Existence of solutions to a system of SDEs with mean-field drift and jump random measures
aa r X i v : . [ m a t h . P R ] F e b Existence of solutions to a system of SDEs with mean-fielddrift and jump random measures
Ying Jiao ∗ and Nikolaos Kolliopoulos †‡ February 24, 2021
Abstract
We study the well-posedness of a system of multi-dimensional SDEs which are cor-related through a non-homogeneous mean-field term in each drift and also by drivingBrownian motions and jump random measures. Supposing the drift coefficients arenon-Lipschitz, we prove for the system the existence of strong, L -integrable, c`adl`agsolution which can be obtained as monotone limit of solutions to some approximatingsystems, extending existing results for one-dimensional jump SDE with non-Lipschitzcoefficients. We show in addition that the solutions are positive. Systems of correlated stochastic differential equations (SDEs) with mean-field interactionhave been widely adopted in both theoretical and application fields, see for examplesthe book by Carmona and Delarue [5, 6]. In this paper, we are interested in a multi-dimensional generalisation with mean-field drift coefficients and more general pure jumpterms of the following one-dimensional SDE dλ t = a ( b − λ t ) dt + σ p λ t dB t + σ Z λ /αt − dZ t , t ≥ a, b, σ, σ Z ≥ B = ( B t , t ≥
0) is a Browinan motion and Z = ( Z t , t ≥
0) isan independent spectrally positive α -stable compensated L´evy process with parameter α ∈ (1 , λ t = λ + a Z t ( b − λ s ) ds + σ Z t Z λ s W ( ds, du ) + σ Z Z t Z λ s − Z R + ζ e N ( ds, dv, dζ ) , (1.2)where W ( ds, du ) is a white noise on R with intensity dsdu , e N ( ds, dv, dζ ) is an indepen-dent compensated Poisson random measure on R with intensity dsdvµ ( dζ ) with µ ( dζ )being a L´evy measure on R + and satisfying R ∞ ( ζ ∧ ζ ) µ ( dζ ) < ∞ . ∗ Universit´e Claude Bernard - Lyon 1, Institut de Science Financi`ere et d’Assurances, 50 Avenue TonyGarnier, 69007 Lyon, France. Email: [email protected]. † Peking University, Beijing International Centre for Mathematical Research, Beijing, China. ‡ Carnegie Mellon University, Department of Mathematical Sciences, Pittsburgh, PA 15213, USA.Email: [email protected] (corresponding author). α -CIR process in Jiao et al. [15]. In arecent paper, Frikha and Li [9] study the well-posedness and numerical approximationof a time-inhomogeneous jump SDE with generally non-Lipschitz coefficients which, asa generalisation of (1.1), has a drift term involving the law of the solution and can beviewed as a mean-field limit of an individual particle evolving within a system. Theconsequences of assuming generally non-Lipschitz coefficients in these many settings aretwofold. On one hand, the well-posedness becomes challenging since the classic iterationmethod fails to apply (see [10, 4, 9]). On the other hand, such CIR-like processes arenon-negative and thus constitute good candidates for financial modelling, see Hamblyand Kolliopoulos [13, 14] for a system of correlated SDEs used to describe a large creditportfolio with stochastic volatility.In this paper, we focus on a system of finite number of jump SDEs where the drift termof each component is characterized by a mean-field function depending on other compo-nents of the system. Such SDEs, with diffusion coefficients being non-Lipschitz, can beadopted to model interacting financial quantities. We refer to works by Bo and Capponi[2], Fouque and Ichiba [8] and Giesecke et al. [12] for the mean-field modelling of largedefault-sensitive portfolios in credit and systemic risks. The main contributions of thispaper are twofold. First, each component of the system contains a jump part driven bygeneral random measures which allows to include various jump processes such as Poissonor compound Poisson processes, L´evy processes or indicator default processes. Second,we impose mild conditions on the dependence among components. In particular there isno need for the driven processes, that is, Brownian motions and jump random measures,of the associated SDEs to be independent. So the system can admit a flexible structureof correlation which will be useful for potential modelling of correlated inhomogeneoussystem.To prove the strong well-posedness of the multi-dimensional system, we construct asequence of approximating solutions whose drifts are defined by a piecewise projectionof the minimal drift processes of all the components. We show that the approximatingsystems are monotone by using a comparison theorem from Gal’chuk [11] (see also Abdel-ghani and Melnikov [1]) who considered SDEs with respect to continuous martingales andjump random measures where the coefficients of the semimartingale are not Lipschitz.We then use the monotone convergence to establish that the family of limit processessolves our system of SDEs. We also provide a key lemma on one-dimensional SDEs witha general drift coefficient. This result is essential to deal with the approximating solutionssince their drifts are defined by conditional expectations so that standard assumptions inliterature fail to hold.The rest of the paper is organized as follows. In Section 2, we present the system ofSDEs, the assumptions on the coefficients and our main result. Section 3 provides thekey lemma and the proof of the main existence result is given in Section 4. We fix a filtered probability space (Ω , F , F = ( F t ) t ≥ , P ) which satisfies the usual condi-tions. Let U and U be two locally compact and separable metric spaces. For N ∈ N ,2e study the system of SDEs of the following form λ it = λ i + a i Z t (cid:0) b i (cid:0) s, λ s , λ s , ..., λ Ns (cid:1) − λ is (cid:1) ds + Z t σ i ( λ is ) dW is + Z t Z U g i, (cid:0) λ is − , u (cid:1) ˜ N i, ( ds, du )+ Z t Z U g i, (cid:0) λ is − , u (cid:1) N i, ( ds, du ) , i ∈ { , , ..., N } (2.1)where λ i ≥ a i ≥ W i = ( W it ) t ≥ is an F -adapted Brownian motion, N i, ( ds, du ) and N i, ( ds, du ) are Poisson random measures associated to two F -adapted point processes p i, : Ω × R + −→ U and p i, : Ω × R + −→ U with compensator measures µ i, ( du ) dt and µ i, ( du ) dt respectively. Let ˜ N i, ( ds, du ) = N i, ( ds, du ) − µ i, ( du ) dt be the compensatedmeasure of p i, ( · ). For every i ∈ { , , ..., N } , we suppose that W i , p i, and p i, aremutually independent but we do not require the triplet to be independent for different i, j ∈ { , , ..., N } . Example 2.1.
The typical example of a mean-field drift function is b i ( t, x , · · · , x N ) = N P Nk =1 x k , which is the same for every i ∈ { , · · · , N } . Let Z = ( Z t ) t ≥ be an α -stable L´evy process and Z i = ( Z it ) t ≥ be an independent α i -stalbe L´evy process with α , α i ∈ (1 , U = R , Z i = ( Z , Z i ) with compensated measure e N i, ( dt, du ) with u = ( u , u i ) ∈ U . Let the diffusion coefficient functions be given as σ i ( x ) = σ i · x / and g i, ( x, u ) = σ Z, u · x /α + σ Z,i u i · x /α i where σ i ≥ σ Z, , σ Z,i ≥
0. In this example, the process Z represents a com-mon external factor which affects significantly the whole market such as financial crisisand pandemics (like the recent Covid pandemic crisis), or can also include more com-mon events which lead to more frequent and smaller jumps. Whereas the process Z i isassociated to the individual shocks without affecting others.For the coefficients appearing in the diffusion terms, we assume the following condi-tions are satisfied for all the components of the system (2.1). The regularity conditionsof Assumption 2.2 are motivated by the one-dimensional case in [10]. These conditionswill be crucial to prove Lemma 3.1. Assumption 2.2.
We assume the conjunction of the following conditions for the param-eters ( σ, g , g , N , N ):(1) σ : R → R is a continuous function such that σ ( x ) = 0 for x ≤
0. Moreover, thereexists a non-negative and increasing function ρ ( · ) on R + such that Z x dzρ ( z ) = + ∞ (2.2)for any x > | σ ( x ) − σ ( y ) | ≤ ρ ( | x − y | ) for all x, y ≥ N is the Poisson random measure of an F -adapted point process with compensatormeasure µ and g : R × U → R is a Borel function, such that3i) for each fixed u ∈ U , the function g ( · , u ) : x g ( x, u ) is increasing, andsatisfies the inequality g ( x, u )+ x ≥ x ≥ g ( x, u ) = 0when x ≤ x ∈ R , the function u g ( x, u ) is locally integrable with respectto the measure µ ,(iii) the function x Z U | g ( x, u ) | ∧ | g ( x, u ) | µ ( du )is locally bounded,(iv) for any m ∈ N , there exists a non-negative and increasing function x −→ ρ m ( x )on R + such that Z x dzρ m ( z ) = + ∞ (2.3)for any x > Z U | g ( x, u ) ∧ m − g ( y, u ) ∧ m | µ ( du ) ≤ ρ m ( | x − y | ) (2.4)for all 0 ≤ x, y ≤ m .(3) N is the Poisson random measure of an F -adapted point process with compensatormeasure µ , and g : R × U → R is a Borel function, such that(i) for any ( x, u ) ∈ R × U , g ( x, u ) + x ≥ x −→ Z U | g ( x, u ) | µ ( du ) (2.5)is locally bounded and has at most a linear growth when x → + ∞ ,(iii) there exists a Borel set U ⊂ U with µ ( U \ U ) < + ∞ , and for any m ∈ N ,a concave and increasing function x −→ r m ( x ) on R + such that Z x dzr m ( z ) = + ∞ (2.6)for all x > Z U | g ( x, u ) ∧ m − g ( y, u ) ∧ m | µ ( du ) ≤ r m ( | x − y | ) (2.7)for all 0 ≤ x, y ≤ m .In particular, the inequality (2.2) can be compared to some H¨older condition in [9].It is satisfied if σ ( x ) are α -H¨older continuous in x for some α ∈ [1 / , . In order to construct appropriate monotone approximations in the multi-dimensionalcase, extra monotonicity and continuity conditions are required in Assumption 2.3.4 ssumption 2.3.
The function σ is either bounded or increasing on R + , the functions g and g are left continuous in x ∈ R , and the function g is either increasing in x ∈ R or bounded by some function ( x, u ) −→ G ( u ) with Z U | G ( u ) | µ ( du ) ∨ Z U G ( u ) µ ( du ) < + ∞ . (2.8)Note that Assumption 2.3 allows to admit some discontinuity for g and g . Forexample, following (1.2), g can take the form u = ( v, ζ ) ∈ R and g ( x, v, ζ ) = 1 { v Consider the system of SDEs (2.1) and suppose for all i ∈ { , · · · , N } that (1) the parameter a i is non-negative and the mean-field function b i : R + × R N −→ R isnon-negative, increasing and Lipschitz continuous in each of its last N variables, (2) the coefficients ( σ i , g i, , g i, , N i, , N i, ) satisfy Assumption 2.2 and 2.3.Then (2.1) has a c`adl`ag F -adapted solution ( λ t , · · · , λ Nt ) t ≥ , with λ i · non-negative and E [ R T λ it dt ] < + ∞ for any T ≥ . Before proving the main result, we need the following lemma on the auxiliary one-dimensional SDE with a more general drift coefficient. Lemma 3.1. Let T > . Consider the SDE Y t = Y + a Z t ( b s − Y s ) ds + Z t σ ( Y s ) dW s + Z t Z U g ( Y s − , u ) N ( ds, du )+ Z t Z U g ( Y s − , u ) ˜ N ( ds, du ) , t ∈ [0 , T ] (3 . b where a > and b = ( b t ) t ∈ [0 ,T ] is a non-negative F -adapted c`adl`ag process. If Assumption2.2 and 2.3 hold for the coefficients ( σ, g , g , N , N ) in (3.1) b , then, the above SDE hasa non-negative F -adapted c`al`ag solution Y = ( Y t ) t ∈ [0 , T ] . We provide the proof of the lemma below. Note that in the above equation (3.1) b , thesymbol “ b ” is attached as a subscript to its label in order to emphasize the dependence ofthe equation on the drift coefficient process b . The process b could be replaced by someauxiliary processes in the following proof and the subscript will be changed accordingly. Proof. The idea is to approximate the drift coefficient b from below by a pointwise increas-ing sequence of adapted, piecewise constant processes, and use the comparison theoremfrom Gal’chuk [11], together with the monotone convergence theorem.5 tep 1: Discretization in time of the process b . For each n ∈ N + we define t n = 0, b n = b − n , and recursively for k ∈ N : t nk +1 = inf { t > t nk : b t nk − n > b t } ∧ (cid:18) t nk + 1 n (cid:19) ∧ T, (3.2) b nt = b t nk − n , t ∈ (cid:2) t nk , t nk +1 (cid:1) . Obviously, we have b nt ≤ b t for all t ∈ [0 , T ] and n ∈ N . We also define b t = 0 and¯ b nt = max { b mt : 0 ≤ m ≤ n } for all 0 ≤ t ≤ T . By definition, for any fixed positive integer n and ω ∈ Ω, t nk ( ω ) is increasing in k . We have in addition the following assertion. Claim A. For any ω ∈ Ω, one has t nk ( ω ) = T for sufficiently large k . Proof of the Claim A. We prove by contradiction. Suppose that t nk ( ω ) takes infinitelymany values, then t nk +1 ( ω ) = inf { t > t nk ( ω ) : b t nk ( ω ) ( ω ) − n > b t ( ω ) } for all large enough k . By the right continuity of the process b , we also have b t nk ( ω ) ( ω ) − n ≥ b t nk +1 ( ω ) ( ω )for all such k . Moreover, t nk ( ω ) increase to a finite limit t n ( ω ) as k → + ∞ , and since thefunction t b t ( ω ) has a left limit ℓ n ( ω ) at t n ( ω ), we have ℓ n ( ω ) = lim k → + ∞ b t nk +1 ( ω ) ( ω ) ≤ lim k → + ∞ b t nk ( ω ) ( ω ) − n = ℓ n ( ω ) − n which is a contradiction. Therefore, t nk ( ω ) only takes finitely many values in [0 , T ] when k varies. In particular, there exists ℓ n ( ω ) ∈ [0 , T ] and k ∈ N such that t nk ( ω ) = ℓ n ( ω )for any k ∈ N with k ≥ k . Note that ℓ n ( ω ) should equal T since otherwise by theright continuity of the process b we would have t nk +1 ( ω ) > t nk ( ω ), which leads again to acontradiction. Step 2. Resolution of the equation with discretized drift coefficients. Note that t nk isa stopping time for each n and each k , and if we define ¯ t nk to be the k th smallest elementof the set { t mk : k ∈ N , m ∈ { , , . . . , n }} , then for any n , { ¯ t nk } k ∈ N is an increasingsequence of stopping times, with ¯ b nt being constant on each stochastic interval of the form[[¯ t nk , ¯ t nk +1 [[. Moreover, we obtain by Claim A that, for each fixed ω ∈ Ω, ¯ t nk ( ω ) = T for alllarge enough k . Assuming that we can find a non-negative semimartingale ( Y nt ) t ∈ [[0 , ¯ t nk ]] Y nt = Y + a Z t (cid:0) ¯ b ns − Y ns (cid:1) ds + Z t σ ( Y ns ) dW s + Z t Z U g (cid:0) Y ns − , u (cid:1) N ( ds, du )+ Z t Z U g (cid:0) Y ns − , u (cid:1) ˜ N ( ds, du ) (3.3)on the stochastic interval [[0 , ¯ t nk ]], we claim that we can extend the solution to the stochasticinterval [[0 , ¯ t nk +1 ]]. Indeed, we only need to find a non-negative solution to the SDE Y nt = Y n ¯ t nk + a Z t ¯ t nk (cid:0) ¯ b ns − Y ns (cid:1) ds + Z t ¯ t nk σ ( Y ns ) dW s + Z t ¯ t nk Z U g (cid:0) Y ns − , u (cid:1) N ( ds, du )+ Z t ¯ t nk Z U g (cid:0) Y ns − , u (cid:1) ˜ N ( ds, du ) (3.4)on ]]¯ t nk , ¯ t nk +1 ]] given F ¯ t nk , in which case ¯ t nk , Y n ¯ t nk and ¯ b ns = ¯ b n ¯ t nk ≥ t nk +1 is a stopping time. This is possible by recalling the results of [10] to solve Y nt = Y n ¯ t nk + a Z t ¯ t nk (cid:16) ¯ b n ¯ t nk − Y ns (cid:17) ds + Z t ¯ t nk σ ( Y ns ) dW s + Z t ¯ t nk Z U g (cid:0) Y ns − , u (cid:1) N ( ds, du )+ Z t ¯ t nk Z U g (cid:0) Y ns − , u (cid:1) ˜ N ( ds, du ) (3.5)on ]]¯ t nk , T ]] given F ¯ t nk , and then stopping at time ¯ t nk +1 . This inductive argument defines anon-negative c`adl`ag semimartingale Y n · which solves the equation (3.1) ¯ b n on [0 , T ]. Byconstruction we have b t ≥ ¯ b n +1 t ≥ ¯ b nt ≥ t ∈ [0 , T ] and n ∈ N . Therefore, by thecomparison theorem from Gal’chuk [11, Theorem 1], we have Y n +1 t ≥ Y nt for all t ∈ [0 , T ]and n ∈ N . Step 3. Convergence of the drift coefficients and associated solutions. We begin withthe following claim. Claim B. The sequence ( Y n ) n ∈ N defined in Step 2 converges pointwise from below toan F -adapted process Y . Proof of Claim B. We first show that the sequence is pointwisely bounded from above.For this purpose, we apply the construction of Step 1 to the process − b as follows. Wedefine s = 0, ˜ b = b + 1, and recursively on k ∈ N , s k +1 = inf { t > s k : b s k + 1 < b t } ∧ ( s k + 1) ∧ T, ˜ b t = b s k + 1 , t ∈ [[ s k , s k +1 [[ . 7y definition, one has ˜ b t ≥ b t for all t ∈ [0 , ω ∈ Ω, s k ( ω ) is increasing in k and s k ( ω ) = T for sufficiently large k . By the same argument asin Step 2 , we obtain that the equation (3.1) ˜ b admits a solution, which we denote by ˜ Y .Still by the comparison theorem of [11], we deduce from the relations˜ b ≥ b ≥ ¯ b n +1 ≥ ¯ b n ≥ Y t ≥ Y n +1 t ≥ Y nt ≥ t ∈ [0 , T ] and n ∈ N . Therefore, the sequence ( Y n ) n ∈ N , n ≥ converges pointwise toa limite process Y , which is clearly F -adapted.We now show that, for any ω ∈ Ω, and any point of continuity t of the function s b s ( ω ), the sequence ¯ b nt ( ω ) converges from below to b t ( ω ) as n → + ∞ . Indeed, forany any positive integer n , there exists a k ( n ) ∈ N such that t ∈ [[ t nk ( n ) ( ω ) , t nk ( n )+1 ( ω )[[and thus | t − t nk ( n ) ( ω ) | ≤ | t nk ( n )+1 ( ω ) − t nk ( n ) ( ω ) | ≤ n , (3.6)which means that t nk ( n ) ( ω ) → t from below as n → + ∞ . Hence, by the continuity of b · ( ω )at t and the definition of b n · , we have b nt ( ω ) = b t nk ( n ) ( ω ) ( ω ) − n −→ b t ( ω ) as n → + ∞ . Recalling then that b t ( ω ) ≥ ¯ b nt ( ω ) ≥ b nt ( ω ) for all n ∈ N , we deduce that ¯ b nt ( ω ) → b t ( ω )as n → + ∞ . Step 4. Resolution of the initial equation. Finally, we will show that a c`adl`ag versionof the process Y solves (3.1) in [0 , T ] by taking n → + ∞ on (3.3) and by exploiting theconvergence results we have just obtained. For s ∈ [0 , T ], we denote by Y s − the limit ofthe increasing sequence ( Y ns − ) n ∈ N + . First, we recall the monotone convergence theoremwhich gives Z t (cid:0) ¯ b ns − Y ns (cid:1) ds = Z t ¯ b ns ds − Z t Y ns ds −→ Z t b s ds − Z t Y s ds = Z t ( b s − Y s ) ds (3.7)as n −→ + ∞ , for any t ∈ [0 , T ]. Next, for every n ∈ N , we consider a sequence { τ m,n } m ∈ N of F -stopping times such that lim m −→ + ∞ τ m,n = + ∞ and also Z T ∧ τ m,n ( σ ( Y ns ) − σ ( Y s )) ds ≤ m, Z T ∧ τ m,n Z U (cid:0) g (cid:0) Y ns − , u (cid:1) − g ( Y s − , u ) (cid:1) µ ( du ) ds ≤ m, T ∧ τ m,n Z U (cid:0) g (cid:0) Y ns − , u (cid:1) − g ( Y s − , u ) (cid:1) µ ( du ) ds ≤ m (3.8)and Z T ∧ τ m,n Z U (cid:12)(cid:12) g (cid:0) Y ns − , u (cid:1) − g ( Y s − , u ) (cid:12)(cid:12) µ ( du ) ds ≤ m (3.9)for each m ∈ N . Since Y n is increasing in n , by the monotonicity of g and Assumption 2.3,we can choose, for each m ∈ N , the F -stopping times τ m,n = τ m to be independent of n .Then, by using the Burkholder-Davis-Gundy inequality (see [7]) we have E sup t ∈ [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t ∧ τ m σ ( Y ns ) dW s − Z t ∧ τ m σ ( Y s ) dW s (cid:12)(cid:12)(cid:12)(cid:12)! = E sup t ∈ [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t ∧ τ m ( σ ( Y ns ) − σ ( Y s )) dW s (cid:12)(cid:12)(cid:12)(cid:12)! ≤ C E (cid:20)Z T ∧ τ m ( σ ( Y ns ) − σ ( Y s )) ds (cid:21) where we can recall the continuity of σ and either the monotone convergence theorem orthe dominated convergence theorem (depending on whether σ is bounded or increasing)to deduce that the RHS tends to zero as n −→ + ∞ . Next, writing ˜ N ( ds, du ) for thecompensated measure N ( ds, du ) − µ ( du ) ds , where µ ( du ) ds is the compensator of N ( ds, du ), by using the Burkholder-Davis-Gundy inequality once more we have E "(cid:18) sup t ∈ [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12) Z t ∧ τ m Z U g (cid:0) Y ns − , u (cid:1) ˜ N ( ds, du ) − Z t ∧ τ m Z U g ( Y s − , u ) ˜ N ( ds, du ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) = E "(cid:18) sup t ∈ [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12) Z t ∧ τ m Z U (cid:0) g (cid:0) Y ns − , u (cid:1) − g ( Y s − , u ) (cid:1) ˜ N ( ds, du ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ C E (cid:20)Z T ∧ τ m Z U (cid:0) g (cid:0) Y ns − , u (cid:1) − g ( Y s − , u ) (cid:1) µ ( du ) ds (cid:21) where the quantity (cid:0) g (cid:0) Y ns − , u (cid:1) − g ( Y s − , u ) (cid:1) is either monotone or bounded by 4 G ( u ),with the last being integrable due to (2.8), so by monotone or dominated convergence andby the continuity of g , the RHS of the above tends also to zero as n −→ + ∞ . Moreover,by a similar argument we have always Z t ∧ τ m Z U g (cid:0) Y ns − , u (cid:1) µ ( du ) ds −→ Z t ∧ τ m Z U g ( Y s − , u ) µ ( du ) ds (3.10)for all t ∈ [0 , T ] as n −→ + ∞ , and combining this with the previous convergence resultwe deduce that almost surely we have Z t ∧ τ m Z U g (cid:0) Y ns − , u (cid:1) N ( ds, du ) −→ Z t ∧ τ m Z U g ( Y s − , u ) N ( ds, du ) (3.11)9or all t ∈ [0 , T ] (in a subsequence). Finally, using the Burkholder-Davis-Gundy inequal-ity and the monotone convergence theorem as we did for the integral with respect to˜ N ( ds, du ), we find that E "(cid:18) sup t ∈ [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12) Z t ∧ τ m Z U g (cid:0) Y ns − , u (cid:1) ˜ N ( ds, du ) − Z t ∧ τ m Z U g ( Y s − , u ) ˜ N ( ds, du ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) tends also to zero as n −→ + ∞ . It follows that almost surely, we can take limits on bothsides of (3.3) and obtain (3.1) when t is replaced by t ∧ τ m , for any t ∈ [0 , T ]. Then, wecan finish the proof by letting m −→ + ∞ . In this section, we provide the proof of Theorem 2.4 on the existence of a strong solutionto the system of SDEs (2.1) under Assumption 2.2 and 2.3. The idea is to construct asequence of approximating solutions whose drift contains a piecewise conditional expec-tation with respect to the minimal of all pre-determined drift processes. The key lemmain the previous section allows to prove the existence of solutions for the approximatingsystem. We then use monotone convergence to establish that the limit processes solvesour system of SDEs. Proof. Without loss of generality, we show that the equation admits a solution ( λ t , · · · , λ Nt )for t ∈ [0 , T ], with λ it non-negative and E [ R T λ it dt ] < + ∞ for a given T > 0. Then thesolution can be extended to R + without difficulty. Step 1. Construction of the approximating systems and monotonicity. For n ∈ N ,we construct a partition 0 = t n < t n < . . . < t n n − = T of [0 , T ] as follows: We startwith t = 0 and t = T and, for any integer n , define inductively t n +12 j = t nj for all j ∈ { , . . . , n − } and t n +12 j +1 = ( t nj + t nj +1 ) / j ∈ { , . . . , n − − } . Next, for each i ∈ { , , ..., N } , let λ i, · to be the solution to the SDE λ i, t = λ i − a i Z t λ i, s ds + Z t σ i ( λ i, s ) dW is + Z t Z U g i, (cid:16) λ i, s − , u (cid:17) N i, ( ds, du )+ Z t Z U g i, (cid:16) λ i, s − , u (cid:17) ˜ N i, ( ds, du ) , (4.1)which exists and is unique as shown in [10]. Then, having λ i,n · defined for some n ≥ i ∈ { , , ..., N } , we define: b i,nk = inf s ∈ [ t nk , t nk +1 ] b i ( s, λ ,ns , λ ,ns , ..., λ N,ns ) (4.2)10nd λ i,n +1 · in (cid:2) t nk , t nk +1 (cid:3) for any k ∈ { , , ..., n − − } by solving the SDE λ i,n +1 t = λ i,n +1 t nk + a i Z tt nk (cid:16) E h b i,nk |F s i − λ i,n +1 s (cid:17) ds + Z tt nk σ i ( λ i,n +1 s ) dW is + Z tt nk Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) N i, ( ds, du )+ Z tt nk Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) ˜ N i, ( ds, du ) (4.3)for t ∈ (cid:2) t nk , t nk +1 (cid:3) , which also has a solution by Lemma 3.1.We will show now that for any n ≥ λ i,n +1 t ≥ λ i,nt for all i ∈ { , , ..., N } and all t ∈ [0 , T ] by induction on n . For the initial case, that is, λ i, t ≥ λ i, t , we onlyneed to recall that E [ b i,nk |F s ] ≥ b i in (4.2) is a non-negative function, andthen use the comparison theorem from [11]. Suppose now that for some n ≥ λ i,n +1 t ≥ λ i,nt for all i ∈ { , , ..., N } and t ∈ [0 , T ]. Then, by the monotonicity of each b i we have b i,n +12 k = inf s ∈ [ t n +12 k , t n +12 k +1 ] b i ( s, λ ,n +1 s , λ ,n +1 s , ..., λ N,n +1 s ) ≥ inf s ∈ [ t n +12 k , t n +12 k +1 ] b i ( s, λ ,ns , λ ,ns , ..., λ N,ns ) ≥ inf s ∈ h t n +12 k , t n +12( k +1) i b i ( s, λ ,ns , λ ,ns , ..., λ N,ns )= inf s ∈ [ t nk , t nk +1 ] b i ( s, λ ,ns , λ ,ns , ..., λ N,ns )= b i,nk (4.4)for n ≥ 1, all i ∈ { , , ..., N } and all k ∈ { , , ..., n − − } , and also b i,n +12 k +1 = inf s ∈ [ t n +12 k +1 , t n +12 k +2 ] b i ( s, λ ,n +1 s , λ ,n +1 s , ..., λ N,n +1 s ) ≥ inf s ∈ [ t n +12 k +1 , t n +12 k +2 ] b i ( s, λ ,ns , λ ,ns , ..., λ N,ns ) ≥ inf s ∈ [ t n +12 k , t n +12 k +2 ] b i ( s, λ ,ns , λ ,ns , ..., λ N,ns )= inf s ∈ [ t nk , t nk +1 ] b i ( s, λ ,ns , λ ,ns , ..., λ N,ns )= b i,nk (4.5)for n ≥ 1, all i ∈ { , , ..., N } and all k ∈ { , , ..., n − − } . We will use these twoinequalities to show that λ i,n +2 t ≥ λ i,n +1 t for all i ∈ { , , ..., N } and t ∈ [0 , T ]. This isdone by applying a second induction as follows: For t ∈ [ t n +10 , t n +11 ] = [0 , t n +11 ] ⊂ [0 , t n ]we have λ i,n +2 t = λ i,n +20 + a i Z t (cid:16) E h b i,n +10 |F s i − λ i,n +2 s (cid:17) ds + Z t σ i ( λ i,n +2 s ) dW is Z t Z U g i, (cid:16) λ i,n +2 s − , u (cid:17) N i, ( ds, du )+ Z t Z U g i, (cid:16) λ i,n +2 s − , u (cid:17) ˜ N i, ( ds, du ) (4.6)and λ i,n +1 t = λ i,n +10 + a i Z t (cid:16) E h b i,n |F s i − λ i,n +1 s (cid:17) ds + Z t σ i ( λ i,n +1 s ) dW is + Z t Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) N i, ( ds, du )+ Z t Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) ˜ N i, ( ds, du ) (4.7)and since E [ b i,n +10 |F s ] ≥ E [ b i,n |F s ] (by taking conditional expectations in (4.4) for k = 0and n replaced by n + 1) the comparison theorem implies that λ i,n +2 t ≥ λ i,n +1 t for all t ∈ [ t n +10 , t n +11 ] = [0 , t n +11 ] . Suppose now that for some k ′ ∈ { , , . . . , n − } wehave λ i,n +2 t = λ i,n +1 t for all t ∈ [ t n +10 , t n +1 k ′ ] = (cid:2) , t n +1 k ′ (cid:3) . Then for k ′ = 2 k with k ∈{ , , ..., n − − } we have t n +1 k ′ = t nk and t n +1 k ′ +1 = ( t nk + t nk +1 ) / 2, while for k ′ = 2 k + 1 with k ∈ { , , ..., n − − } we have t n +1 k ′ = ( t nk + t nk +1 ) / t n +1 k ′ +1 = t nk +1 , so in both cases itholds that [ t n +1 k ′ , t n +1 k ′ +1 ] ⊂ [ t nk , t nk +1 ] and for any t ∈ [ t n +1 k ′ , t n +1 k ′ +1 ] we have both λ n +2 t = λ i,n +2 t n +1 k ′ + a i Z tt n +1 k ′ (cid:16) E h b i,n +1 k ′ |F s i − λ i,n +2 s (cid:17) ds + Z tt n +1 k ′ σ i ( λ i,n +2 s ) dW is + Z tt n +1 k ′ Z U g i, (cid:16) λ i,n +2 s − , u (cid:17) N i, ( ds, du )+ Z tt n +1 k ′ Z U g i, (cid:16) λ i,n +2 s − , u (cid:17) ˜ N i, ( ds, du ) (4.8)and λ n +1 t = λ i,n +1 t n +1 k ′ + a i Z tt n +1 k ′ (cid:16) E h b i,nk |F s i − λ i,n +1 s (cid:17) ds + Z tt n +1 k ′ σ i ( λ i,n +1 s ) dW is + Z tt n +1 k ′ Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) N i, ( ds, du )+ Z tt n +1 k ′ Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) ˜ N i, ( ds, du ) (4.9)with E h b i,n +1 k ′ |F s i ≥ E h b i,nk |F s i (by taking expectations given F s in (4.4) and (4.5)).Thus, the comparison theorem implies that λ i,n +2 t ≥ λ i,n +1 t for all t ∈ (cid:2) t n +1 k ′ , t n +1 k ′ +1 (cid:3) , whichmeans that the same inequality holds for all t ∈ (cid:2) t n +10 , t n +1 k ′ +1 (cid:3) ≡ (cid:2) , t n +1 k ′ +1 (cid:3) . This completesthe second induction and gives λ i,n +2 t ≥ λ i,n +1 t for all t ∈ [0 , T ], and the last completesthe initial induction giving λ i,n +1 t ≥ λ i,nt for all t ∈ [0 , T ] and all n ≥ Step 2. Finiteness of the monotone limits. We have shown in the previous step thatthe family of processes { λ i,nt } t ∈ [0 ,T ] is pointwise increasing in n , we will show that almost12urely, lim n −→ + ∞ λ i,nt is finite for almost all t ∈ [0 , T ] and every i ∈ { , , . . . , N } . This willfollow by Fatou’s lemma if we can show thatsup ≤ i ≤ N E (cid:20)Z T λ i,nt dt (cid:21) (4.10)is bounded in n ∈ N . For the last, we recall that by the Lipschitz property of each b i ,there exist constants B, L > b i (cid:0) s, λ ,ns , λ ,ns , ..., λ N,ns (cid:1) ≤ B + L N X i =1 λ i,ns (4.11)for every i ∈ { , , . . . , N } , so taking infimum on the LHS for s ∈ (cid:2) t nk , t nk +1 (cid:3) and thenconditioning on F s we obtain E h b i,nk | F s i ≤ B + L N X i =1 λ i,ns (4.12)for all s ∈ (cid:2) t nk , t nk +1 (cid:3) and i ∈ { , , ..., N } . Plugging the above in (4.3), localizing ifneeded, taking expectations and then supremum in i and finally using (2.5), we caneasily getsup ≤ i ≤ N E h λ i,n +1 t i ≤ sup ≤ i ≤ N E h λ i,n +1 t nk i + ¯ aB ( t − t nk ) + ¯ aLN Z tt nk sup ≤ i ≤ N E (cid:2) λ i,ns (cid:3) ds + K Z tt nk sup ≤ i ≤ N E (cid:2) λ i,ns (cid:3) + 1 ! ds (4.13)for ¯ a := sup ≤ i ≤ N a i , which can be written assup ≤ i ≤ N E h λ i,n +1 t i ≤ sup ≤ i ≤ N E h λ i,n +1 t nk i + B ′ ( t − t nk ) + L ′ Z tt nk sup ≤ i ≤ N E (cid:2) λ i,ns (cid:3) ds (4.14)for B ′ = ¯ aB + K and L ′ = ¯ aLN + K , so replacing k with k ′ < k and taking t = t nk ′ +1 weget also sup ≤ i ≤ N E h λ i,n +1 t nk ′ +1 i ≤ sup ≤ i ≤ N E h λ i,n +1 t nk ′ i + B ′ ( t nk ′ +1 − t nk ′ ) + L ′ Z t nk ′ +1 t nk ′ sup ≤ i ≤ N E (cid:2) λ i,ns (cid:3) ds. (4.15)Summing (4.14) with (4.15) for k ′ ∈ { , , ..., k − } we obtainsup ≤ i ≤ N E h λ i,n +1 t i ≤ sup ≤ i ≤ N E (cid:2) λ i (cid:3) + B ′ t + L ′ Z t sup ≤ i ≤ N E (cid:2) λ i,ns (cid:3) ds (4.16)13nd since k was arbitrary, the above holds for any t ∈ [0 , T ]. Take now a constant M > ≤ i ≤ N E h λ i, t i ≤ M for all t ∈ [0 , T ], which is possible by recalling the estimate(2 . M is large enough, we will show by induction on n thatsup ≤ i ≤ N E h λ i,nt i ≤ M e L ′ t (4.17)for all n ∈ N and t ∈ [0 , T ]. The base case is trivial, and if M is large enough such that M > sup ≤ i ≤ N E (cid:2) λ i (cid:3) + B ′ T , plugging sup ≤ i ≤ N E (cid:2) λ i,ns (cid:3) ≤ M e L ′ s in (4.16) we find thatsup ≤ i ≤ N E h λ i,n +1 t i ≤ sup ≤ i ≤ N E (cid:2) λ i (cid:3) + B ′ t + L ′ M Z t e L ′ s ds = sup ≤ i ≤ N E (cid:2) λ i (cid:3) + B ′ t + M e L ′ t − M ≤ M e L ′ t (4.18)which completes the induction. Integrating then (4.17) for t ∈ [0 , T ] we obtain the desiredboundedness. Step 3. Limit processes as solution to the system (2.1) and positivity. Now that wehave the pointwise monotone convergence of n λ i,nt o t ∈ [0 , T ] to a finite process (cid:8) λ it (cid:9) t ∈ [0 , T ] for all i ∈ { , , ..., N } , we will show that these limiting processes solve our system ofSDEs. The first step is to fix an i ∈ { , , ..., N } , and for each n ∈ N and s ∈ [0 , T ]take k n ( s ) ∈ { , , ..., n − − } such that s ∈ h t nk n ( s ) , t nk n ( s )+1 i . Obviously, if we take s n ∈ h t nk n ( s ) , t nk n ( s )+1 i for all n ∈ N , we will have s n −→ s as n −→ + ∞ since | s n − s | ≤| t nk n ( s ) − t nk n ( s )+1 | = O (2 − n ). Taking s ∈ D with D denoting the set of points where λ j,n · is continuous for all j and n , for an arbitrary ǫ > b i ( s n , λ ,ns n , λ ,ns n , ..., λ N,ns n ) − ǫ ≤ b i,nk n ( s ) ≤ b i ( s, λ ,ns , λ ,ns , ..., λ N,ns ) (4.19)for some s n ∈ h t nk n ( s ) , t nk n ( s )+1 i (by the definition of infimum). For an m ∈ N , recallingthe pointwise monotonicity of each λ i,n · in n ∈ N and the monotonicity of each mean-fieldfunction b i in each of its arguments, the previous double inequality easily gives b i ( s n , λ ,ms n , λ ,ms n , ..., λ N,ms n ) − ǫ ≤ b i,nk n ( s ) ≤ b i ( s, λ ,ns , λ ,ns , ..., λ N,ns ) (4.20)for all n ≥ m . Since s ∈ D , taking n −→ + ∞ in the above and recalling that each b i iscontinuous, we obtain b i ( s, λ ,ms , λ ,ms , ..., λ N,ms ) − ǫ ≤ lim inf n −→ + ∞ b i,nk n ( s ) ≤ lim sup n −→ + ∞ b i,nk n ( s ) ≤ b i ( s, λ s , λ s , ..., λ Ns ) . (4.21)Taking now m −→ + ∞ we get b i ( s, λ s , λ s , ..., λ Ns ) − ǫ ≤ lim inf n −→ + ∞ b i,nk n ( s ) ≤ lim sup n −→ + ∞ b i,nk n ( s ) ≤ b i ( s, λ s , λ s , ..., λ Ns ) . ǫ > n −→ + ∞ b i,nk n ( s ) = b i ( s, λ s , λ s , ..., λ Ns ),where the convergence is obviously monotone. Next, for any t ∈ [0 , T ], recalling (4.3)and that for all k ∈ { , , ..., n − − } we have k = k n ( s ) for all s ∈ (cid:2) t nk , t nk +1 (cid:3) , for any i ∈ { , , ..., N } we can write λ i,n +1 t = λ i + k n ( t ) − X k =0 (cid:16) λ i,n +1 t nk +1 − λ i,n +1 t nk (cid:17) + (cid:16) λ i,n +1 t − λ i,n +1 t nkn ( t ) (cid:17) = λ i + a i k n ( t ) − X k =0 Z t nk +1 t nk (cid:16) E h b i,nk n ( s ) |F s i − λ i,n +1 s (cid:17) ds + a i Z tt nkn ( t ) (cid:16) E h b i,nk n ( s ) |F s i − λ i,n +1 s (cid:17) ds + k n ( t ) − X k =0 Z t nk +1 t nk σ i ( λ i,n +1 s ) dW is + Z tt nkn ( t ) σ i ( λ i,n +1 s ) dW is + k n ( t ) − X k =0 Z t nk +1 t nk Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) N i, ( ds, du )+ Z tt nkn ( t ) Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) N i, ( ds, du )+ k n ( t ) − X k =0 Z t nk +1 t nk Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) ˜ N i, ( ds, du )+ Z tt nkn ( t ) Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) ˜ N i, ( ds, du )= λ i + a i Z t (cid:16) E h b i,nk n ( s ) |F s i − λ i,n +1 s (cid:17) ds + Z t σ i ( λ i,n +1 s ) dW is + Z t Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) N i, ( ds, du )+ Z t Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) ˜ N i, ( ds, du ) (4.23)and taking n −→ + ∞ in the above for all i we derive the desired system of SDEs satisfiedby the limiting processes (cid:8) λ i · : i ∈ { , , ..., N } (cid:9) . Indeed, since [0 , T ] /D is obviously acountable random subset of [0 , T ], the monotone convergence theorem gives Z t (cid:16) E h b i,nk n ( s ) |F s i − λ i,n +1 s (cid:17) ds = Z t E h b i,nk n ( s ) |F s i ds − Z t λ i,n +1 s ds −→ Z t E (cid:2) b i ( s, λ s , λ s , ..., λ Ns ) |F s (cid:3) ds − Z t λ is ds = Z t (cid:0) b i ( s, λ s , λ s , ..., λ Ns ) − λ is (cid:1) ds, (4.24)15nd then we have E sup t ∈ [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t ∧ τ m σ i (cid:0) λ i,n +1 s (cid:1) dW is − Z t ∧ τ m σ i (cid:0) λ is (cid:1) dW is (cid:12)(cid:12)(cid:12)(cid:12)! = E sup t ∈ [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t ∧ τ m (cid:0) σ i (cid:0) λ i,n +1 s (cid:1) − σ i (cid:0) λ is (cid:1)(cid:1) dW is (cid:12)(cid:12)(cid:12)(cid:12)! ≤ C E (cid:20)Z T ∧ τ m (cid:0) σ i (cid:0) λ i,n +1 s (cid:1) − σ i (cid:0) λ is (cid:1)(cid:1) ds (cid:21) and for j ∈ { , } also E "(cid:18) sup t ∈ [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12) Z t ∧ τ m Z U j g i,j (cid:16) λ i,n +1 s − , u (cid:17) ˜ N i,j ( ds, du ) − Z t ∧ τ m Z U j g i,j (cid:0) λ is − , u (cid:1) ˜ N i,j ( ds, du ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) = E "(cid:18) sup t ∈ [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12) Z t ∧ τ m Z U j (cid:16) g i,j (cid:16) λ i,n +1 s − , u (cid:17) − g i,j (cid:0) λ is − , u (cid:1)(cid:17) ˜ N i,j ( ds, du ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ C E "Z T ∧ τ m Z U j (cid:16) g i,j (cid:16) λ i,n +1 s − , u (cid:17) − g i,j (cid:0) λ is − , u (cid:1)(cid:17) µ i,j ( du ) ds by the Burkholder-Davis-Gundy inequality (see [7]), with the sequence { τ m } m ∈ N of stop-ping times selected as in the proof of Lemma 3.1 to ensure that the RHS in the lasttwo estimates is finite for all n ∈ N , and these RHS tending to zero by the monotonepointwise convergence of λ i,n · to λ i · , the continuity of σ i and g i,j , the monotonicity orboundedness of these functions and the corresponding convergence theorem. Finally, asimilar argument shows that Z t ∧ τ m Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) µ i, ( du ) ds −→ Z t ∧ τ m Z U g i, (cid:0) λ is − , u (cid:1) µ i, ( du ) ds (4.25)surely for all t ∈ [0 , T ] as n −→ + ∞ , and combining this with the previous convergenceresult for the integral with respect to ˜ N i, we deduce that almost surely we have Z t ∧ τ m Z U g i, (cid:16) λ i,n +1 s − , u (cid:17) N i, ( ds, du ) −→ Z t ∧ τ m Z U g i, (cid:0) λ is − , u (cid:1) N i, ( ds, du ) (4.26)as n −→ + ∞ for any t ∈ [0 , T ]. The desired system of SDEs is obtained by observingthat almost surely we have t = t ∧ τ m for large enough m . The proof is now completesince for every i and all t ≥ λ it ≥ λ i, t with λ i, t being non-negativein the one-dimensional case, and since we can integrate (4.17) and use Fatou’s lemma todeduce that λ i · is L - integrable for each i . Acknowledgement The second-named author’s work was supported financially by the Boya PostdoctoralFellowship of Peking University, and by the Beijing International Center for MathematicalResearch (BICMR). 16 eferences [1] Abdelghani, M. and Melnikov, A. A comparison theorem for stochastic equations ofoptional semimartingales, Stoch. Dyn. , , (2018), 1850029, 21.[2] Bo, L. and Capponi, A. Systemic risk in interbanking networks. SIAM J. FinancialMath. , (2015), 386–424[3] Bass, F. R. Stochastic differential equations driven by symmetric stable processes. S´eminaire de probabilit´es de Strasbourg , (2002), 302–313[4] Dawson, D. A. and Li, Z. Skew convolution semigroups and affine Markov processes. Ann. Probab. , (2006), 1103–1142[5] Carmona, R. and Delarue, F. Probabilistic theory of mean field games with applica-tions I: Mean field FBSDEs, control, and games . Probability Theory and StochasticModeling. Springer, Cham (2018).[6] Carmona, R. and Delarue, F. Probabilistic theory of mean field games with appli-cations II: Mean field games with common noise and master equations . ProbabilityTheory and Stochastic Modeling. Springer, Cham (2018).[7] Cohen, S. N. and Elliot, R. J. Stochastic Calculus and Applications . Second edition.Probability and Its Applications. Birkhauser, New York, NY (2015).[8] Fouque, J. P. and Ichiba, T. Stability in a model of interbank lending. SIAM J.Financial Math. , (2013), 784–803.[9] Frikha, N. and Li, L. Well-posedness and approximation of some one-dimensionalL´evy-driven non-linear SDEs. Stochastic Process. Appl. , (2021), 76–107.[10] Fu, Z. and Li, Z. Stochastic equations of non-negative processes with jumps. Stochas-tic Process. Appl. , (2010), 306–330.[11] Gal’chuk, L. I. A Comparison Theorem for Stochastic Equations with Integrals withRespect to Martingales and Random Measures. Theor. Probab. Appl. , (1983),450–460.[12] Giesecke, K.; Spiliopoulos, K and Sowers, R. B. Default clustering in large portfolios:typical events. Ann. Appl. Probab. , (2013), 348–385.[13] Hambly, B. and Kolliopoulos, N. Stochastic evolution equations for large portfoliosof stochastic volatility models. SIAM J. Financial Math. , (2017), 962–1014.[14] Hambly, B. and Kolliopoulos, N. Erratum: Stochastic evolution equations for largeportfolios of stochastic volatility models. SIAM J. Financial Math. , (2019), 857–876.[15] Jiao, Y.; Ma, C. and Scotti, S. Alpha-CIR model with branching processes insovereign interest rate modeling. Finance Stoch. , (2017), 789–813.[16] Li, Z. and Mytnik, L. Strong solutions for stochastic differential equations withjumps. Ann. Inst. Henri Poincar´e Probab. Stat. ,47(4)