v : . [ q -f i n . M F ] J un Financial asset bubbles in banking networks
Francesca Biagini ∗† Andrea Mazzon ∗ Thilo Meyer-Brandis ∗ June 6, 2018
Abstract
We consider a banking network represented by a system of stochastic differentialequations coupled by their drift. We assume a core-periphery structure, and that thebanks in the core hold a bubbly asset. The banks in the periphery have not directaccess to the bubble, but can take initially advantage from its increase by investingon the banks in the core. Investments are modeled by the weight of the links, whichis a function of the robustness of the banks. In this way, a preferential attachmentmechanism towards the core takes place during the growth of the bubble. We theninvestigate how the bubble distort the shape of the network, both for finite and infinitelylarge systems, assuming a non vanishing impact of the core on the periphery. Due tothe influence of the bubble, the banks are no longer independent, and the law of largenumbers cannot be directly applied at the limit. This results in a term in the driftof the diffusions which does not average out, and that increases systemic risk at themoment of the burst. We test this feature of the model by numerical simulations.
Keywords : Bubbles, Systemic risk, Financial networks, Mean field models
Contagion mechanisms within a banking system and corresponding measurement and man-agement of systemic risk have become central topics in macroprudential regulation in partic-ular since the last financial crisis. The urge for the development of new quantitative methodsto deal with these topics has triggered various lines of active research on systemic risk. ∗ Workgroup Financial and Insurance Mathematics, Department of Mathematics, Ludwig-MaximiliansUniversit¨at, Theresienstraße 39, 80333 Munich, Germany. Emails: [email protected], [email protected], [email protected]. † Secondary affiliation: Department of Mathematics, University of Oslo, Box 1053, Blindern, 0316, Oslo,Norway. dX it = λn n X j =1 ( X jt − X it ) dt + σdW it , ≤ t < ∞ , (1.1)where W = ( W t , . . . , W nt ) t ≥ is a standard n -dimensional Brownian motion and λ, σ > X i stand for the log-monetary reserves of banks, and the drift terms λ ( X jt − X it )represent the connections between banks in the network. In this case, the borrowing andlending rate λ is supposed to be the same for every couple of banks. When the network size n grows towards infinity, it is a well-know result (see Sznitman [41]) that due to law-of-large-number effects the diffusions in (1.1) converge towards their mean-field limit d ¯ Y it = λ (cid:0) E [ ¯ Y t ] − ¯ Y it (cid:1) dt + σdW it , ≤ t < ∞ . Thus, for large networks propagation of chaos applies and the evolution of the X i asymp-totically de-couples due to averaging effects, which allows to asymptotically describe thecomplex system by a representative particle evolution. The simple model in (1.1) to study2ystemic risk has been generalized in various ways in a number of articles, see e.g. Carmonaet al. [14, 15] where mean-field games are considered, Fouque and Sun [26] where the prob-ability distributions of multiple default times is approximated, Garnier et al. [28, 29] andBattiston et al. [5] where a tradeoff between individual and systemic risk in a banking net-work is described, and Chong and Kl¨uppelberg [17], Kley et al. [34] where partial mean-fieldlimits are studied.In this paper the main objective is to extend the model in (1.1) such that the effect of afinancial speculation bubble on the evolution of the network and the evolving systemic riskcan be studied. It is a common understanding that bubbles are intimately connected withfinancial crises, and many historical crises indeed originated after the burst of a bubble (e.g.the Great Depression of the 1930s and the financial crisis of 2007-2008). This causality isinvestigated for example in Brunnermeier [11] and statistically confirmed in Brunnermeierand Schanabel [13]. However, it seems that literature on mathematical models that dealwith this question is very scarce.We here take a first step towards filling this gap and specify a model for the network of financial robustness of the institutions, introduced by Battiston et al. [5] and Hull and White[32] as an indicator of agent’s creditworthiness or distance to default and also considered inKley et al. [34], by a system of coupled diffusions. The banks affect each other’s robustnessby being financially exposed to each other, for example because of cross-holdings, whichresults in a coupling of the drift terms. Following the setting in Battiston [4], we thenassume that a fixed number of banks are directly investing in a bubble that affects theirfinancial robustness. The remaining banks have the possibility to participate in the bubble byinvesting in the bubble banks. This results in a typical core/periphery structure for financialnetworks, where here the core is formed by the banks holding the bubble. Contrarily to theliterature on mean-field models mentioned above, where the coupling drift rates representingthe weighted network connections are constant, we allow for heterogeneity of the drift ratesin that they depend on the robustness of the institution. More precisely, the rates dependon the robustness of the attracting institution with a delay δ >
0, where the delay reflectsthe fact that the banks’ investment do not immediately react to changes in the system.This results in a preferential attachment mechanism where the attractiveness of a node doesnot depend on its degree, but on its “fitness”, as proposed by Bianconi and Barab`asi [9].Due to this behavior, the bubble causes a distortion in the network evolution: during theexpanding phase of the bubble, the network structure shifts towards an increasingly intenseand centralized connectivity due to the strong growth of the bubbly banks’ robustness, whichthen causes instability in case the bubble bursts.We then study the behaviour of the system when its size gets large. More precisely, we let thenumber of periphery banks go towards infinity, but keep the number of core banks holdingthe bubble constant and assume that their impact on the system does not vanish when the3otal number of banks goes to infinity. In this way the bubble produces a common stochasticsource in the system that does not not average out even for large networks. Our main resultthen determines a partial mean-field limit for the system where the influence of the bubble isrepresented via stochastic interaction with the core banks even in the limit. Because of thisterm, also the banks in the periphery are affected by a potential bubble burst. This effect isamplified by the impossibility to immediately desinvest when the robustness of some banksdecreases due to the delay δ . We also refer to Chong and Kl¨uppelberg [17] where the authorsinvestigate partial mean-field limits in a different setting, without taking into account thedelay and the influence of the bubble.The remaining part of the paper is organized as follows. In Section 2 we introduce our modeland some technical results. In Section 3 we define the limit system and prove a convergenceresult, whereas in Section 4 we perform Monte Carlo simulations both in the finite and inthe limit systems in order to numerically investigate the impact of the bubble on systemicrisk. Let (Ω , F , F , P ) be a filtered probability space endowed with a ( m + n + 2)-dimensionalBrownian motion ¯ W = ( W t , . . . , W nt , W B, t , . . . , W B,mt , B t , B t ) t ≥ , m, n ∈ N , where F =( F t ) t ∈ R + is the natural filtration of ¯ W . We consider a network of m + n banks, consistingof m banks holding a bubbly asset in their portfolio (also referred to as core ), and n banksthat do not directly hold the bubbly asset (also referred to as periphery ).By following a similar approach as in Kley et al. [34], we model the robustness of the banksin the system. This coefficient dynamically evolves and represents a measure of how healthya bank remains in stress situations. Let ρ i,n = ( ρ i,nt ) t ≥ , i = 1 , . . . n , and ρ k,B = ( ρ k,Bt ) t ≥ , k = 1 , . . . , m , be the robustness of banks not holding and holding the bubble, respectively.We assume that they satisfy the following system of stochastic differential delay equations(SDDEs) for t ≥ δ , δ > dρ i,nt = n − n X j =1 ,j = i f P ( ρ j,nt − δ − A nt − δ )( ρ j,nt − A nt ) + 1 m m X k =1 f B ( ρ k,Bt − δ − A nt − δ ) ( ρ k,Bt − A nt ) ! + λ ( A nt − ρ i,nt ) dt + σ dW it , (2.1) dρ k,Bt = n n X i =1 f P ( ρ i,nt − δ − A nt − δ )( ρ i,nt − A nt ) + 1 m − m X ℓ =1 ,ℓ = k f B ( ρ ℓ,Bt − δ − A nt − δ )( ρ ℓ,nt − A nt ) ! dt + λ ( A nt − ρ k,Bt ) dt + σ dW k,Bt + dβ t , (2.2)4here λ > σ > σ > A nt = 1 m + n n X r =1 ρ r,nt + m X h =1 ρ h,Bt ! , t ≥ δ, (2.3)is the mean of the robustness of all the banks in the network at time t . For t ∈ [0 , δ ),we assume that ( ρ i,ns ) s ∈ [0 ,δ ) , ( ρ k,Bs ) s ∈ [0 ,δ ) , i = 1 , . . . , n , k = 1 , . . . , m , satisfy (2.1)-(2.2) with δ = 0, by following the approach of Mao [36]. We also suppose that ρ i,n = ρ > i = 1 , . . . , n .The process β = ( β t ) t ≥ in (2.2) represents the influence of the asset price bubble on therobustness of core banks and has dynamics dβ t = µ t dt + σ B dB t , t ≥ , (2.4)where σ B > µ is an adapted process satisfying dµ t = ˜ b ( µ t ) dt + ˜ σ ( µ t ) dB t , t ≥ , (2.5)where ˜ b , ˜ σ fulfill the usual Lipschitz and sublinear growth conditions such that there existsa unique solution of (2.5) , satisfying Z t E [ | µ s | ] ds < ∞ , ≤ t < ∞ . (2.6)Later on in Section 4 we will specify a concrete model for the bubbly evolution in (2.4).The interdependencies of the banks’ robustness and corresponding contagion effects are spec-ified through the drifts in (2.1) and (2.2). The term λ ( A nt − ρ i,nt ) represents an attraction ofthe individual robustness towards the average robustness of the system with rate λ as in theclassical mean-field model (1.1). In addition to the homogeneous average term, we introducethe terms of type f P ( ρ j,nt − δ − A nt − δ )( ρ j,nt − A nt ) and f B ( ρ k,Bt − δ − A nt − δ ) ( ρ k,Bt − A nt ) that representa robustness-dependent evolution of the network connectivity: for typically positive and in-creasing f B and f P , bank i is the more connected to bank j the higher bank j ’s robustness isabove the average. In this way, the evolution of the bubble alters the connectivity structureof the network according to a model of preferential attachment. Moreover, the propensityof a node i to attract future links not only depends on the current level of robustness of i , but also on the robustness of the banks already connected to i . This induces a form of preferential preferential attachment , which creates a strong clustering effect. This changein network structure then comes along with an increasing systemic risk and instability incase the bubble burst, as noted by Battiston [4]. Further we introduce the delay δ > f P = λ , the system(2.1)-(2.2) collapses to the basis mean-field model in (1.1).We assume the following hypothesis on f B and f P .5 ssumption 2.1. The functions f B , f P : ( R , B ( R )) → ( R + , B ( R + )) are measurable, Lip-schitz continuous and such that also the functions F B ( x ) := xf B ( x ) , F P ( x ) := xf P ( x ) , x ∈ R , are Lipschitz continuous, i.e. | f ℓ ( x ) − f ℓ ( y ) | ≤ K | x − y | , x, y ∈ R , ℓ = B, P, (2.7) and | xf ℓ ( x ) − yf ℓ ( y ) | ≤ K | x − y | , x, y ∈ R , ℓ = B, P, (2.8) with < K , K < ∞ . Note that (2.8) implies that f B and f P are bounded, since if f ( x ) x is Lipschitz then | f ( x ) x | = | f ( x ) x − f (0) · | ≤ K | x | . (2.9) Example 2.2.
We have that f ( x ) = 1 + 2 arctan( x ) /π satisfies Assumption 2.1: f takesvalues in [0 , , and both f and F ( x ) = xf ( x ) are Lipschitz, because they have boundedderivative.In particular, f is increasing, so that if ρ jt > ρ it then the link towards j is bigger then thelink towards i . If the robustness ρ jt of bank j is equal to the average A nt in (2.3) , then thelink towards bank j has weight f (0) = 1 , if ρ jt > A nt the link has weight bigger than and if ρ jt < A nt the link has weight less than . If all the banks have the same robustness, we havean homogenous network, where all the links have weight equal to .Furthermore, any constant function clearly satisfies Assumption 2.1. For such a choice, wehave a static and homogenous network. Proposition 2.3.
Under Assumption 2.1, for every δ ≥ there exists a unique strongsolution for the system of SDEs (2.1)-(2.2). Moreover, it holds sup ≤ s ≤ t E [ | ρ is | ] < ∞ , < t < ∞ , i = 1 , . . . , n, (2.10)sup ≤ s ≤ t E [ | ρ k,Bs | ] < ∞ , < t < ∞ , k = 1 , . . . , m. (2.11) Proof.
Suppose by simplicity λ = 1. We start by proving existence and uniqueness of thestrong solution of (2.1)-(2.2) when δ = 0. In this case we can write the system of SDEs givenby (2.1),(2.2) and (2.5) as an ( m + n + 1)-dimensional SDE dX t = b ( X t ) dt + σ ( X t ) d ¯ W t , t ≥ , (2.12)6here b ( x ) = n − P nj =2 f P ( x j − ¯ x )( x j − ¯ x ) + m P m + nk = n +1 f B ( x k − ¯ x )( x k − ¯ x ) + ¯ x − x , ... n − P n − j =1 f P ( x j − ¯ x )( x j − ¯ x ) + m P m + nk = n +1 f B ( x k − ¯ x )( x k − ¯ x ) + ¯ x − x n n P nj =1 f P ( x j − ¯ x )( x j − ¯ x ) + m − P m + nk = n +2 f B ( x k − ¯ x )( x k − ¯ x ) + ¯ x − x n +1 ... n P nj =1 f P ( x j − ¯ x )( x j − ¯ x ) + m − P m + n − k = n +1 f B ( x k − ¯ x )( x k − ¯ x ) + ¯ x − x m + n ˜ b ( x m +2 ) , (2.13)with ¯ x = m + n P m + ni =1 x i . Here σ ( x ) is a ( n + m + 1) × ( n + m + 1) block matrix of the form σ ( x ) = Σ ( x ) 0 00 Σ ( x ) 00 0 ˜ σ ( x m +2 ) , (2.14)where Σ ( x ) is a n × n diagonal matrix with diagonal ( σ , . . . , σ ) and Σ ( x ) is the m × ( m +1)matrix Σ ( x ) = σ . . . σ B σ . . . σ B ... ... . . . 0 σ B . . . σ σ B . We use Theorem 9.11 in Pascucci [39] to prove existence and uniqueness of the strong solutionof (2.12), and that the second moments of the solution are finite. To this purpose, we showthat b ( · ) and σ ( · ) defined in (2.13) and (2.14), respectively, are Lipschitz continuous in x and that there exists some C such that k σ ( x ) k + k b ( x ) k ≤ C (1 + k x k ) . We begin by proving the first condition. The Lipschitz property clearly holds for σ ( · ), since˜ σ ( · ) is Lipschitz by hypothesis. Given x = ( x , . . . , x m + n ) , x ′ = ( x ′ , . . . , x ′ m + n ) ∈ R m + n , weshow that there exists a constant ¯ K ∈ (0 , ∞ ) such that k b ( x ) − b ( x ′ ) k ≤ ¯ K k x − x ′ k . For the first entry of (2.13) we have | b ( x ) − b ( x ′ ) | ≤ n − n X j =2 | f P ( x j − ¯ x )( x j − ¯ x ) − f P ( x ′ j − ¯ x ′ )( x ′ j − ¯ x ′ ) | + 1 m m + n X k = n +1 | f B ( x k − ¯ x )( x k − ¯ x ) − f B ( x ′ k − ¯ x ′ )( x ′ k − ¯ x ′ ) | + | ¯ x − ¯ x ′ | + | x − x ′ | , | b ( x ) − b ( x ′ ) | ≤ K n − n X j =2 | ( x j − ¯ x ) − ( x ′ j − ¯ x ′ ) | + K m m + n X k = n +1 | ( x k − ¯ x ) − ( x ′ k − ¯ x ′ ) | + | ¯ x − ¯ x ′ | + | x − x ′ |≤ K n − n X j =2 | x j − x ′ j | + 1 m m + n X k = n +1 | x k − x ′ k | ! + (2 K + 1) | ¯ x − ¯ x ′ | + | x − x ′ |≤ K n − n X j =2 | x j − x ′ j | + 1 m m + n X k = n +1 | x k − x ′ k | ! + 2 K + 1 m + n m + n X i =2 | x i − x ′ i | + | x − x ′ | . Then, since for z , . . . , z N ∈ R it holds (cid:16)P Ni =1 | z i | (cid:17) ≤ N P Ni =1 | z i | , we have | b ( x ) − b ( x ′ ) | ≤ m + n ) ( K ) n − n X j =2 | x j − x ′ j | + 1 m m + n X k = n +1 | x k − x ′ k | !! + 6( m + n ) (2 K + 1) m + n ) m + n X i =1 | x i − x ′ i | + | x − x ′ | ! ≤ C k x − x ′ k , for a suitable constant C >
0. Similarly, | b i ( x ) − b i ( x ′ ) | ≤ C i k x − x ′ k , ≤ i ≤ m + n, for a suitable constant C i >
0, whereas | b m +2 ( x ) − b m +2 ( x ′ ) | = | ˜ b ( x m +2 ) − ˜ b ( x ′ m +2 ) | ≤ K µ | x m +2 − x ′ m +2 | , where K µ is the Lipschitz constant for the function ˜ b ( · ) in (2.5). Then we obtain k b ( x ) − b ( x ′ ) k = m + n +1 X i =1 | b i ( x ) − b i ( x ′ ) | ≤ m + n +1 X i =1 C i + K µ ! k x − x ′ k . (2.15)The second condition, i.e. k σ ( x ) k + k b ( x ) k ≤ C (1 + k x k ) , (2.16)for some C >
0, holds because of Assumption 2.1 and the hypothesis on ˜ σ ( · ).Inequalities (2.10) and (2.11) then follow by Theorem 9.11 in Pascucci [39], and in particular8y estimation (A.2) in the Appendix.When δ >
0, equation (2.12) becomes dX t = ¯ b ( X t , X t − δ ) dt + ¯ σ ( X t , X t − δ ) dW t , t ≥ δ, (2.17)where ¯ σ ( x, y ) = σ ( x ) as in (2.14) and b ( x, y ) = n − P nj =2 f P ( y j − ¯ y )( x j − ¯ x ) + m P m + nk = n +1 f B ( y k − ¯ y )( x k − ¯ x ) + ¯ x − x , ... n − P n − j =1 f P ( y j − ¯ y )( x j − ¯ x ) + m P m + nk = n +1 f B ( y k − ¯ y )( x k − ¯ x ) + ¯ x − x n n P nj =1 f P ( y j − ¯ y )( x j − ¯ x ) + m − P m + nk = n +2 f B ( y k − ¯ y )( x k − ¯ x ) + ¯ x − x n +1 ... y n P nj =1 f P ( y j − ¯ y )( x j − ¯ x ) + m − P m + n − k = n +1 f B ( y k − ¯ y )( x k − ¯ x ) + ¯ x − x m + n ˜ b ( x m +2 ) . By Theorem 3.1 in Mao [36, chapter 5], to prove existence and uniqueness of the solution itsuffices to show that the linear growth condition k ¯ b ( x, y ) k ≤ C (1 + k x k + k y k ) (2.18)holds and that ¯ b is Lipschitz in the variable x uniformly in y , i.e. that there exists a constant˜ K ∈ (0 , ∞ ) such that k ¯ b ( x, y ) − ¯ b ( x ′ , y ) k ≤ ˜ K k x − x ′ k (2.19)for all y ∈ R , x, x ′ ∈ R m + n . Property (2.18) can be proven by computations similar to theones used for showing (3.14). For the Lipschitz condition we have | ¯ b ( x, y ) − ¯ b ( x ′ , y ) | ≤ n − n X j =2 | f P ( y j − ¯ y ) || ( x j − ¯ x ) − ( x ′ j − ¯ x ′ ) | + 1 m m + n X k = n +1 | f B ( y k − ¯ y ) || ( x k − ¯ x ) − ( x ′ k − ¯ x ′ ) | + | ¯ x − ¯ x ′ | + | x − x ′ | . Hence, as f B and f P are bounded by K , the computations to show (2.19) are identical tothe ones for (2.15).In order to prove (2.10) and (2.11), we apply the same argument used in the proof of Theorem3.1 in Mao [36, chapter 5]: on [0 , δ ] we have by hypothesis a classic stochastic differentialequation, and by inequality (9.15) in Theorem 9.11 in Pascucci [39] it holds E [ sup ≤ s ≤ δ k X s k ] < ∞ . (2.20)9n the interval [ δ, δ ], we can write equation (2.17) as dX t = ¯ b ( X t , ξ t ) dt + ¯ σ ( X t , ξ t ) dW t , δ ≤ t ≤ δ, where ξ t = X t − δ . Once the solution on [0 , δ ] is known, this is again a classic SDE (withoutdelay) with initial value X δ = ξ , so that by Theorem 9.11 in Pascucci [39], there exists aconstant C δ > E [ sup δ ≤ s ≤ δ k X s k ] ≤ C δ (cid:0) E [ k X δ k ] (cid:1) e δC δ , (2.21)which is finite by (2.20). Repeating this argument on the interval [2 δ, δ ], we obtain E [ sup δ ≤ s ≤ δ k X s k ] ≤ C δ (cid:0) E [ k X δ k ] (cid:1) e δC δ ≤ C δ (cid:18) E [ sup δ ≤ s ≤ δ k X s k ] (cid:19) e δC δ < ∞ by (2.21). Recursively we have E [ sup ( k − δ ≤ s ≤ kδ k X s k ] < ∞ . Then, sup ≤ s ≤ t E [ k X s k ] = sup s ∈ [¯ kδ, (¯ k +1) δ ] E [ k X s k ] < ∞ , (2.22)for some ¯ k with [¯ kδ, (¯ k + 1) δ ] ⊆ [0 , t ]. ✷ We now study a mean field limit for the system of banks (2.1)-(2.2) for large n .Define the processes ˜ ρ i = ( ˜ ρ it ) t ≥ , i = 1 , . . . , n , ¯ ρ k,B = ( ¯ ρ k,Bt ) t ≥ , k = 1 , . . . , m , and ν = ( ν t ) t ≥ as the solutions of the following system of SDEs for t ≥ δ : d ˜ ρ it = − λ ˜ ρ it dt + σ dW it , (3.1) dν t = ϕ ( t, t − δ ) + 1 m m X k =1 f B (cid:16) ¯ ρ k,Bt − δ − ν t − δ − E [ ˜ ρ it − δ ] (cid:17) (cid:16) ¯ ρ k,Bt − ν t − E [ ˜ ρ it ] (cid:17) + λ E [ ˜ ρ it ] ! dt, (3.2) d ¯ ρ k,Bt = ϕ ( t, t − δ ) + 1 m − m X ℓ =1 ,ℓ = k f B (cid:16) ¯ ρ ℓ,Bt − δ − ν t − δ − E [ ˜ ρ it − δ ] (cid:17) (cid:16) ¯ ρ ℓ,Bt − ν t − E [ ˜ ρ it ] (cid:17)! dt + (cid:16) µ t + λ ( E [ ˜ ρ it ] + ν t − ¯ ρ k,Bt ) (cid:17) dt + σ dW k,Bt + σ B dB t , (3.3)10ith ϕ ( t, t − δ ) := E (cid:2) f P (cid:0) ˜ ρ it − δ − E [ ˜ ρ it − δ ] (cid:1) (cid:0) ˜ ρ it − E [ ˜ ρ it ] (cid:1)(cid:3) , t ≥ δ. (3.4)For t ∈ [0 , δ ] we assume that ( ˜ ρ t ) ≤ t ≤ δ , ( ν t ) ≤ t ≤ δ and ( ¯ ρ k,Bt ) ≤ t ≤ δ satisfy (3.1)-(3.3) for δ = 0,with initial conditions ˜ ρ i = ρ ∈ R , ν = 0, ¯ ρ k,B = ρ k,B ∈ R .Note that in equation (3.2) the expression of ϕ is independent of the choice of ˜ ρ i since ˜ ρ i , i = 1 , . . . , n , are identically distributed. For the same reason, the process ν in (3.2) does notdepend on ˜ ρ i .Set ¯ ρ i := ˜ ρ i + ν, i = 1 , . . . , n. (3.5)In particular, ¯ ρ it = ¯ ρ iδ + Z tδ ϕ ( s, s − δ ) + 1 m m X k =1 f B (¯ ρ k,Bs − δ − ν s − δ − E [˜ ρ is − δ ]) (cid:16) ¯ ρ k,Bs − ν s − E [˜ ρ is ] (cid:17) + λ ( E [˜ ρ is ] − ˜ ρ is ) ! ds + σ W is , t ≥ δ. (3.6) Proposition 3.1.
Under Assumption 2.1, for every δ ≥ there exists a unique strongsolution of the system of SDEs (3.1)-(3.3). In particular, it holds sup ≤ s ≤ t E [ | ν s | ] < ∞ , < t < ∞ , (3.7)sup ≤ s ≤ t E [ | ρ k,Bs | ] < ∞ , < t < ∞ , k = 1 , . . . , m. (3.8) Proof.
For the sake of simplicity we take λ = 1. It is well known that (3.1) admits a uniquestrong solution. As before, we start by proving existence and uniqueness of the strongsolution of (3.2)-(3.3) when δ = 0. The system given by (3.2), (3.3) and (2.5) can be writtenas an ( m + 2)-dimensional SDE dX t = b ( t, X t ) dt + σ ( t, X t ) dW t , t ≥ , (3.9)where W = ( W B, t , . . . , W B,mt , B t , B t ) t ≥ , and b ( t, x ) = ϕ ( t ) + m P mk =1 f B ( x k − x − ψ ( t ))( x k − x − ψ ( t )) + ψ ( t ) ,ϕ ( t ) + m − P m +1 ℓ =3 f B ( x ℓ − x − ψ ( t ))( x ℓ − x − ψ ( t )) + x + x m +2 − x + ψ ( t ) , ... ϕ ( t ) + m − P mℓ =2 f B ( x ℓ − x − ψ ( t ))( x ℓ − x − ψ ( t )) + x + x m +2 − x m +1 + ψ ( t ) , ˜ b ( x m +2 ) (3.10)11ith ψ ( t ) = E [ ˜ ρ it ] and ϕ ( t ) := E (cid:2) f P (cid:0) ˜ ρ it − E [ ˜ ρ it ] (cid:1) (cid:0) ˜ ρ it − E [ ˜ ρ it ] (cid:1)(cid:3) , t ≥ . (3.11)The ( m + 2) × ( m + 2) matrix σ ( x ) has the form σ ( t, x ) = . . . σ . . . σ B σ . . . σ B σ B
00 0 . . . σ σ B
00 0 . . . σ ( x m +2 ) . (3.12)As before, we rely on Theorem 9.11 in Pascucci [39]. We have to show that b and σ definedin (3.10) and (3.12) respectively are Lipschitz continuous in x uniformly in t and that foreach constant T > C such that for all t ∈ [0 , T ] it holds k σ ( t, x ) k + k b ( t, x ) k ≤ ˜ C (1 + k x k ) . We begin by proving the first condition. The Lipschitz property clearly holds for σ , since ˜ σ is Lipschitz by hypothesis. Take now x = ( x , . . . , x m +2 ), x ′ = ( x ′ , . . . , x ′ m +2 ). We have that | b ( t, x ) − b ( t, x ′ ) |≤ m m X k =1 | f B ( x k − x − ψ ( t )) ( x k − x − ψ ( t )) − f B ( x ′ k − x ′ − ψ ( t )) ( x ′ k − x ′ − ψ ( t )) |≤ K m m X k =1 | ( x k − x − ψ ( t )) − ( x ′ k − x ′ − ψ ( t )) | = K m m X k =1 | x k − x ′ k | + | x − x ′ | ! . Similarly, for k = 2 , . . . , m + 1 we have | b k ( t, x ) − b k ( t, x ′ ) | ≤ K m − m X ℓ =1 ,ℓ = k | x ℓ − x ′ ℓ | + | x − x ′ | + | x k − x ′ k | ! . With computations as in the proof of Proposition 2.3 we obtain that for t ≥ k b ( t, x ) − b ( t, x ′ ) k ≤ ¯ C k x − x ′ k , (3.13)for some appropriate ¯ C .We now show the second condition, i.e. that for t ∈ [0 , T ] it holds k σ ( t, x ) k + k b ( t, x ) k ≤ ˜ C (1 + k x k ) , (3.14)12or some ˜ C >
0. By (3.12) we can focus only on k b ( t, x ) k . The computations are here thesame as in Proposition 2.3, but we have to estimate the term ϕ ( t ) from (3.11). Since | ϕ ( t ) | = (cid:12)(cid:12) E (cid:2) f P (cid:0) ˜ ρ it − E [ ˜ ρ it ] (cid:1) (cid:0) ˜ ρ it − E [ ˜ ρ it ] (cid:1)(cid:3)(cid:12)(cid:12) ≤ K E [ | ˜ ρ it − E [ ˜ ρ it ] | ] , ≤ K (cid:0) E [ | ˜ ρ it − E [ ˜ ρ it ] | ] (cid:1) / ≤ K (cid:18) σ − e − t ) (cid:19) / ≤ K σ √ , (3.14) follows by the proof of Proposition 2.3.Inequalities (3.7) and (3.8) follow since, by Theorem 9.11 in Pascucci [39], (3.13) and (3.14)guarantee that the second moments of the solution of (3.9) are finite.The proof for the case δ >
0, based on Theorem 3.1 in Mao [36, chapter 5], is analogous tothe one of Proposition 2.3. ✷ Denote | x − y | ∗ t = sup s ≤ t | x s − y s | . We have the following Theorem 3.2.
Fix i ∈ N . Under Assumption 2.1, for any t ∈ [0 , ∞ ) and δ ≥ it holds lim n →∞ (cid:0) E (cid:2) | ρ i,n − ¯ ρ i | ∗ t (cid:3) + E [ | ρ k,B − ¯ ρ k,B | ∗ t ] (cid:1) = 0 , k = 1 , . . . , m, where ρ i,n , ¯ ρ i , ρ k,B , ¯ ρ k,B are defined in (2.1), (3.6), (2.2), (3.3) respectively. Before proving Theorem 3.2, we give the following
Proposition 3.3.
Under Assumption 2.1, for ≤ δ < ∞ , it holds lim n →∞ Z δ E (cid:20)(cid:12)(cid:12)(cid:12) n n X i =1 f P ( ¯ ρ is − ¯ A ns )( ¯ ρ is − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds = 0 , (3.15) and lim n →∞ Z tδ E (cid:20)(cid:12)(cid:12)(cid:12) n n X i =1 f P ( ¯ ρ is − δ − ¯ A ns − δ )( ¯ ρ is − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − δ − E [ ˜ ρ is − δ ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds = 0 , for ≤ δ ≤ t < ∞ , where ˜ ρ i and ¯ ρ i satisfy (3.1) and (3.6), respectively, and ¯ A nt = 1 m + n n X r =1 ¯ ρ rt + m X h =1 ¯ ρ h,Bt ! , t ≥ . (3.16) Proof . We limit ourselves to prove the second limit, since the first one follows as a particular13ase. Let us write, for t ≥ δ > E (cid:20)(cid:12)(cid:12)(cid:12) n n X i =1 f P ( ¯ ρ it − δ − ¯ A nt − δ )( ¯ ρ it − ¯ A nt ) − E (cid:2) f P (cid:0) ˜ ρ it − δ − E [ ˜ ρ it − δ ] (cid:1) (cid:0) ˜ ρ it − E [ ˜ ρ it ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ≤ n n X i =1 E (cid:20)(cid:12)(cid:12)(cid:12) f P ( ¯ ρ it − δ − ¯ A nt − δ )( ¯ ρ it − ¯ A nt ) − f P ( ˜ ρ it − δ − E [ ˜ ρ it − δ ])( ˜ ρ it − E [ ˜ ρ it ]) (cid:12)(cid:12)(cid:12)(cid:21) + E (cid:20)(cid:12)(cid:12)(cid:12) n n X i =1 f P ( ˜ ρ it − δ − E [ ˜ ρ it − δ ])( ˜ ρ it − E [ ˜ ρ it ]) − E (cid:2) f P (cid:0) ˜ ρ it − δ − E [ ˜ ρ it − δ ] (cid:1) (cid:0) ˜ ρ it − E [ ˜ ρ it ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) , since ¯ ρ i , i = 1 , . . . , n are identically distributed and the same holds for ˜ ρ i , i = 1 , . . . , n .By (3.5) we have that¯ A nt = 1 m + n n X r =1 ¯ ρ rt + m X h =1 ¯ ρ h,Bt ! = 1 m + n nν t + n X r =1 ˜ ρ rt + m X h =1 ¯ ρ h,Bt ! , so that lim n →∞ ¯ A nt = ν t + lim n →∞ m + n n X r =1 ˜ ρ rt = ν t + E [ ˜ ρ it ] , a.s., by (2.11) and the law of large numbers, as ˜ ρ i , i = 1 , . . . , n , are independent and identicallydistributed. Then we havelim n →∞ f P ( ¯ ρ it − δ − ¯ A nt − δ )( ¯ ρ it − ¯ A nt ) = f P (cid:0) ν t − δ + ˜ ρ it − δ − ( ν t − δ + E [ ˜ ρ it − δ ]) (cid:1) (cid:0) ν t + ˜ ρ it − ( ν t + E [ ˜ ρ it ]) (cid:1) = f P (cid:0) ˜ ρ it − δ − E [ ˜ ρ it − δ ] (cid:1) (cid:0) ˜ ρ it − E [ ˜ ρ it ] (cid:1) a.s. (3.17)We now prove that the family of random variables { n P ni =1 f P ( ¯ ρ is − δ − ¯ A ns − δ )( ¯ ρ is − ¯ A ns ) } n ∈ N isuniformly integrable for every s ∈ [ δ, t ], so that convergence almost surely implies convergencein L .By point (iii) of Theorem 11 in Protter [40, chapter 1] it is enough to prove that for every s ∈ [ δ, t ], sup n E n n X i =1 f P ( ¯ ρ is − δ − ¯ A ns − δ )( ¯ ρ is − ¯ A ns ) ! < ∞ . (3.18)14or every s ∈ [ δ, t ], we have that E (cid:20)(cid:16) n n X i =1 f P ( ¯ ρ is − δ − ¯ A ns − δ )( ¯ ρ is − ¯ A ns ) (cid:17) (cid:21) ≤ ( K ) E (cid:20)(cid:16) n n X i =1 | ¯ ρ is − ¯ A ns | (cid:17) (cid:21) ≤ ( K ) E (cid:20)(cid:16) (1 − n/ ( m + n )) | ν s | + (cid:12)(cid:12) ˜ ρ is (cid:12)(cid:12) + 1 m + n n X r =1 | ˜ ρ rs | + 1 m + n m X h =1 (cid:12)(cid:12) ¯ ρ h,Bs (cid:12)(cid:12) (cid:17) (cid:21) ≤ ( K ) E (cid:20)(cid:16) | ν s | + (cid:12)(cid:12) ˜ ρ is (cid:12)(cid:12) + 1 n n X r =1 | ˜ ρ rs | + 1 m m X h =1 (cid:12)(cid:12) ¯ ρ h,Bs (cid:12)(cid:12) (cid:17) (cid:21) ≤ K ) (cid:18) E h | ν s | + | ˜ ρ is | + m X k =1 | ¯ ρ k,Bs | i + E h(cid:16) n n X r =1 | ˜ ρ rs | (cid:17) i(cid:19) ≤ K ) (cid:18) E h | ν s | + | ˜ ρ is | + m X k =1 | ¯ ρ k,Bs | i + 1 n E (cid:20) n X r =1 | ˜ ρ rs | (cid:21)(cid:19) . ≤ K ) (cid:18) E h | ν s | + | ˜ ρ is | + m X k =1 | ¯ ρ k,Bs | i + E [ | ˜ ρ is | ] (cid:19) < ∞ , by (3.7) and (3.8) and because E | ˜ ρ is | ] < ∞ . Hence, { n P ni =1 f P ( ¯ ρ is − δ − ¯ A ns − δ )( ¯ ρ is − ¯ A ns ) } n ∈ N is uniformly integrable and we obtain therefore by (3.17) thatlim n →∞ E (cid:20)(cid:12)(cid:12)(cid:12) f P ( ¯ ρ it − δ − ¯ A nt − δ )( ¯ ρ it − ¯ A nt ) − f P ( ˜ ρ it − δ − E [ ˜ ρ it − δ ])( ˜ ρ it − E [ ˜ ρ it ]) (cid:12)(cid:12)(cid:12)(cid:21) = 0 . Moreover, for δ ≤ s ≤ t it holds E (cid:20)(cid:12)(cid:12)(cid:12) f P ( ¯ ρ it − δ − ¯ A nt − δ )( ¯ ρ it − ¯ A nt ) − f P ( ˜ ρ it − δ − E [ ˜ ρ it − δ ])( ˜ ρ it − E [ ˜ ρ it ]) (cid:12)(cid:12)(cid:12)(cid:21) ≤ K ( E [ | ¯ ρ it − ¯ A nt | ] + E [ | ˜ ρ it − E [ ˜ ρ it | ]) , where the second term belongs to L ([ δ, t ]) and does not depend on n . On the other hand,we have Z t E [ | ¯ ρ is − ¯ A ns | ] ds ≤ Z t E " | ˜ ρ is | + (1 − n/ ( m + n )) | ν s | + 1 m + n n X r =1 | ˜ ρ rs | + 1 m + n m X h =1 | ¯ ρ h,Bs | ds ≤ Z t E (cid:2) | ˜ ρ is | + | ν s | + | ¯ ρ h,Bs | (cid:3) ds ≤ t sup ≤ s ≤ t E (cid:2) | ˜ ρ is | + | ν s | + | ¯ ρ h,Bs | (cid:3) < ∞ , (3.19)by (3.7) and (3.8). We can then apply the dominated convergence theorem to obtain, for t ∈ [ δ, ∞ ),lim n →∞ Z tδ E (cid:20)(cid:12)(cid:12)(cid:12) f P ( ¯ ρ is − δ − ¯ A ns − δ )( ¯ ρ is − ¯ A ns ) − f P ( ˜ ρ is − δ − E [ ˜ ρ is − δ ])( ˜ ρ is − E [ ˜ ρ is ]) (cid:12)(cid:12)(cid:12)(cid:21) ds = 0 , t ≥ δ. (3.20)15t remains to show that for t ≥ δ it holdslim n →∞ Z tδ E (cid:20)(cid:12)(cid:12)(cid:12) n n X i =1 f P ( ˜ ρ is − δ − E [ ˜ ρ is − δ ])( ˜ ρ is − E [ ˜ ρ is ]) − E (cid:2) f P (cid:0) ˜ ρ is − δ − E [ ˜ ρ is − δ ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds = 0 . (3.21)Since ˜ ρ i , i = 1 , . . . , n , are independent and identically distributed, we have that, for δ ≤ s ≤ t , lim n →∞ E (cid:20)(cid:12)(cid:12)(cid:12) n n X i =1 f P ( ˜ ρ is − δ − E [ ˜ ρ is − δ ])( ˜ ρ is − E [ ˜ ρ is ]) − E (cid:2) f P (cid:0) ˜ ρ is − δ − E [ ˜ ρ is − δ ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) = 0 . Then limit (3.21) follows by the dominated convergence theorem, by Assumption 2.1 andsince the Ornstein-Uhlenbeck process has finite moments, see the computations in (3.19). ✷ Proof of Theorem 3.2 . We suppose by simplicity λ = 1 and we proceed by steps, startingfrom the case when 0 ≤ t < δ , i.e. when there is no delay in equations (2.1)-(2.2) and(3.2)-(3.3). First step: case ≤ t < δ . For every i = 1 , . . . , n and t ∈ [0 , δ ), we have ρ i,nt − ¯ ρ it = Z t ∆ ns ds, where∆ ns = 1 n − n X j =1 ,j = i f P ( ρ j,ns − A ns )( ρ j,ns − A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) + 1 m m X k =1 (cid:0) f B ( ρ k,Bs − A ns )( ρ k,Bs − A ns ) − f B ( ¯ ρ k,Bs − ν ns − E [ ˜ ρ is ])( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ]) (cid:1) − ( ρ i,ns − ¯ ρ is ) + ( A ns − ¯ A ns ) + ( ¯ A ns − E [ ˜ ρ is ] − ν s ) . Thus | ρ i,n − ¯ ρ i | ∗ t = sup s ≤ t (cid:12)(cid:12)(cid:12)(cid:12)Z s ∆ nu du (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup s ≤ t Z s | ∆ nu | du = Z t | ∆ nu | du. i = 1 , . . . , n and t ≥
0, we have E [ | ρ i,n − ¯ ρ i | ∗ t ] ≤ E (cid:20)Z t | ∆ ns | ds (cid:21) ≤ Z t E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i ( f P ( ρ j,ns − A ns )( ρ j,ns − A ns ) − f P ( ¯ ρ js − ¯ A ns )( ¯ ρ js − ¯ A ns )) (cid:12)(cid:12)(cid:12)(cid:21) ds + Z t E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i f P ( ¯ ρ js − ¯ A ns )( ¯ ρ js − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds + Z t E (cid:20)(cid:12)(cid:12)(cid:12) m m X k =1 (cid:0) f B ( ρ k,Bs − A ns )( ρ k,Bs − A ns ) − f B ( ¯ ρ k,Bs − ¯ A ns )( ¯ ρ k,Bs − ¯ A ns ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:21) ds + Z t E (cid:20) m m X k =1 (cid:12)(cid:12) f B ( ¯ ρ k,Bs − ¯ A ns )( ¯ ρ k,Bs − ¯ A ns ) − f B ( ¯ ρ k,Bs − ν ns − E [ ˜ ρ is ])( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ]) (cid:12)(cid:12) (cid:21) ds + Z t E [ | ρ i,ns − ¯ ρ is | ] ds + Z t E [ | A ns − ¯ A ns | ] ds + Z t E (cid:2) | ¯ A ns − E [ ˜ ρ is ] − ν s | (cid:3) ds. (3.22)By (2.8) it holds Z t E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i ( f P ( ρ j,ns − A ns )( ρ j,ns − A ns ) − f P ( ¯ ρ js − ¯ A ns )( ¯ ρ js − ¯ A ns )) (cid:12)(cid:12)(cid:12)(cid:21) ds ≤ n − n X j =1 ,j = i Z t E (cid:20)(cid:12)(cid:12)(cid:12) f P ( ρ j,ns − A ns )( ρ j,ns − A ns ) − f P ( ¯ ρ js − ¯ A ns )( ¯ ρ js − ¯ A ns ) (cid:12)(cid:12)(cid:12)(cid:21) ds ≤ K n − n X j =1 ,j = i Z t E (cid:20)(cid:12)(cid:12)(cid:12) ( ρ j,ns − A ns ) − ( ¯ ρ js − ¯ A ns ) (cid:12)(cid:12)(cid:12)(cid:21) ds ≤ K n − n X j =1 ,j = i Z t E (cid:2)(cid:12)(cid:12) ρ j,ns − ¯ ρ js (cid:12)(cid:12) + (cid:12)(cid:12) A ns − ¯ A ns (cid:12)(cid:12)(cid:3) ds = K Z t E (cid:2)(cid:12)(cid:12) ρ i,ns − ¯ ρ is (cid:12)(cid:12)(cid:3) ds + K Z t E (cid:2)(cid:12)(cid:12) A ns − ¯ A ns (cid:12)(cid:12)(cid:3) ds, t ≥ . (3.23)By (2.3) and (3.16) we have that Z t E (cid:2)(cid:12)(cid:12) A ns − ¯ A ns (cid:12)(cid:12)(cid:3) ds ≤ Z t E h m + n n X r =1 | ρ r,ns − ¯ ρ rs | i ds + Z t E h m + n m X k =1 (cid:12)(cid:12) ρ h,Bs − ¯ ρ h,Bs (cid:12)(cid:12) i ds ≤ Z t E (cid:2)(cid:12)(cid:12) ρ i,ns − ¯ ρ is (cid:12)(cid:12)(cid:3) ds + Z t E (cid:2)(cid:12)(cid:12) ρ k,Bs − ¯ ρ k,Bs (cid:12)(cid:12)(cid:3) ds, t ≥ , (3.24)because all ρ i , i = 1 , . . . , n , and ρ k,B , k = 1 , . . . , m , are identically distributed, respectively.17e can conclude by (3.23) and (3.24) that Z t E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i ( f P ( ρ j,ns − A ns )( ρ j,ns − A ns ) − f P ( ¯ ρ js − ¯ A ns )( ¯ ρ js − ¯ A ns )) (cid:12)(cid:12)(cid:12)(cid:21) ds ≤ K Z t E (cid:2)(cid:12)(cid:12) ρ i,ns − ¯ ρ is (cid:12)(cid:12)(cid:3) ds + K Z t E (cid:2)(cid:12)(cid:12) ρ k,Bs − ¯ ρ k,Bs (cid:12)(cid:12)(cid:3) ds ≤ K Z t E (cid:2)(cid:12)(cid:12) ρ i,n − ¯ ρ i (cid:12)(cid:12) ∗ s (cid:3) ds + K Z t E h(cid:12)(cid:12) ρ k,B − ¯ ρ k,B (cid:12)(cid:12) ∗ s i ds, t ≥ . (3.25)Similarly, Z t E (cid:20)(cid:12)(cid:12)(cid:12) m m X k =1 (cid:0) f B ( ρ k,Bs − A ns )( ρ k,Bs − A ns ) − f B ( ¯ ρ k,Bs − ¯ A ns )( ¯ ρ k,Bs − ¯ A ns ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:21) ds ≤ K Z t E (cid:2)(cid:12)(cid:12) ρ i,n − ¯ ρ i (cid:12)(cid:12) ∗ s (cid:3) ds + 2 K Z t E h(cid:12)(cid:12) ρ k,B − ¯ ρ k,B (cid:12)(cid:12) ∗ s i ds t ≥ . (3.26)From (3.22), (3.24), (3.25) and (3.26) we have that for t ≥ E [ | ρ i,n − ¯ ρ i | ∗ t ] ≤ (3 K + 2) Z t E (cid:2)(cid:12)(cid:12) ρ i,n − ¯ ρ i (cid:12)(cid:12) ∗ s (cid:3) ds + (3 K + 1) Z t E h(cid:12)(cid:12) ρ k,B − ¯ ρ k,B (cid:12)(cid:12) ∗ s i ds + Z t E (cid:20) (cid:12)(cid:12) f B ( ¯ ρ k,Bs − ¯ A ns )( ¯ ρ k,Bs − ¯ A ns ) − f B ( ¯ ρ k,Bs − ν ns − E [ ˜ ρ is ])( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ]) (cid:12)(cid:12) (cid:21) ds + Z t E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i f P ( ¯ ρ js − ¯ A ns )( ¯ ρ js − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds + Z t E (cid:2) | ¯ A ns − E [ ˜ ρ is ] − ν s | (cid:3) ds, t ≥ . (3.27)Proceeding as before, we find E [ | ρ k,B − ¯ ρ k,B | ∗ t ] ≤ (3 K + 1) Z t E (cid:2)(cid:12)(cid:12) ρ i,n − ¯ ρ i (cid:12)(cid:12) ∗ s (cid:3) ds + (3 K + 2) Z t E h(cid:12)(cid:12) ρ k,B − ¯ ρ k,B (cid:12)(cid:12) ∗ s i ds + Z t E (cid:20) (cid:12)(cid:12) f B ( ¯ ρ k,Bs − ¯ A ns )( ¯ ρ k,Bs − ¯ A ns ) − f B ( ¯ ρ k,Bs − ν ns − E [ ˜ ρ is ])( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ]) (cid:12)(cid:12) (cid:21) ds + Z t E (cid:20)(cid:12)(cid:12)(cid:12) n n X i =1 f P ( ¯ ρ is − ¯ A ns )( ¯ ρ is − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds + Z t E (cid:2) | ¯ A ns − ν s − E [˜ ν is | ] (cid:3) ds, (3.28)18o that, summing up (3.27) and (3.28), we have E [ | ρ i,n − ¯ ρ i | ∗ t ] + E [ | ρ k,B − ¯ ρ k,B | ∗ t ] ≤ (6 K + 3) Z t E (cid:2)(cid:12)(cid:12) ρ i,n − ¯ ρ i (cid:12)(cid:12) ∗ s (cid:3) ds + (6 K + 3) Z t E h(cid:12)(cid:12) ρ k,B − ¯ ρ k,B (cid:12)(cid:12) ∗ s i ds + 2 Z t E (cid:20) (cid:12)(cid:12) f B ( ¯ ρ k,Bs − ¯ A ns )( ¯ ρ k,Bs − ¯ A ns ) − f B ( ¯ ρ k,Bs − ν ns − E [ ˜ ρ is ])( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ]) (cid:12)(cid:12) (cid:21) ds + Z t E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i f P ( ¯ ρ js − ¯ A ns )( ¯ ρ js − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds + Z t E (cid:20)(cid:12)(cid:12)(cid:12) n n X i =1 f P ( ¯ ρ is − ¯ A ns )( ¯ ρ is − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds + 2 Z t E (cid:2) | ¯ A ns − ν s − E [˜ ν is | ] (cid:3) ds, t ≥ . (3.29)We can now apply Gronwall’s Lemma and obtain E [ | ρ i,n − ¯ ρ i | ∗ t ] + E [ | ρ k,Bt − ¯ ρ k,Bt | ∗ s ] ≤ e (6 K +3) t Z t E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i f P ( ¯ ρ js − ¯ A ns )( ¯ ρ js − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds + e (6 K +3) t Z t E (cid:20)(cid:12)(cid:12)(cid:12) n n X i =1 f P ( ¯ ρ is − ¯ A ns )( ¯ ρ is − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds + 2 e (6 K +3) t Z t E (cid:20) (cid:12)(cid:12) f B ( ¯ ρ k,Bs − ¯ A ns )( ¯ ρ k,Bs − ¯ A ns ) − f B ( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ])( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ]) (cid:12)(cid:12) (cid:21) ds + 2 e (6 K +3) t Z t E (cid:2) | ¯ A ns − ν s − E [˜ ν is | ] (cid:3) ds, t ≥ . (3.30)We can write Z t E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i f P ( ¯ ρ js − ¯ A ns )( ¯ ρ js − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds ≤ (cid:18) n − − n (cid:19) Z t E (cid:20)(cid:12)(cid:12)(cid:12) n X j =1 ,j = i f P ( ¯ ρ js − ¯ A ns )( ¯ ρ js − ¯ A ns ) (cid:12)(cid:12)(cid:12)(cid:21) ds + Z t E (cid:20)(cid:12)(cid:12)(cid:12) n n X i =1 f P ( ¯ ρ is − ¯ A ns )( ¯ ρ is − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds + 1 n Z t E (cid:2) f P (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) ds t ≥ , (cid:18) n − − n (cid:19) Z t E (cid:20)(cid:12)(cid:12)(cid:12) n X j =1 ,j = i f P ( ¯ ρ is − ¯ A ns )( ¯ ρ is − ¯ A ns ) (cid:12)(cid:12)(cid:12)(cid:21) ds ≤ n ( n − Z t n X j =1 ,j = i E [ | f P ( ¯ ρ is − ¯ A ns )( ¯ ρ is − ¯ A ns ) | ] ds = 1 n Z t E [ | f P ( ¯ ρ is − ¯ A ns )( ¯ ρ is − ¯ A ns ) | ] ds ≤ K n Z t E [ | ¯ ρ is − ¯ A ns | ] ds, t ≥ , where the last term tends to zero when n → ∞ by (3.19).Since it can be shown, for t ≥
0, thatlim n →∞ Z t E (cid:20) (cid:12)(cid:12) f B ( ¯ ρ k,Bs − ¯ A ns )( ¯ ρ k,Bs − ¯ A ns ) − f B ( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ])( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ]) (cid:12)(cid:12) (cid:21) ds = 0 , and lim n →∞ Z t E (cid:2) | ¯ A ns − ν s − E [˜ ν is | ] (cid:3) ds = 0 , t ≥ , with the same proof as for (3.20), then by (3.15) we obtain the result for t ∈ [0 , δ ). Second step: case t ∈ [ δ, δ ) . For every i = 1 , . . . , n and t ≥ δ , we have | ρ i,nt − ¯ ρ it | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z δ ( ρ i,ns − ¯ ρ is ) ds + Z tδ ∆ δ,ns ds (cid:12)(cid:12)(cid:12)(cid:12) , where∆ δ,ns = 1 n − n X j =1 ,j = i f P ( ρ j,ns − δ − A ns − δ )( ρ j,ns − A ns ) − E (cid:2) f P (cid:0) ˜ ρ it − δ − E [ ˜ ρ it − δ ] (cid:1) (cid:0) ˜ ρ it − E [ ˜ ρ it ] (cid:1)(cid:3) + 1 m m X k =1 (cid:16) f B ( ρ k,Bs − δ − A ns − δ )( ρ k,Bs − A ns ) − f B ( ¯ ρ k,Bs − δ − ν s − δ − E [ ˜ ρ is − δ ])( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ]) (cid:17) − ( ρ i,ns − ¯ ρ is ) + ( A ns − ¯ A ns ) + ( ¯ A ns − E [ ˜ ρ is ] − ν s | ) . Thus | ρ i,n − ¯ ρ i | ∗ t = sup s ≤ t (cid:12)(cid:12)(cid:12)(cid:12)Z δ ( ρ i,nu − ¯ ρ iu ) du + Z sδ ∆ δ,nu du (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z δ | ρ i,nu − ¯ ρ iu | du + sup δ ≤ s ≤ t Z sδ (cid:12)(cid:12) ∆ δ,nu (cid:12)(cid:12) du = Z δ | ρ i,nu − ¯ ρ iu | du + Z tδ (cid:12)(cid:12) ∆ δ,nu (cid:12)(cid:12) du, δ ≤ t. (3.31)20or every i = 1 , . . . , n , we have E (cid:20)Z tδ | ∆ δ,ns | ds (cid:21) ≤ Z tδ E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i ( f P ( ρ j,ns − δ − A ns − δ )( ρ j,ns − A ns ) − f P ( ¯ ρ js − δ − ¯ A ns − δ )( ¯ ρ js − ¯ A ns )) (cid:12)(cid:12)(cid:12)(cid:21) ds + Z tδ E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i f P ( ¯ ρ js − δ − ¯ A ns − δ )( ¯ ρ js − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − δ − E [ ˜ ρ is − δ ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds + Z tδ E (cid:20)(cid:12)(cid:12)(cid:12) m m X k =1 (cid:16) f B ( ρ k,Bs − δ − A ns − δ )( ρ k,Bs − A ns ) − f B ( ¯ ρ k,Bs − δ − ¯ A ns − δ )( ¯ ρ k,Bs − ¯ A ns ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:21) ds + Z tδ E (cid:20) m m X k =1 (cid:12)(cid:12)(cid:12) f B ( ¯ ρ k,Bs − δ − ¯ A ns − δ )( ¯ ρ k,Bs − ¯ A ns ) − f B ( ¯ ρ k,Bs − δ − ν s − δ − E [ ˜ ρ is − δ ])( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ]) (cid:12)(cid:12)(cid:12) (cid:21) ds + Z tδ E [ | ρ i,ns − ¯ ρ is | ] ds + Z t E [ | A ns − ¯ A ns | ] ds + Z t E (cid:2) | ¯ A ns − E [ ˜ ρ is ] − ν s | (cid:3) ds, δ ≤ t. (3.32)By (2.8) it holds Z tδ E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i ( f P ( ρ j,ns − δ − A ns − δ )( ρ j,ns − A ns ) − f P ( ¯ ρ js − δ − ¯ A ns − δ )( ¯ ρ js − ¯ A ns )) (cid:12)(cid:12)(cid:12)(cid:21) ds ≤ n − n X j =1 ,j = i Z tδ E (cid:20)(cid:12)(cid:12)(cid:12) f P ( ρ j,ns − δ − A ns − δ ) (cid:0) ( ρ j,ns − A ns ) + ( ¯ ρ js − ¯ A ns ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:21) ds + 1 n − n X j =1 ,j = i Z tδ E (cid:20)(cid:12)(cid:12)(cid:12) ( ¯ ρ js − ¯ A ns ) (cid:0) f P ( ρ j,ns − δ − A ns − δ ) − f P ( ¯ ρ js − δ − ¯ A ns − δ ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:21) ds ≤ K Z tδ E [ | ρ i,ns − ¯ ρ is | ] ds + K Z tδ E [ | A ns − ¯ A ns | ds + Z tδ E (cid:2)(cid:12)(cid:12) ¯ ρ is − ¯ A ns (cid:12)(cid:12) (cid:12)(cid:12) f P ( ρ i,ns − δ − A ns − δ ) − f P ( ¯ ρ is − δ − ¯ A ns − δ ) (cid:12)(cid:12)(cid:3) ds. (3.33)We have that for δ ≤ t Z tδ E (cid:2)(cid:12)(cid:12) ¯ ρ is − ¯ A ns (cid:12)(cid:12) (cid:12)(cid:12) f P ( ρ i,ns − δ − A ns − δ ) − f P ( ¯ ρ is − δ − ¯ A ns − δ ) (cid:12)(cid:12)(cid:3) ds ≤ Z tδ (cid:16) E (cid:2)(cid:12)(cid:12) ¯ ρ is − ¯ A ns (cid:12)(cid:12) (cid:3) ds (cid:17) / (cid:16) E h(cid:12)(cid:12) f P ( ρ i,ns − δ − A ns ) − f P ( ¯ ρ is − δ − ¯ A ns ) (cid:12)(cid:12) i(cid:17) / ds ≤ (cid:18)Z tδ E h(cid:12)(cid:12) ¯ ρ is − ¯ A ns (cid:12)(cid:12) i ds (cid:19) / (cid:18)Z tδ E h(cid:12)(cid:12) f P ( ρ i,ns − δ − A ns − δ ) − f P ( ¯ ρ is − δ − ¯ A ns − δ ) (cid:12)(cid:12) i ds (cid:19) / (cid:18)Z tδ E h(cid:12)(cid:12) ¯ ρ is − ¯ A ns (cid:12)(cid:12) i ds (cid:19) / (cid:18)Z tδ E (cid:2)(cid:12)(cid:12) f P ( ρ i,ns − δ − A ns − δ ) − f P ( ¯ ρ is − δ − ¯ A ns − δ ) (cid:12)(cid:12)(cid:3) ds (cid:19) / ≤ p K (cid:18)Z tδ E h(cid:12)(cid:12) ¯ ρ is − ¯ A ns (cid:12)(cid:12) i ds (cid:19) / (cid:18)Z tδ E (cid:2)(cid:12)(cid:12) f P ( ρ i,ns − δ − A ns − δ ) − f P ( ¯ ρ is − δ − ¯ A ns − δ ) (cid:12)(cid:12)(cid:3) ds (cid:19) / ≤ p K K (cid:18)Z tδ E h(cid:12)(cid:12) ¯ ρ is − ¯ A ns (cid:12)(cid:12) i ds (cid:19) / (cid:18)Z tδ E (cid:2)(cid:12)(cid:12) ρ i,ns − δ − ¯ ρ is − δ (cid:12)(cid:12) + (cid:12)(cid:12) A ns − δ − ¯ A ns − δ (cid:12)(cid:12)(cid:3) ds (cid:19) / , where we have used that | a − b | ≤ | a − b | for a, b ∈ R + .Then, setting G n ( t ) := (cid:16)R tδ E h(cid:12)(cid:12) ¯ ρ is − ¯ A ns (cid:12)(cid:12) i ds (cid:17) / , by (3.33) we have Z tδ E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i ( f P ( ρ j,ns − δ − A ns − δ )( ρ j,ns − A ns ) − f P ( ¯ ρ js − δ − ¯ A ns − δ )( ¯ ρ js − ¯ A ns )) (cid:12)(cid:12)(cid:12)(cid:21) ds ≤ K Z tδ E [ | ρ i,ns − ¯ ρ is | ] ds + K Z tδ E [ | A ns − ¯ A ns | ds + p K K G n ( t ) (cid:18)Z tδ E (cid:2)(cid:12)(cid:12) ρ i,ns − δ − ¯ ρ is − δ (cid:12)(cid:12) + (cid:12)(cid:12) A ns − δ − ¯ A ns − δ (cid:12)(cid:12)(cid:3) ds (cid:19) / , δ ≤ t. (3.34)Since Z tδ E (cid:2)(cid:12)(cid:12) ρ i,ns − δ − ¯ ρ is − δ (cid:12)(cid:12) + (cid:12)(cid:12) A ns − δ − ¯ A ns − δ (cid:12)(cid:12)(cid:3) ds = E (cid:20)Z tδ (cid:0)(cid:12)(cid:12) ρ i,ns − δ − ¯ ρ is − δ (cid:12)(cid:12) + (cid:12)(cid:12) A ns − δ − ¯ A ns − δ (cid:12)(cid:12)(cid:1) ds (cid:21) = E (cid:20)Z t − δ (cid:0)(cid:12)(cid:12) ρ i,nu − ¯ ρ iu (cid:12)(cid:12) + (cid:12)(cid:12) A nu − ¯ A nu (cid:12)(cid:12)(cid:1) du (cid:21) ≤ Z δ E [ (cid:12)(cid:12) ρ i,nu − ¯ ρ iu (cid:12)(cid:12) + (cid:12)(cid:12) A nu − ¯ A nu (cid:12)(cid:12) ] du, δ ≤ t < δ, we can rewrite (3.34) as Z tδ E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i ( f P ( ρ j,ns − δ − A ns − δ )( ρ j,ns − A ns ) − f P ( ¯ ρ js − δ − ¯ A ns − δ )( ¯ ρ js − ¯ A ns )) (cid:12)(cid:12)(cid:12)(cid:21) ds ≤ K Z tδ E [ | ρ i,ns − ¯ ρ is | ] ds + K Z tδ E [ | A ns − ¯ A ns | ds + p K K G n ( t ) (cid:18)Z δ E (cid:2)(cid:12)(cid:12) ρ i,ns − ¯ ρ is (cid:12)(cid:12) + (cid:12)(cid:12) A ns − ¯ A ns (cid:12)(cid:12)(cid:3) ds (cid:19) / , δ ≤ t. (3.35)Similarly, Z t E (cid:20)(cid:12)(cid:12)(cid:12) m m X k =1 (cid:16) f B ( ρ k,Bs − δ − A ns − δ )( ρ k,Bs − A ns ) − f B ( ¯ ρ k,Bs − δ − ¯ A ns − δ )( ¯ ρ k,Bs − ¯ A ns ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:21) ds ≤ K Z tδ E [ | ρ k,Bs − ¯ ρ k,Bs | ] ds + K Z tδ E [ | A ns − ¯ A ns | ds + p K K G n ( t ) (cid:18)Z δ E (cid:2)(cid:12)(cid:12) ρ k,Bs − ¯ ρ k,Bs (cid:12)(cid:12) + (cid:12)(cid:12) A ns − ¯ A ns (cid:12)(cid:12)(cid:3) ds (cid:19) / , δ ≤ t. (3.36)22ith G n ( t ) := (cid:16)R tδ E h(cid:12)(cid:12) ¯ ρ k,Bs − ¯ A ns (cid:12)(cid:12) i ds (cid:17) / . From (3.24), (3.31), (3.32), (3.35) and (3.36) we obtain E [ | ρ i,n − ¯ ρ i | ∗ t ] ≤ (3 K + 2) Z tδ E (cid:2)(cid:12)(cid:12) ρ i,n − ¯ ρ i (cid:12)(cid:12) ∗ s (cid:3) ds + (3 K + 1) Z tδ E [ | ρ k,B − ¯ ρ k,B | ∗ s ] ds + p K K G n ( t ) (cid:18)Z δ E (cid:2)(cid:12)(cid:12) ρ i,ns − ¯ ρ is (cid:12)(cid:12) + (cid:12)(cid:12) A ns − ¯ A ns (cid:12)(cid:12)(cid:3) ds (cid:19) / + p K K G n ( t ) (cid:18)Z δ E (cid:2)(cid:12)(cid:12) ρ k,Bs − ¯ ρ k,Bs (cid:12)(cid:12) + (cid:12)(cid:12) A ns − ¯ A ns (cid:12)(cid:12)(cid:3) ds (cid:19) / + Z t E (cid:20) (cid:12)(cid:12)(cid:12) f B ( ¯ ρ k,Bs − δ − ¯ A ns − δ )( ¯ ρ k,Bs − ¯ A ns ) − f B ( ¯ ρ k,Bs − δ − ν s − δ − E [ ˜ ρ is − δ ])( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ]) (cid:12)(cid:12)(cid:12) (cid:21) ds + Z t E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i f P ( ¯ ρ js − δ − ¯ A ns − δ )( ¯ ρ js − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − δ − E [ ˜ ρ is − δ ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds + Z δ E [ | ρ i,ns − ¯ ρ is | ] ds + Z t E (cid:2) | ¯ A ns − E [ ˜ ρ is ] − ν s | (cid:3) ds, δ ≤ t < δ. (3.37)At the same way, by (2.2) and (3.3) we have E [ | ρ k,B − ¯ ρ k,B | ∗ t ] ≤ (3 K + 1) Z tδ E (cid:2)(cid:12)(cid:12) ρ i,n − ¯ ρ i (cid:12)(cid:12) ∗ s (cid:3) ds + (3 K + 2) Z tδ E [ | ρ k,B − ¯ ρ k,B | ∗ s ] ds + p K K G n ( t ) (cid:18)Z δ E (cid:2)(cid:12)(cid:12) ρ i,ns − ¯ ρ is (cid:12)(cid:12) + (cid:12)(cid:12) A ns − ¯ A ns (cid:12)(cid:12)(cid:3) ds (cid:19) / + p K K G n ( t ) (cid:18)Z δ E (cid:2)(cid:12)(cid:12) ρ k,Bs − ¯ ρ k,Bs (cid:12)(cid:12) + (cid:12)(cid:12) A ns − ¯ A ns (cid:12)(cid:12)(cid:3) ds (cid:19) / + Z t E (cid:20) (cid:12)(cid:12)(cid:12) f B ( ¯ ρ k,Bs − δ − ¯ A ns − δ )( ¯ ρ k,Bs − ¯ A ns ) − f B ( ¯ ρ k,Bs − δ − ν s − δ − E [ ˜ ρ is − δ ])( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ]) (cid:12)(cid:12)(cid:12) (cid:21) ds + Z t E (cid:20)(cid:12)(cid:12)(cid:12) n n X i =1 f P ( ¯ ρ is − δ − ¯ A ns − δ )( ¯ ρ is − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − δ − E [ ˜ ρ is − δ ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds + Z δ E [ | ρ k,Bs − ¯ ρ k,Bs | ] ds + Z t E (cid:2) | ¯ A ns − ν s − E [˜ ν is | ] (cid:3) ds, δ ≤ t < δ. (3.38)23umming up (3.37) and (3.38) we find E [ | ρ i,n − ¯ ρ i | ∗ t ] + E [ | ρ k,B − ¯ ρ k,B | ∗ t ] ≤ (6 K + 3) Z t (cid:0) E [ | ρ i,n − ¯ ρ i | ∗ s ] + E [ | ρ k,B − ¯ ρ k,B | ∗ s ] (cid:1) ds + p K K ( G n ( t ) + G n ( t )) (cid:18)Z δ (cid:0) E [ | ρ i,ns − ¯ ρ is | ] + E [ | ρ k,Bs − ¯ ρ k,Bs | ] + E (cid:2)(cid:12)(cid:12) A ns − ¯ A ns (cid:12)(cid:12)(cid:3)(cid:1) ds (cid:19) / (3.39)+ Z t E (cid:20) (cid:12)(cid:12)(cid:12) f B ( ¯ ρ k,Bs − δ − ¯ A ns − δ )( ¯ ρ k,Bs − ¯ A ns ) − f B ( ¯ ρ k,Bs − δ − ν s − δ − E [ ˜ ρ is − δ ])( ¯ ρ k,Bs − ν s − E [ ˜ ρ is ]) (cid:12)(cid:12)(cid:12) (cid:21) ds + Z t E (cid:20)(cid:12)(cid:12)(cid:12) n − n X j =1 ,j = i f P ( ¯ ρ js − δ − ¯ A ns − δ )( ¯ ρ js − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − δ − E [ ˜ ρ is − δ ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds + Z t E (cid:20)(cid:12)(cid:12)(cid:12) n n X i =1 f P ( ¯ ρ is − δ − ¯ A ns − δ )( ¯ ρ is − ¯ A ns ) − E (cid:2) f P (cid:0) ˜ ρ is − δ − E [ ˜ ρ is − δ ] (cid:1) (cid:0) ˜ ρ is − E [ ˜ ρ is ] (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:21) ds + 2 Z t E (cid:2) | ¯ A ns − ν s − E [˜ ν is | ] (cid:3) ds, δ ≤ t < δ. (3.40)With the same computations used in the first step of the proof, we show that the lastfour terms of (3.40) converge to zero when n → ∞ by the proof of Proposition 3.3. Theterm in (3.39) also goes to zero when n → ∞ , by the first step of the proof and becauselim n →∞ [ G n ( t ) + G n ( t )] < ∞ , by (3.19). Then applying Gronwall’s Lemma to (3.40) we provethe result for t ∈ [ δ, δ ).The result then follows by proceeding in the same way for all the steps t ∈ [ kδ, ( k + 1) δ ) ,k ≥ ✷ We now study by numerical simulations how the system described in Section 3 reacts tothe growth and the burst of a bubble. In particular, we investigate how a bank not holdingthe bubbly asset can be affected by a bubble burst through contagion mechanisms. We firstconsider the case of (2.1)-(2.2), i.e. of a network with a finite number of banks, and then weanalyze the limit system (3.1)-(3.3).The bubble has the dynamics specified in Biagini et al. [8], i.e. it solves (2.4) with µ t = M t Λ t ( − kβ t + 2¯ µ t ) , σ t = 2¯ σM t Λ t , t ≥ , where M = ( M t ) t ∈ [0 ,T ] , Λ = (Λ t ) t ∈ [0 ,T ] are respectively a measure of illiquidity and the socalled resiliency , ¯ µ = (¯ µ t ) t ≥ is the drift of the signed volume of market orders (buy market24rders minus sell market orders) and ¯ σ >
0. Here, the illiquidity M is supposed to be ageometric Brownian motion, i.e. dM t = M t ( µ M dt + σ M dB t ) , t ≥ . with µ M ∈ R and σ M >
0. We choose the same function f for both core and peripherybanks in (2.1)-(2.2), i.e. f B = f P = f . In particular, we take f ( x ) = 1 + 2 arctan( x ) /π , asin Example 2.2. We first focus on the system (2.1)-(2.2). We investigate how the first bank reacts whenbanks holding the bubble are in trouble. Specifically, we here introduce and compute therisk measure
Risk iα = − sup x ∈ R (" N s N s X k =1 n ( ρ i,n,kτk +∆ − ρ i,n,kτk ) /ρ i,n,kτk ≤ x o ≤ α ) , (4.1)with α >
0, where N s is the number of simulations of the processes in (2.1)-(2.2), τ k is thevalue at the k -th simulation of the bursting time τ of the bubble, and ρ i,n,kt is the value of ρ i,nt computed in the k -th simulation.The risk measure Risk iα as defined in (4.1) is analogous to the CoVar of a bank without thebubble with respect to a bank with the bubble (for a definition of CoVar see e.g. Biaginiet al. [6] and Brunnermeier and Oehmke [12]). Note that, since the banks not holding thebubble are identically distributed, we only compute the risk for one bank.From now on, we set α = 0 .
05 in (4.1). We perform N s = 10000 simulations of Risk . inthe case when there are n = 6 banks not holding the bubble and m = 2 banks holding it.We consider different values of λ and of the delay δ .The results are given in Table 1: δ = 0 δ = 0 . δ = 0 . δ = 0 . δ = 0 . δ = 0 . δ = 0 . λ = 0 . .
283 0 .
390 0 .
451 0 .
716 0 .
925 0 .
916 0 . λ = 1 0 .
281 0 .
385 0 .
434 0 .
661 0 .
886 0 .
879 0 . λ = 2 0 .
280 0 .
377 0 .
422 0 .
641 0 .
851 0 .
824 0 . Risk . in the case when the robustness is given by (2.1)-(2.2), with parameters σ = σ = 0 .
2, ∆ = 0 . ρ i, = ρ k,B = 0 . i = 1 , . . . , k = 1 , .
1, the risk is still big but it decreases.25his depends on the fact that we check the robustness of the banks at time τ + 0 .
1: at thistime, when δ = 0 . , . f is smaller than in the case δ = 0 . λ . Indeed, it follows by (2.1) that ρ i,n reverts to A nt + 1 λ n n X i =1 f ( ρ i,nt − δ − A nt − δ )( ρ i,nt − A nt ) + 1 m − m X ℓ =1 ,ℓ = k f ( ρ ℓ,Bt − δ − A nt − δ )( ρ ℓ,nt − A nt ) ! , so that for large λ the term involving the network, and then the direct effects of the banksholding the bubbly asset, is less significative.We now consider (2.1)-(2.2) when β is replaced by ¯ β , where d ¯ β t = t ≤ τ, ρ , ¯ βτ ρ ,βτ dβ t for t ≥ τ, (4.2)where ρ ,β is the robustness of bank 1 when there is a bubble in the network, and ρ , ¯ β isthe robustness of bank 1 when there is no bubble. In this way we model the case when thebanks that used to hold the bubbly asset are subject at time τ to the same (relative) shock,but without having experienced the growth of the bubble. The results are given in Table 2,for the same parameters as in Table 1. δ = 0 δ = 0 . δ = 0 . δ = 0 . δ = 0 . δ = 0 . δ = 0 . λ = 0 . .
281 0 .
383 0 .
388 0 .
415 0 .
505 0 .
499 0 . λ = 1 0 .
280 0 .
381 0 .
385 0 .
403 0 .
502 0 .
494 0 . λ = 2 0 .
280 0 .
371 0 .
380 0 .
399 0 .
500 0 .
490 0 . Risk . in the case when the robustness is given by (2.1)-(2.2) with no bubble inthe system, but with the same shock at time τ , for parameters σ = σ = 0 .
2, ∆ = 0 . ρ i, = ρ k,B = 0 . i = 1 , . . . , k = 1 , δ = 0 there is not any significant difference with the case when there is abubble in the system, since the banks are able to disinvest immediately at the time whenthe shock hits the banks with the bubble. Anyway, this difference increases with the delay.When the delay is big, the banks with no bubble are much more in trouble in the first case,i.e when they are attached to banks holding the bubbly asset.We can then conclude that the increase of the value of the bubbly asset can put the networkin trouble, because it makes the system more centralized on the riskier banks, due to thepreferential attachment mechanism implied by (2.1)-(2.2).26his can also be seen by considering a static network, i.e. by taking f B = f P = 1 in (2.1)-(2.2). In this case, we obtain the following values of the risk for different values of λ : λ = 0 . λ = 1 λ = 20 .
670 0 .
626 0 . Risk . with ∆ = 0 . f B = f P = 1 and withparameters σ = σ = 0 .
2, ∆ = 0 . ρ i, = ρ k,B = 0 . i = 1 , . . . , k = 1 , f B and f P . Comparing this result with Table 1, one can see that when δ in (2.1)-(2.2) is small,then the fact that banks are able to quickly disinvest makes the system safer than in the caseof a static network. On the other hand, for big values of δ , a centralized network towardsthe banks holding the bubble and the impossibility to disinvest quickly after the burst giverise to a more dangerous system than in the static case. We now consider the case of the limit system (3.1)-(3.3). We compute
Risk . = − sup x ∈ R (" N s N s X k =1 n (¯ ρ ,kτk +∆ − ¯ ρ ,kτk ) / ¯ ρ ,kτk ≤ x o ≤ . ) , (4.3)where N s and τ k are the number of simulations and the time of the burst of the bubble inthe k -th simulation, respectively, and ¯ ρ ,kt is the value of ¯ ρ t computed in the k -th simulation.As before, we consider m = 2 banks holding the bubble and we make N s = 10000 simulationsof (3.1)-(3.3) taking different values of λ and δ .We compute φ ( t, t − δ ) = E (cid:2) f (cid:0) ˜ ρ it − δ − E [ ˜ ρ it − δ ] (cid:1) ( ˜ ρ it − E [ ˜ ρ it ]) (cid:3) in (3.2) and (3.3) via MonteCarlo simulations of the trajectories of the Ornstein-Uhlenbeck process in (3.1). Note that E [ ˜ ρ it ] = ρ e − λt . The results are gathered in Table 4. δ = 0 δ = 0 . δ = 0 . δ = 0 . δ = 0 . δ = 0 . δ = 0 . λ = 0 . .
305 0 .
367 0 .
563 0 .
908 1 .
281 1 .
251 1 . λ = 1 0 .
302 0 .
360 0 .
521 0 .
765 1 .
170 1 .
125 1 . λ = 2 0 .
302 0 .
356 0 .
503 0 .
647 0 .
908 0 .
907 0 . Risk . with ∆ = 0 . σ = σ = 0 . ρ k,B = 0 . k = 1 , δ = 0 . λ , since ¯ ρ it λ ϕ ( t, t − δ ) + 1 m m X k =1 f (cid:16) ¯ ρ k,Bt − δ − ν t − δ − E [ ˜ ρ it − δ ] (cid:17) (cid:16) ¯ ρ k,Bt − ν t − E [ ˜ ρ it ] (cid:17)! + E [ ˜ ρ it ] − ˜ ρ it , so that a large λ diminishes the influence of the banks holding the bubbly asset.We can also see that the risk is bigger at the limit by comparing (2.1) and (3.6): since ν t − δ + E [ ˜ ρ it ] < A nt − δ , because the first term is the average robustness of banks not holding thebubble, the argument of f is bigger in (3.6). This leads to a bigger weight multiplying theloss at the moment of the burst at the limit.In Table 5, we report the results for the case when β is replaced by ¯ β as in (4.2), i.e. whenthere is no bubble in the network. δ = 0 δ = 0 . δ = 0 . δ = 0 . δ = 0 . δ = 0 . δ = 0 . λ = 0 . .
303 0 .
355 0 .
468 0 .
682 0 .
698 0 .
659 0 . λ = 1 0 .
302 0 .
347 0 .
410 0 .
528 0 .
640 0 .
628 0 . λ = 2 0 .
300 0 .
340 0 .
395 0 .
455 0 .
612 0 .
561 0 . Risk . with ∆ = 0 . σ = σ = 0 . ρ k,B = 0 . k = 1 , f B = f P = 1, the results, shown in Table 6, agree with the ones obtained in the case ofthe finite network: for small delays the dynamic network is less exposed to systemic risk withrespect to the static one, whereas when the delay increases and the banks in the dynamicnetwork are slower in disinvesting, the risk is bigger than for the static network. λ = 0 . λ = 1 λ = 21 .
001 0 .
910 0 . Risk . with ∆ = 0 . f B = f P = 1 in themean field limit, with parameters σ = σ = 0 .
2, ∆ = 0 . ρ k,B = 0 . k = 1 , A Existence and uniqueness theorems
For the reader’s convenience we report here the results, which we have used in the paperto prove existence and uniqueness of a strong solution of a system of stochastic differential28quations (SDEs) and of stochastic differential delay equations (SDDEs). These theoremsalso guarantee the finiteness of the second moments of the strong solution.In the following, let (Ω , F , P ) be a complete probability space with a filtration F := ( F t ) t ≥ satisfying the usual conditions, and B t = ( B t , . . . , B mt ) t ≥ , be an m -dimensional F -Brownianmotion defined on (Ω , F , P ).We begin by the following existence and uniqueness result for a system of SDEs, given inTheorem 9.11 in Pascucci [39]. Theorem A.1.
Let X be an F t -measurable R d -valued random variable such that E [ X ] < ∞ . Consider the d -dimensional stochastic differential equation of Itˆo type dX t = f ( t, X t ) dt + g ( t, X t ) dB t , t ≤ t ≤ T, (A.1) with X t = X , where f : [ t , T ] × R d → R d and g : [ t , T ] × R d → R d × m are both Borelmeasurable.Assume that there there exist two positive constants K and K such that:1. (Lipschitz condition) for all x, y ∈ R d and t ∈ [ t , T ] , k f ( t, x ) − f ( t, y ) k + k g ( t, x ) − g ( t, y ) k ≤ K k x − y k ;
2. (Linear growth condition) for all ( t, x ) ∈ [ t , T ] × R d , k f ( t, x ) k + k g ( t, x ) k ≤ K (1 + k x k ) . Then there exists a unique solution X = ( X t ) x ∈ [ t ,T ] to equation ( A. and it holds E (cid:20) sup t ≤ s ≤ t k X s k (cid:21) ≤ C (1 + E (cid:2) k X k (cid:3) ) e Ct , t ∈ [ t , T ] , (A.2) where C is a constant depending on K and T only. We now recall Theorem 3.1 in Mao [36, chapter 5], that provides the existence and uniquenessresults for SDDEs.
Theorem A.2.
Let F : [ t , T ] × R d × R d → R d and G : [ t , T ] × R d × R d → R d × m beBorel-measurable. Consider the delay equation dX t = F ( t, X t , X t − τ ) dt + G ( t, X t , X t − τ ) dB t , (A.3) with initial data { X s : t − τ ≤ s ≤ t } , such that X s is F t -measurable for all s ∈ [ t − τ, t ] and E [ k X s k ] < ∞ for all s ∈ [ t − τ, t ] .Assume that there exists two positive constants ˜ K and ˜ K such that . (Linear growth condition) for all ( t, x, y ) ∈ [ t , T ] × R d × R d , k F ( t, x, y ) k + k G ( t, x, y ) k ≤ ˜ K (1 + k x k + k y k );
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