Fluctuation Analysis for the Loss From Default
Konstantinos Spiliopoulos, Justin A. Sirignano, Kay Giesecke
aa r X i v : . [ m a t h . P R ] F e b FLUCTUATION ANALYSIS FOR THE LOSS FROM DEFAULT
KONSTANTINOS SPILIOPOULOS, JUSTIN A. SIRIGNANO, AND KAY GIESECKE
Abstract.
We analyze the fluctuation of the loss from default around its large portfolio limit in a class ofreduced-form models of correlated firm-by-firm default timing. We prove a weak convergence result for thefluctuation process and use it for developing a conditionally Gaussian approximation to the loss distribution.Numerical results illustrate the accuracy and computational efficiency of the approximation. Introduction
Reduced-form point process models of correlated firm-by-firm default timing are widely used to measurethe credit risk in portfolios of defaultable assets such as loans and corporate bonds. In these models,defaults arrive at intensities governed by a given system of stochastic differential equations. Computing thedistribution of the loss from default in these models is often challenging. Transform methods are tractableonly in special cases, for example, if defaults are assumed to be conditionally independent. Monte Carlosimulation methods have a much wider scope but can be slow for the large portfolios and longer timehorizons common in industry practice. A major US bank might easily have 20 ,
000 wholesale loans, 50 , ,
000 mid-market and commercial loans, and derivatives trades with 10 ,
000 to 20 ,
000 different legalentities. Simulation of such large pools is extremely burdensome.This paper develops a conditionally Gaussian approximation to the distribution of the loss from defaultin large pools. The approximation applies to a broad class of empirically motivated reduced-form models inwhich a name defaults at an intensity that is influenced by an idiosyncratic risk factor process, a systematicrisk factor process X common to all names in the pool, and the portfolio loss. It is based on an analysisof the fluctuation of the loss around its large portfolio limit, i.e., a central limit theorem (CLT). Moreprecisely, we show that the signed-measure-valued process describing the fluctuation of the loss around itslaw of large numbers (LLN) limit weakly converges to a unique distribution-valued process in a certainweighted Sobolev space. The limiting fluctuation process satisfies a stochastic evolution equation driven bythe Brownian motion governing the systematic risk X and a distribution-valued martingale that is centeredGaussian given X . The fluctuation limit, and thus the resulting approximation to the loss distribution, isGaussian only in the special case that the names in the pool are not sensitive to X . In the general case, theapproximation is conditionally Gaussian.The weak convergence result proven in this paper extends the LLNs for the portfolio loss establishedin our earlier work Giesecke, Spiliopoulos & Sowers (2013) and Giesecke, Spiliopoulos, Sowers & Sirignano(2012). The fluctuation analysis performed here involves challenging topological issues that do not arise inthe analysis of the LLN. Firstly, the fluctuations process takes values in the space of signed measures. Thisspace is not well suited for the study of convergence in distribution; the weak topology, which is the naturaltopology to consider here, is in general not metrizable (see, for example, Del Barrio, Deheuvels & Geer(2007)). We address this issue by analyzing the convergence of the fluctuations as a process taking values inthe dual of a suitable Sobolev space of test functions. Weights are introduced in order to control the normof the fluctuations and establish tightness. Related ideas appear in M´etivier (1985), Fernandez & M´el`eard(1997), Kurtz & Xiong (2004) and other articles, but for other systems and under different assumptions.Additional issues are the growth and degeneracies of the coefficients of our stochastic system, which makeit difficult to identify the appropriate weights for the Sobolev space in which the convergence is establishedand uniqueness of the limiting fluctuation process is proved. Purtukhia (1984), Gy¨ongy & Krylov (1990), Date : September 18, 2018. We are grateful to Andrew Abrahams, Jose Blanchet, Terry Lyons, and Marek Musiela forinsightful comments. We also thank participants of seminars at Columbia University, University of Michigan, Rutgers University,the Oxford-Man Institute, the University of Southern California, the Office of Financial Research, the SIAM Conference onFinancial Mathematics and Engineering, and the INFORMS Annual Meeting for comments. rylov & Lototsky (1998), Lototsky (2001), Kim (2009), and others face similar issues in settings that aredifferent from ours. The approach we pursue is inspired by the class of weights that were introduced byPurtukhia (1984) and further analyzed by Gy¨ongy & Krylov (1990).The fluctuation analysis leads to a second-order approximation to the loss distribution that can be sig-nificantly more accurate than an approximation obtained from just the LLN. Our numerical results, whichare based on a method of moments for solving the stochastic evolution equation governing the fluctuationlimit, confirm that the second-order approximation is much more accurate. The second-order approximationis even accurate for relatively small portfolios in the order of hundreds of names or when the influence of thesystematic risk process X is relatively low. The second-order approximation also improves the accuracy inthe tails. The effort of computing the second-order approximation exceeds that of computing the first-orderapproximation, but is still much lower than directly simulating the pool.Prior research has established various weak convergence results for interacting particle systems representedby reduced-form models of correlated default timing. Dai Pra, Runggaldier, Sartori & Tolotti (2009) andDai Pra & Tolotti (2009) study mean-field models in which an intensity is a function of the portfolio loss. In amodel with local interaction, Giesecke & Weber (2006) take the intensity of a name as a function of the stateof the names in a specified neighborhood of that name. In these formulations, the impact of a default on thedynamics of the surviving constituent names, a contagion effect, is permanent. In our mean-field model, anintensity depends on the path of the portfolio loss. Therefore, the impact of a default may be transient andfade away with time. There is a recovery effect. Moreover, the system which we analyze includes firm-specificsources of default risk and addresses an additional source of default clustering, namely the exposure of aname to a systematic risk factor process. This exposure generates a random limiting behavior for the LLNand leads to a fluctuation limit which is only conditionally Gaussian.There are several other related articles. Cvitani´c, Ma & Zhang (2012) prove a LLN for a mean-field systemwith permanent default impact, taking an intensity as a function of an idiosyncratic risk factor, a systematicrisk factor, and the portfolio loss rate. The risk factors follow diffusion processes whose coefficients maydepend on the portfolio loss. Bush, Hambly, Haworth, Jin & Reisinger (2011) establish a LLN for a systemin which defaults occur at first hitting times of correlated diffusion processes. Conditional on a correlatingsystematic risk factor governed by a Brownian motion, defaults occur independently of one another. Garnier,Papanicolaou & Yang (2012) analyze large deviations in a system of diffusion processes that interact throughtheir empirical mean and have a stabilizing force acting on each of them. Fouque & Ichiba (2012) studydefaults and stability in an interacting diffusion model of inter-bank lending.The rest of this paper is organized as follows. Section 2 describes the class of reduced-form models ofcorrelated default timing which we analyze. Section 3 reviews the law of large numbers proved by Gieseckeet al. (2012) for the portfolio loss in these models. Section 4 states our main weak convergence result for thefluctuation process, essentially a central limit theorem for the loss. Section 5 provides a numerical methodfor solving the stochastic evolution equation governing the fluctuation limit and provides numerical results.Sections 6-9 are devoted to the proof of the main result. There is an appendix.2. Model and Assumptions
We analyze a class of reduced-form point process models of correlated default timing in a pool of firms(“names”). Models of this type have been studied by Cvitani´c et al. (2012), Dai Pra et al. (2009), Dai Pra& Tolotti (2009), Giesecke et al. (2013), Giesecke et al. (2012), and others. In these models, a default isgoverned by the first jump of a point process. The point process intensities follow a system of SDEs.Fix a probability space (Ω , F , P ). Let { W n } n ∈ N be a countable collection of independent standardBrownian motions. Let { e n } n ∈ N be an i.i.d. collection of standard exponential random variables which areindependent of the W n ’s. Finally, let V be a standard Brownian motion which is independent of the W n ’s and e n ’s. Each W n will represent a source of risk which is idiosyncratic to a specific name. Each e n will representa normalized default time for a specific name. The process V will drive a systematic risk factor processto which all names are exposed. Define V t = σ ( V s , ≤ s ≤ t ) ∨ N and F nt = σ (( V s , W ns ) , ≤ s ≤ t ) ∨ N ,where N contains the P -null sets. Lastly, we will denote by P Y the conditional law given Y , where Y mayrepresent V t , for example. ix N ∈ N , n ∈ { , , . . . , N } and consider the following system:(1) dλ N,nt = − α N,n ( λ N,nt − ¯ λ N,n ) dt + σ N,n q λ N,nt dW nt + β CN,n dL Nt + β SN,n λ N,nt dX t , λ N,n = λ ◦ ,N,n dX t = b ( X t ) dt + σ ( X t ) dV t , X = x ◦ τ N,n = inf (cid:26) t ≥ Z t λ N,ns ds ≥ e n (cid:27) L Nt = 1 N N X n =1 χ { τ N,n ≤ t } . Here, χ is the indicator function. The initial value x ◦ of X is fixed. The α N,n , ¯ λ N,n , σ
N,n , β
CN,n , β
SN,n , λ ◦ ,N,n are parameters. The process L N is the fraction of names in default, which we loosely call the “loss rate” orsimply the “loss.” The process λ N,n represents the intensity, or conditional default rate, of the n -th namein the pool. More precisely, λ N,n is the density of the Doob-Meyer compensator to the default indicator1 − m N,nt , where m N,nt = χ { τ N,n >t } . That is, a martingale is given by m N,nt + R t λ N,ns m N,ns ds.
The results inSection 3 of Giesecke et al. (2013) imply that the system (1) has a unique solution such that λ N,nt ≥ N ∈ N , n ∈ { , , . . . , N } and t ≥
0. Thus, the model is well-posed.The default timing model (1) addresses several channels of default clustering. An intensity is influencedby an idiosyncratic source of risk represented by a Brownian motion W n , and a source of systematic riskcommon to all firms–the diffusion process X . Movements in X cause correlated changes in firms’ intensitiesand thus provide a channel for default clustering emphasized by Das, Duffie, Kapadia & Saita (2007) forcorporate defaults in the U.S. The sensitivity of λ N,n to changes in X is measured by the parameter β SN,n ∈ R .The second channel for default clustering is modeled through the feedback (“contagion”) term β CN,n dL Nt .A default causes a jump of size β CN,n /N in the intensity λ N,n , where β CN,n ∈ R + = [0 , ∞ ). Due to themean-reversion of λ N,n , the impact of a default fades away with time, exponentially with rate α N,n ∈ R + .Azizpour, Giesecke & Schwenkler (2010) have found self-exciting effects of this type to be an importantchannel for the clustering of defaults in the U.S., over and above any clustering caused by the exposure offirms to systematic risk factors.Figure ?? illustrates the behavior of the system when the systematic risk X follows an OU process. Itshows sample paths of the processes λ N,n m N,n and L N for a pool with N = 4 names. Between defaults, theintensities λ N,n of the surviving names evolve as correlated diffusion processes, where the co-movement isdriven by the systematic risk X . At a default, the process λ N,n m N,n associated with the defaulting namedrops to 0. At the same time, the processes λ N,n m N,n associated with the surviving names jump by 1 / L N increases by 1 / p N,n def = ( α N,n , ¯ λ N,n , σ
N,n , β
CN,n , β
SN,n ) . The p N,n ’s take values in parameter space P def = R × R . For each N ∈ N , defineˆ p N,nt def = ( p N,n , λ
N,nt )for all n ∈ { , , . . . , N } and t ≥
0. Define ˆ P def = P × R + . The vector ˆ p N = (ˆ p N, , . . . , ˆ p N,N ) represents arandom environment that addresses the heterogeneity of the system. Condition 2.1.
We assume that the ˆ p N,n are i.i.d. random variables with common law ν and that ν hascompact support in ˆ P . In particular, and to avoid inessential complications, we assume that all elements ofthe random vector p N,n are bounded in absolute values by some K , for all n ∈ { , , . . . , N } . Moreover, weassume that { ˆ p N,n } is independent of { W n } , { e n } and V . Giesecke & Schwenkler (2011) develop and analyze likelihood estimators of the parameters of point process models such as(1), given a realization of L N over some sample period. The term “loss rate” can be taken literally if a name corresponds to a unit-notional position in the portfolio and the recoveryat default is 0. his formulation of heterogeneity generalizes that of Giesecke et al. (2012). They assume that π =lim N →∞ N P Nn =1 δ p N,n and Λ ◦ = lim N →∞ N P Nn =1 δ λ ◦ ,N,n exist (in P ( P ) and P ( R + ), respectively). Theirformulation is a special case of the one proposed here, where the law ν = π × Λ ◦ . Condition 2.2.
We assume that E R t (cid:2) | b ( X s ) | + | σ ( X s ) | (cid:3) ds < ∞ for all t ≥ . Law of large numbers
Except for special cases of little practical interest, the distribution of the portfolio loss L N in the system(1) is difficult to compute. We are interested in an approximation to this distribution for the large portfolioscommon in practice, i.e., for the case that N is large. Giesecke et al. (2012) prove a law of large numbers(LLN) for the loss in the system (1) and use it for developing a first-order approximation. We first reviewthis result and then extend it in Section 4 by analyzing the fluctuations of the loss around its LLN limit.The fluctuation analysis will allow us to construct a more accurate second-order approximation.To outline the LLN, define µ Nt def = 1 N N X n =1 δ ˆ p N,nt m N,nt ;this is the empirical distribution of the type and intensity for those names which are still “alive.” We notethat µ Nt is a sub-probability measure. Since L Nt = 1 − µ Nt ( ˆ P ) , (2)it suffices to study the limiting behavior of the measure-valued process { µ Nt , t ∈ [0 , T ] } N ∈ N for some fixedhorizon T >
0. Let E be the collection of sub-probability measures (i.e., defective probability measures) onˆ P ; i.e., E consists of those Borel measures ν on ˆ P such that ν ( ˆ P ) ≤
1. Topologizing E in the usual way (byprojecting onto the one-point compactification of ˆ P ; see Chapter 9.5 of Royden (1988)) we obtain that E isa Polish space. Thus, µ N is an element of D E [0 , ∞ ) where D is the Skorokhod space (i.e., D E [0 , ∞ ) is theset of RCLL processes on [0 , ∞ ) taking values in E ).Further, for ˆ p = ( p , λ ) where p = ( α, ¯ λ, σ, β C , β S ) ∈ P and f ∈ C b ( ˆ P ) (the space of twice continuouslydifferentiable, bounded functions), define, similarly to (Giesecke et al. 2013), the operators ( L f )(ˆ p ) = 12 σ λ ∂ f∂λ (ˆ p ) − α ( λ − ¯ λ ) ∂f∂λ (ˆ p ) − λf (ˆ p )( L f )(ˆ p ) = β C ∂f∂λ (ˆ p )( L x f )(ˆ p ) = β S λb ( x ) ∂f∂λ (ˆ p ) + 12 ( β S ) λ σ ( x ) ∂ f∂λ (ˆ p )( L x f )(ˆ p ) = β S λσ ( x ) ∂f∂λ (ˆ p ) . Also define Q (ˆ p ) def = λ. The generator L corresponds to the diffusive part of the intensity with killing rate λ , and L is the macro-scopic effect of contagion on the surviving intensities at any given time. The operators L x and L x are relatedto the exogenous systematic risk X . For a measure ν ∈ D E [0 , ∞ ), we also specify the inner product (cid:10) f, ν (cid:11) E = Z ˆ P f (ˆ p ) dν (ˆ p ) . The law of large numbers of Giesecke et al. (2012) states that µ Nt weakly converges to ¯ µ t in D E [0 , T ]. Torigorously formulate the result we need to use the weak form (see also Lemma 8.3). In particular, for all At this point we would like to remark that there is a typo in the formulation of the corresponding operators in (Gieseckeet al. 2013). In particular, it is mentioned there that ( L f )( p ) = ∂f∂λ ( p ) and Q ( p ) = β C λ , where it should have been ( L f )( p ) = β C ∂f∂λ ( p ) and Q ( p ) = λ . ∈ C b ( ˆ P ), the evolution of ¯ µ · is governed by the measure evolution equation(3) d h f, ¯ µ t i E = n hL f, ¯ µ t i E + hQ , ¯ µ t i E hL f, ¯ µ t i E + D L X t f, ¯ µ t E E o dt + D L X t f, ¯ µ t E E dV t , a.s.The LLN suggests an approximation to the distribution of the loss L N in large pools by the large pool limit: L Nt d ≈ L t = 1 − ¯ µ t ( ˆ P ) . (4) 4. Main Result: Fluctuations Theorem
In order to improve the first-order approximation (4), we analyze the fluctuations of µ N around its largepool limit ¯ µ . As is indicated by the proof of Lemmas 8.4 and 8.5 (see also Giesecke et al. (2012)), for anappropriate metric, the sequence {√ N ( µ Nt − ¯ µ t ) : N < ∞} is stochastically bounded. Hence, it is reasonableto define the scaled fluctuation process Ξ N by(5) Ξ Nt = √ N ( µ Nt − ¯ µ t ) . In Theorem 4.1 below we will show that the signed-measure-valued process Ξ N weakly converges to afluctuation limit process ¯Ξ in an appropriate space.The analysis of the limiting behavior of the fluctuation process (5) involves issues that do not occur inthe treatment of the LLN. In particular, even though the fluctuation process is a signed-measure-valuedprocess, its limit process { ¯Ξ · } is distribution-valued in an appropriate space. The space of signed measuresendowed with the weak topology is not metrizable. The difficulty is then to identify a rich enough space,where tightness and uniqueness can be proven. It turns out that we have to consider the convergence inweighted Sobolev spaces. Here, several technical challenges arise. These are mainly due to the growth anddegeneracies of the coefficients of the system (1), which make it difficult to identify the correct weights. Thespaces that we consider are Hilbert spaces. For the sake of clarity of presentation the appropriate Hilbertspaces will be defined in detail in Section 7. For the moment, we mention that the space in question isdenoted by W J ( w, ρ ), with w and ρ the appropriate weight functions, J ∈ N and W − J ( w, ρ ) will be its dual.A precise definition is given in Section 7.We need to introduce several additional operators to state our weak convergence result. For ˆ p = ( p , λ ) ∈ ˆ P ⊂ R and f ∈ C b ( ˆ P ), we define( G x,µ f )(ˆ p ) = ( L f )(ˆ p ) + ( L x f )(ˆ p ) + hQ , µ i E ( L f )(ˆ p ) + hL f, µ i E Q (ˆ p )( L ( f, g ))(ˆ p ) = σ ∂f∂λ (ˆ p ) ∂g∂λ (ˆ p ) λ ( L ( f, g ))(ˆ p ) = f (ˆ p ) g (ˆ p ) λ ( L f )(ˆ p ) = f (ˆ p ) λ The main result of this paper is the following theorem.
Theorem 4.1.
Let D = dim ( ˆ P )2 = 3 . For J > large enough (in particular for J > D + 1 ) and forweight functions ( w, ρ ) such that Condition 7.1 holds, the sequence { Ξ Nt , t ∈ [0 , T ] } N ∈ N is relatively compact In general, a topological space is Polish if and only if its topology can be defined by a metric for which it is completeand separable. A locally compact space is metrizable if and only if it has a countable base in which it is Polish. The weakconvergence of probability measures (i.e., finite non-negative measures) on a separable metric space can be defined by theProkhorov metric. Thus, the weak topology on the space of probability measures on a locally compact space E is metrizableif and only if E is Polish. However, issues arise if one replaces the space of finite non-negative measures with the space offinite signed measures. The space of bounded and continuous functions on E , endowed with the sup-norm is a Banach space.Its topological dual endowed with the weak ∗ -topology coincides with the set of finite signed measures on E endowed withthe weak topology. However, the topological dual of an infinite dimensional Banach space is not metrizable, even though, bythe Banach-Alaoglu theorem, any weak ∗ -compact subset of such a topological dual will be metrizable. For a more thoroughdiscussion of these issues, see Del Barrio et al. (2007), Remark 1.2. n D W − J ( w,ρ ) [0 , T ] . For any subsequence of this sequence, there exists a subsubsequence that converges indistribution with limit { ¯Ξ t , t ∈ [0 , T ] } . Any accumulation point ¯Ξ satisfies the stochastic evolution equation (6) (cid:10) f, ¯Ξ t (cid:11) = (cid:10) f, ¯Ξ (cid:11) + Z t (cid:10) G X s , ¯ µ s f, ¯Ξ s (cid:11) ds + Z t D L X s f, ¯Ξ s E dV s + (cid:10) f, ¯ M t (cid:11) , a.s.for any f ∈ W J ( w, ρ ) , where ¯ M is a distribution-valued martingale with predictable variation process [ (cid:10) f, ¯ M (cid:11) ] t = Z t h hL ( f, f ) , ¯ µ s i + hL ( f, f ) , ¯ µ s i + hL f, ¯ µ s i hQ , ¯ µ s i − hL f, ¯ µ s i hL f, ¯ µ s i i ds. Moreover, conditional on the σ -algebra V t , ¯ M t is centered Gaussian with covariance function, for f, g ∈ W J ( w, ρ ) , given by Cov h(cid:10) f, ¯ M t (cid:11) , (cid:10) g, ¯ M t (cid:11) (cid:12)(cid:12)(cid:12) V t ∨ t i = E (cid:20) Z t ∧ t [ hL ( f, g ) , ¯ µ s i + hL ( f, g ) , ¯ µ s i + hL f, ¯ µ s i hL g, ¯ µ s i hQ , ¯ µ s i− hL g, ¯ µ s i hL f, ¯ µ s i − hL f, ¯ µ s i hL g, ¯ µ s i ] ds (cid:12)(cid:12)(cid:12) V t ∨ t (cid:21) . (7) Finally, the limiting stochastic evolution equation (6) has a unique solution in W − J ( w, ρ ) and thus the limitaccumulation point ¯Ξ · is unique. Remark 4.2.
Clearly, when β SN,n = 0 for all
N, n , then the limiting distribution-valued martingale ¯ M iscentered Gaussian with covariance operator given by the (now deterministic) term within the expectation in(7). Also, we remark that the operators G x,µ and L x in the stochastic evolution equation (6) are linear.Conditionally on the systematic risk X , equation (6) is linear. The proof of Theorem 4.1 is developed in Sections 6 through 9. In Section 6, we identify the limitingequation and prove the convergence theorem based on the tightness and uniqueness results of Sections 8and Section 9. In Section 7, we discuss the Sobolev spaces we are using. In Section 8, we prove that thefamily { Ξ N,nt , t ∈ [0 , T ] } N ∈ N is relatively compact in D W − J ( w,ρ ) [0 , T ], see Lemma 8.8. In Section 9, we proveuniqueness of (6) in W − J ( w, ρ ), see Theorem 9.7.The fluctuation analysis leads to a second-order approximation to the distribution of the portfolio loss L N in large pools. The weak convergence established in Theorem 4.1 implies that P ( √ N ( L Nt − L t ) ≥ ℓ ) ≈ P (¯Ξ t ( ˆ P ) ≤ − ℓ )for large N . Theorem 4.1 and its proof in Section 6 imply that (Ξ N , V, ¯ µ ) weakly converges to (¯Ξ , V, ¯ µ ). Thismotivates the approximation µ Nt = 1 √ N Ξ Nt + ¯ µ t d ≈ √ N ¯Ξ t + ¯ µ t , which implies a second-order approximation for the portfolio loss: L Nt d ≈ L t − √ N ¯Ξ t ( ˆ P ) . (8)The next section develops, implements and tests a numerical method for computing the distribution of L t − ¯Ξ t ( ˆ P ) / √ N , and numerically demonstrates the accuracy of the approximation (8).5. Numerical Solution
The numerical solution of the stochastic evolution equation for the fluctuation limit is not standard. Theresurprisingly exists little literature on solving this class of problems. Solutions do not exist in L , so standardGalerkin methods are not applicable. An example of the unique challenges posed by these equations is the simple case of i.i.d. Brownian motions starting from zeroon the real line. The fluctuation limit ζ t ( x ) for this system satisfies the stochastic evolution equation d (cid:10) f, ζ t (cid:11) = (cid:10) f ′′ , ζ t (cid:11) dt + d (cid:10) f, W t (cid:11) , where W t is a Gaussian process with covariance Cov[ (cid:10) f, W t (cid:11) , (cid:10) g, W s (cid:11) ] = R s ∧ t R R f ′ ( x ) g ′ ( x )(2 πu ) − / e − x / (2 u ) dxdu .Suppose there exists a solution ζ ( t, x ) which satisfies the SPDE dζ = ∂ ζ∂x dt + d W t . Then, ζ t ( x ) = ζ ( t, x ) dx and one could solvethe stochastic evolution equation by solving this SPDE. However, challenges are immediately evident. The Green’s function .1. Method of Moments.
We provide a method of moments for solving for the fluctuation limit ¯Ξ( ˆ P ).Along with the LLN limit ¯ µ ( ˆ P ) = 1 − L , the fluctuation limit yields the approximation (8). The methodextends the numerical approach for computing ¯ µ ( ˆ P ) developed by Giesecke et al. (2012). We first present itfor a homogeneous portfolio; i.e., set ν = δ ˆ p where ˆ p ∈ ˆ P . Write ¯Ξ t ( dλ ) for the solution of the stochasticevolution equation (6); for notational convenience, we do not explicitly show the dependence on the fixedparameters. For k ∈ N , we define the “fluctuation moments” v k ( t ) = Z ∞ λ k ¯Ξ t ( dλ ) . We are interested in v ( t ), which is equal to the fluctuation limit ¯Ξ t ( R + ). We also define the moments u k ( t )of the solution ¯ µ t ( dλ ) of the stochastic evolution equation (3), again not explicitly showing the dependenceon the fixed parameters: u k ( t ) = Z ∞ λ k ¯ µ t ( dλ ) . The zero-th moment gives the limiting loss L since L t = 1 − u ( t ). Both the zero-th LLN moment u ( t )and the zero-th fluctuation moment v ( t ) are required to compute the approximation (8). The moments { u k ( t ) } ∞ k =0 satisfy a system of SDEs,(9) du k ( t ) = (cid:8) u k ( t ) (cid:0) − αk + β S b ( X t ) k + 0 . β S ) σ ( X t ) k ( k − (cid:1) + u k − ( t ) (cid:0) . σ k ( k −
1) + α ¯ λk + β C ku ( t ) (cid:1) − u k +1 ( t ) (cid:9) dt + β S σ ( X t ) ku k ( t ) dV t ,u k (0) = Z ∞ λ k ¯ µ ( dλ ) , see Giesecke et al. (2012). The fluctuation moments { v k ( t ) } ∞ k =1 can also be shown to satisfy a system ofSDEs using the stochastic evolution equation (6). Taking a test function f = λ k in (6), we have dv k ( t ) = β C ku k − ( t ) v ( t ) dt + (cid:2) . σ k ( k −
1) + α ¯ λk + kβ C u ( t ) (cid:3) v k − ( t ) dt − v k +1 ( t ) dt + h kβ S b ( X t ) − kα + 0 . k ( k − (cid:0) β S σ ( X t ) (cid:1) i v k ( t ) dt + kβ S σ ( X t ) v k ( t ) dV t + d ¯ M k ( t ) ,v k (0) = Z ∞ λ k ¯Ξ ( dλ ) . (10)where ¯ M k ( t ) = (cid:10) λ k , ¯ M t (cid:11) and its covariation is (cid:2) d ¯ M k ( t ) , d ¯ M j ( t ) (cid:3) = (cid:0) σ kju k + j − ( t ) + u k + j +1 − β C ku k − u j +1 ( t ) − β C ju j − ( t ) u k +1 ( t ) + ( β C ) kju k − ( t ) u j − ( t ) u ( t ) (cid:1) dt. From these covariations we form the covariation matrix Σ M ( t ). Note that this matrix depends upon thepath of X ; the solution to the SDE system (10) is conditionally Gaussian given a path of X .For the derivation of (10) to be rigorous, we must show that λ k belongs to the weighted Sobolev space W J ( w, ρ ) for every k ∈ N . The choice of the weight functions w and ρ is not unique; see Section 7. We willmake a choice that is convenient for our purposes but may not be minimal. Let ρ = 1, and w = exp( − cλ )for c >
0. Then, ρ ℓ − D ℓ ρ and w − ρ ℓ D ℓ w are bounded for every ℓ ≤ J and Condition 7.1 in Section 7 issatisfied. There exists a unique solution to the stochastic evolution equation for ¯Ξ in this chosen space. W J ( w, p ) is the closure of C ∞ in the norm || · || W J ( w,ρ ) . Therefore, to prove λ k belongs to W J ( w, ρ ), weshow that a sequence f km ∈ C ∞ approaches λ k as m → ∞ under the norm || · || W J ( w,ρ ) .More precisely, let f km = λ k B m ( λ ) where B m ( λ ) = 1 on [0 , K m ], 0 ≤ B m ≤ K m , K m + ǫ ), and B m ( λ ) = 0 on [ K m + ǫ, ∞ ). Here, ( K m ) is a sequence of numbers tending to ∞ and ǫ >
0. Furthermore, B m ( λ ) is smooth. Such a function is B m ( λ ) = 1 − R λ −∞ g m ( x ) dx R ∞−∞ g m ( x ) dx , solution ζ ( t, x ) = R t R R dt ′ dx ′ G ( t ′ , x ′ ; t, x ) d W t ′ to the SPDE has infinite variance, indicating that the Green’s solution is notin fact a solution to this SPDE. It is interesting to note that the lack of a Green’s function solution stems directly from thecovariance structure; if one had f ( x ) g ( x ) instead of f ′ ( x ) g ′ ( x ), there would be a finite variance Green’s function solution. m ( x ) = h ( x − K m ) h ( K m + ǫ − x ) , where h ( x ) = e − x when x > x ≤
0. Then, f km is a smooth function with compact support.We also note that the integral R ∞−∞ g m ( x ) dx only depends upon ǫ ; it is not affected by K m . To prove that || λ k − f km ( λ ) || W J ( w,ρ ) → K m → ∞ , one has to show that the following integral tends to zero as K m → ∞ for each ℓ ≤ J .: Z ∞ e − cλ | D ℓ [ λ k − f km ( λ )] | dλ = Z K m e − cλ | D ℓ [ λ k − λ k B m ( λ )] | dλ + Z K m + ǫK m e − cλ | D ℓ [ λ k − λ k B m ( λ )] | dλ + Z ∞ K m + ǫ e − cλ | D ℓ [ λ k − λ k B m ( λ )] | dλ. The first term trivially is zero due to the definition of B m while the third term can be shown to tend to zerousing integration by parts. Turning to the second integral, we have that Z K m + ǫK m e − cλ | D ℓ [ λ k − λ k B m ( λ )] | dλ ≤ Z K m + ǫK m e − cλ | D ℓ λ k | + Z K m + ǫK m e − cλ | D ℓ [ λ k B m ( λ )] | dλ. The first integral again obviously tends to zero by integration by parts. In order to show that the secondterm tends to zero, one has to demonstrate that the | D ℓ B m ( λ ) | is uniformly bounded in m on [ K m , K m + ǫ ].For ℓ = 0, the result is trivial. For ℓ = 1, we have that D B m ( λ ) = C exp (cid:18) − ǫy ( ǫ − y ) (cid:19) ≤ C exp (cid:18) − ǫ (cid:19) , where y = λ − K m and y ∈ [0 , ǫ ]. For ℓ = 2, we have that D B m ( λ ) = C exp (cid:18) − ǫy ( ǫ − y ) (cid:19) (cid:18) ǫ − y ) − y (cid:19) . Exponential decay dominates the polynomial growth, so D B m ( λ ) and its derivative approach zero as y approaches 0 or ǫ . Since [0 , ǫ ] is a bounded domain, D B m ( λ ) is therefore also bounded on [0 , ǫ ]. Higherderivatives can be treated similarly. Therefore, since | D ℓ B m ( λ ) | is uniformly bounded on [ K m , K m + ǫ ], thesecond integral also tends to zero.Unless the volatility parameter σ = 0, the SDE system (10) is not closed. For computational purposes,it must be truncated at some level K . Note that this means one also needs the LLN moments { u k } K +1 k =0 .The system (9) must also be truncated and can be solved using an Euler scheme. Important details for thenumerical solution of (9) as well as an alternative formulation as a random ODE system are described inGiesecke et al. (2012).5.2. Case with no Systematic Risk.
When β S = 0, there exists a semi-analytic solution. Observe thatthe moment system (9) is now a system of ODEs and that (10) is a linear system with a Gaussian forcingterm. Therefore, the system of moments v ( t ) is itself Gaussian. One can directly compute its distribution.We rewrite (10) concisely as d v ( t ) = A ( t ) v ( t ) dt + d ¯ M ( t ) , v (0) = v , (11)where A : [0 , T ] R K +1 ,K +1 and v , ¯ M : [0 , T ] × Ω R K +1 . Now, let Ψ : [0 , T ] R K +1 ,K +1 be thefundamental solution matrix satisfying d Ψ( t ) = A ( t )Ψ( t ) dt, Ψ(0) = I, (12)where I is the identity matrix. If β C = 0, A is a constant matrix and there is an exponential matrix solution.For β C >
0, one can either solve (12) numerically using a method such as Runge Kutta or the Magnus seriesapproximation. We assume that v is deterministic (if it is a Gaussian random variable, the result is verysimilar). Then, v ( t ) = Ψ( t ) v + Ψ( t ) Z t Ψ − ( s ) d ¯ M ( s ) . If σ = 0, the LLN reduces to two SDEs and the fluctuation limit also reduces to two SDEs. .05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450510152025303540 Loss Actual Loss DistributionSecond−Order Approximation β C = 0 β C = 1.5 β C = 3 Figure 1.
Comparison of approximate loss distribution and actual loss distribution in thefinite system at T = 1 for N = 1 , σ = . , α = 4 , λ = . , ¯ λ = . β S = 0.It easily follows that v ( t ) ∼ N (Ψ( t ) v , Σ( t )) whereΣ( t ) = Ψ( t ) (cid:2) Z t Ψ − ( s )Σ M ( s ) (cid:0) Ψ − ( s ) (cid:1) ⊤ ds (cid:3) Ψ( t ) ⊤ . (13)Therefore, we have avoided simulation of (11) by instead developing a semi-analytic approach for the com-putation of the distribution. Also, note that in the case β C = β S = 0, we have a closed-form formula for theapproximating loss since Ψ( t ) = e At .Figure 1 shows a comparison between the approximate loss distribution according to (8) and the actualloss distribution for a pool of N = 1 ,
000 names, for each of several values of the contagion sensitivity β C .Here and in the numerical experiments described below, the actual distribution is estimated by simulatingthe default times in (1) using the discretization method detailed in Giesecke et al. (2012). The second-orderGaussian approximation (8) is a significant improvement over the first-order approximation (4) implied bythe LLN. The latter produces a delta function for this case while the Gaussian approximation is able toaccurately capture a large portion of the remaining noise in the finite system.The linear stochastic evolution equation (6) involves a linearization of the jump term β C dL Nt in (1). It isreasonable to expect that the accuracy of the approximation will decrease as β C increases. This is confirmedby the numerical results. For larger pools, the linearization will become more accurate. In the case of β C = 3, we are taking too small of a pool.Another common concern with linearizations is a loss of accuracy over long time horizons. However, wedo not observe any significant loss of accuracy for the fluctuation limit in our model.5.3. Case with Systematic Risk.
In the case with systematic risk (i.e., β S > X . There are two approaches to treat the general case. We presentthe most obvious approach first and then a second scheme which achieves lower variance by replacing aportion of the simulation with a semi-analytic computation. .3.1. Scheme 1: Direct Simulation.
Here we directly discretize and solve the systems (9) and (10) (forexample, using an Euler method). Alternatively, one could instead numerically solve the random ODEformulation of (9) given in Giesecke et al. (2012) and then solve (10). • Simulate paths X , . . . , X M of the systematic risk process on [0 , T ]. • Conditional upon a path X m , the martingale term ¯ M is Gaussian. – First, solve for the LLN moments u m ( t ) in (9) for a path X m and then calculate the conditionalcovariation matrix Σ m M ( t ). – Perform a spectral decomposition of the covariation matrix Σ m M ( t ) at a discrete set of times ona grid T . – For j = 1 , . . . , J , discretize system (10) on the grid T . At each time t i ∈ T , draw a sample d ¯ M m,j ( t i ) using the spectral decomposition of Σ m M ( t ) and then solve (10) using the Eulermethod. This produces J samples from the conditionally Gaussian solution given the path X m . – Finally, for each j , compute a sample of the “approximate conditional loss” L m,j,Nt for a poolof size N as the difference between the conditional LLN loss L mt = 1 − u m ( t ) and a sample ofthe conditionally Gaussian solution scaled by 1 / √ N . • Approximate the unconditional distribution of L Nt by JM P Mm =1 P Jj =1 δ L m,j,Nt .Let σ = Var[ R ∞ f ( y ) p t ( y | X ) dy ] and σ = E [Var[ f ( L Nt ) | X ]] where p t ( y | X ) is the conditional density of theapproximate loss given X at time t . The total simulation time is M τ + M Jτ , where τ is the time neededto simulate the systematic risk process and solve the LLN moment system while τ is the time requiredto solve the fluctuation moment system. If we want to estimate the expectation of a function f of theapproximate loss at some time t , it is straightforward to show using the first order condition for a minimumthat the optimal allocation of simulation resources M ∗ and J ∗ to minimize the estimator’s variance for afixed computational time τ is M ∗ ≈ − τ σ + p (2 τ σ ) + 4 σ ( σ τ τ − σ τ )2( σ τ τ − σ τ ) τ = M ∗ τJ ∗ ≈ − M ∗ τ τ M ∗ . (14)The optimal number for J does not depend upon the total computational resources.5.3.2. Scheme 2: Conditionally Semi-Analytic Approach.
We propose an alternate scheme that replaces someof the Monte Carlo simulation of the previous scheme with a semi-analytic calculation. First, we rewrite(10) concisely as d v ( t ) = A ( t ) v ( t ) dt + B v ( t ) dX t + d ¯ M ( t ) , v (0) = v , (15)where A ( t ) is a random, time-dependent matrix and B is a constant matrix. In the calculations that follow,we assume that v is deterministic; the computations when v is a Gaussian random variable are similar.The fundamental solution Ψ : [0 , T ] × Ω R K +1 ,K +1 satisfies d Ψ( t ) = A ( t )Ψ( t ) dt + B Ψ( t ) dX t , Ψ(0) = I. (16)Then, since [ dX t , d ¯ M k ( t )] = 0 for any k ≥ v ( t ) = Ψ( t ) v + Ψ( t ) Z t Ψ − ( s ) d ¯ M ( s ) . (17)(It is easy to show that this is a solution to (15) by substituting it back into that SDE.) Unfortunately, evenwhen β C = 0, b ( x ) = b , and σ ( x ) = σ so that A is a constant matrix, there is no exponential solution to(16) since A and B do not commute.To calculate the approximation (8), one must simulate from the joint law of the paths of X and ¯ M over[0 , T ]. We can do this using the following scheme: • Simulate paths X , . . . , X M of the systematic risk process on [0 , T ]. • Conditional upon a path X m , we have a Gaussian solution for the fluctuation moments v m ( t ).Furthermore, we can semi-analytically compute the conditional distribution of v m ( t ). Given a path X m , solve for the LLN moments u m ( t ) (which also yield the conditional LLN loss L m ) and the fundamental solution Ψ m ( t ). – Then, form the conditional covariation matrix Σ m M ( t ). Finally, compute the conditional covari-ance matrix using the closed-form formula (13). This yields Var[ v m ( t )]. • Approximate the unconditional distribution of L Nt by M P Mm =1 P mt , where P mt is a Gaussian measurewith mean L m and variance Var[ v m ( t )] /N .A skeleton of the systematic risk process X for both schemes discussed above can be generated exactly(without discretization bias) using the methods of Beskos & Roberts (2005), Chen & Huang (2012), orGiesecke & Smelov (2012). It is also worthwhile to highlight that either scheme yields an approximation tothe distribution of the loss L Nt for all time horizons t ≤ T (i.e., the “loss surface”) and all portfolio sizes N ∈ N simultaneously.Scheme 2 will have lower variance than Scheme 1. Let ˆ f and ˆ f be estimators of E [ f ( L Nt )] using Schemes1 and 2, respectively. We haveVar[ ˆ f ] = Var (cid:20) JM M X m =1 J X j =1 f ( L m,j,Nt ) (cid:21) = 1 M (cid:0) σ + 1 J σ (cid:1) > Var[ ˆ f ] = 1 M σ . Var[ ˆ f ] = M σ since Scheme 2 generates samples from the random variable R ∞ f ( y ) p t ( y | X ) dy .We note that there is numerical instability for large β S for both schemes, especially over long time horizons.One must use a small time-step to avoid this instability. The instability is caused by the exponential growthterms ( β S σ ( X t )) k ( k − v k ( t ) and ( β S σ ( X t )) k ( k − u k ( t ). When one is only interested in calculatingthe LLN approximation (4) via the moment system (9), the following transformed moments significantlyreduce instability by removing the exponential growth term: w k ( t ) = exp (cid:18) −
12 ( β S ) k ( k − Z t σ ( X s ) ds (cid:19) u k ( t ) . However, when interested in solving (9) and (10) in conjunction, there is no simple transformation. Thebest approach is to solve (10) with a sufficiently small time step such that it is stable and then solve for thetransformed fluctuation moments ˜ w k ( t ) = exp( − ( β S ) k ( k − R t σ ( X s ) ds ) v k ( t ) with a larger time step.Figure 2 compares the approximate loss distribution according to (8) for different truncation levels K ofthe fluctuation moment system (10). We use a time step of 0 .
005 and produce samples from X using anEuler scheme. The approximate loss distribution converges very rapidly in terms of the truncation level.This conforms with previous numerical studies of the LLN moment system (9) which also demonstrated itsquick convergence rate; see Giesecke et al. (2012).We now study the validity of the second-order approximation (8) by comparing it against the actual lossdistribution from simulating the true finite system. In the following numerical studies, we choose a time stepof 0 .
005 and a truncation level of K = 6 for both schemes described above. A time step of 0 .
005 is used forsimulating the finite system. Samples from X are produced using an Euler scheme. Figure 3 compares theapproximate loss distribution according to (8) with the actual loss distribution for β C = 0 and β S = 1, foreach of several portfolio sizes. The approximate loss distribution is extremely accurate, even for a very smallpool with only N = 250. We also observe that the approximation accurately captures the tails of the actualloss distribution. Figure 4 compares the approximate loss distribution with the actual loss distribution for β C = 1 and β S = 1. The approximate loss distribution is again accurate, although not as accurate as when β C = 0. We also show in both Figures 3 and 4 the first-order LLN approximation (4). It is clear that thesecond-order approximation has increased accuracy, especially for smaller portfolios and in the tail of thedistribution. Finally, Figure 5 shows a comparison for the 95 and 99 percent value at risk (VaR) betweenthe actual loss, LLN approximation (4), and approximation (8) for a pool of N = 1 , igure 2. Comparison of approximate loss distribution for different truncation levels at T = 0 .
5. Parameter case is σ = 0 . , α = 4 , λ = ¯ λ = 0 . , β C = 1 , and β S = 1. The system-atic risk X is an OU process with mean 1, reversion speed 2, volatility 1, and initial value1. Approximate loss distribution computed using the conditionally semi-analytic Scheme 2.The fluctuation moments are truncated at level K and the LLN moments are truncated atlevel 3 K .5.4. Approximate Loss Process.
One of the significant advantages of the fluctuation analysis presentedhere is the dynamic approximation it provides for the entire loss process . An approximation of the lossprocess may be used to estimate the probability that the loss per month never exceeds a certain amount, orthe density of the hitting time for the loss exceeding a certain level.One can generate a sample skeleton of the approximate loss process according to the right hand side of (8)using Scheme 1. Although Scheme 2 gives the loss distribution at all time horizons, it does not directly yielda skeleton of the approximate loss process. However, due to the process being conditionally Gaussian, onecan use a slightly modified form of Scheme 2 to simulate such a skeleton on as fine a time grid as desired. Forsmall β S , one can expect that Scheme 2 will be more computationally efficient than Scheme 1 (see AppendixA for numerical results).To simulate skeletons for the moments (10) of the fluctuation limit, one can use the following method.Let v = 0 (the general case is very similar). As usual, begin by simulating a path from the systematic riskprocess X . Given a path of X , we then simulate the loss at t = T from the conditionally Gaussian distributiongiven by the formula (13). To simulate at 0 < s < T , we only need the covariance matrix Cov[ v T , v s |V T ].One can continue to simulate at grid points of one’s choice. For instance, one can simulate at time s where s < s < T given the covariance matrices Cov[ v T , v s |V T ], Cov[ v T , v s |V T ], and Cov[ v s , v s |V T ]. Thesecovariance matrices are easily computed using the fundamental solution (16):Σ( τ , τ ) = Cov[ v τ , v τ |V T ] = Ψ( τ ) (cid:2) Z τ ∧ τ Ψ − ( s )Σ M ( s ) (cid:0) Ψ − ( s ) (cid:1) ⊤ ds (cid:3) Ψ( τ ) ⊤ . N=150N=250 N=1000
Figure 3.
Comparison of approximate loss distribution and actual loss distribution in thefinite system at T = 0 .
5. Parameter case is σ = 0 . , α = 4 , λ = ¯ λ = 0 . , β C = 0, and β S = 1. The systematic risk X is an OU process with mean 1, reversion speed 2, volatility1, and initial value 1. Approximate loss distribution computed using the conditionally semi-analytic Scheme 2.To simulate v s given v τ and v τ where τ < s < τ , we generate a sample from the multivariate Gaussiandistribution N (˜ µ, ˜Σ) where ˜ µ = Σ Σ − Z and ˜Σ = Σ( s, s ) − Σ Σ − Σ . The matrices Σ , Σ , and Z areΣ = (cid:2) Σ( s, τ ) Σ( s, τ ) (cid:3) Σ = (cid:20) Σ( τ , τ ) Σ( τ , τ )Σ( τ , τ ) Σ( τ , τ ) (cid:21) Z = (cid:20) v τ v τ (cid:21) It is worthwhile to note that this scheme requires only a single computation of the fundamental solutionΨ m ( t ) for each path X m . Given Ψ m ( t ) for 0 ≤ t ≤ T , one can generate as many skeletons of the fluctuationmoments as desired for that particular X m via the aforementioned method. In contrast, Scheme 1 requiresone to resolve the system of SDEs (10) for each new realization. In particular, this suggests that this schemewill be more efficient than Scheme 1 for small β S .Once skeletons from the fluctuation limit have been simulated, skeletons of the approximate loss processcan be produced by combining these fluctuation paths with the law of large numbers limit process given by(9) and using the approximation formula (8).5.5. Nonhomogeneous Portfolio.
The moment method can be extended to a nonhomogeneous portfolio.Define the moments v k ( t, p ) = R R + λ k ¯Ξ t ( dλ, p ) and u k ( t, p ) = R R + λ k ¯ µ t ( dλ, p ). Then dv k ( t, p ) = β C ku k − ( s, p ) (cid:18)Z P v ( t, p ) d p (cid:19) dt + (cid:20) σ k ( k −
1) + α ¯ λk + kβ C Z P u ( s, p ) d p (cid:21) v k − ( t, p ) dt N=1000N=250N=150
Figure 4.
Comparison of approximate loss distribution and actual loss distribution in thefinite system at T = 0 .
5. Parameter case is σ = 0 . , α = 4 , λ = ¯ λ = 0 . , β C = 1 , and β S = 1. The systematic risk X is an OU process with mean 1, reversion speed 2, volatility1, and initial value 1. Approximate loss distribution computed using the conditionally semi-analytic Scheme 2.+ v k ( t, p ) (cid:20) − kα + kβ S b ( X t ) + 12 k ( k − (cid:0) β S σ ( X t ) (cid:1) (cid:21) dt − v k +1 ( t, p ) dt + kβ S σ ( X t ) v k ( t, p ) dV t + ¯ M k ( t, p ) ,v k (0 , p ) = Z ∞ λ k ¯Ξ ( dλ, p ) . The covariation (cid:2) d ¯ M k ( t, p ) , d ¯ M j ( t, p ) (cid:3) for p , p ∈ P is given by (cid:2) d ¯ M k ( t, p ) , d ¯ M j ( t, p ) (cid:3) = p = p (cid:0) σ kju k + j − ( t, p ) + u k + j +1 ( t, p ) − β C ku k − ( t, p ) u j +1 ( t, p ) − β C ju j − ( t, p ) u k +1 ( t, p ) + ( β C ) kju k − ( t, p ) u j − ( t, p ) u ( t, p ) (cid:1) dt. The LLN moments are du k ( t, p ) = u k ( t, p ) (cid:20) − αk + β S b ( X t ) k + 12 ( β S ) σ ( X t ) k ( k − (cid:21) dt − u k +1 ( t, p ) dt + u k − ( t, p ) (cid:20) . σ k ( k −
1) + α ¯ λk + β C k Z P u ( t, p ) d p (cid:21) dt + β S σ ( X t ) ku k ( t, p ) dV t u k (0 , p ) = Z ∞ λ k ¯ µ ( dλ, p ) . For numerical implementation, one must discretize the parameter space P . As we allow more parametersto vary, P can become high dimensional. This can become computationally expensive. .25 0.3 0.35 0.4 0.45 0.50.060.070.080.090.10.110.120.130.140.150.16 Time V a R Actual VaRFirst−Order Approximation VaRSecond−Order Approximation for VaR95% VaR99% VaR
Figure 5.
Comparison of approximate and actual value at risks for N = 1 , σ = 0 . , α = 4 , λ = ¯ λ = 0 . , β C = 1 , and β S = 1. The systematic risk X is an OUprocess with mean 1, reversion speed 2, volatility 1, and initial value 1. Approximate lossdistribution computed using the conditionally semi-analytic Scheme 2.6. Proof of Theorem 4.1
In this section we prove the fluctuation Theorem 4.1. The methodology of the proof goes as follows. Aftersome preliminary computations, we obtain a convenient formulation of the equation that { Ξ Nt } satisfies, see(20). Some terms in this equation will vanish in the limit as N → ∞ ; this is Lemma 6.1. Based on tightnessof the involved processes (the topic of Section 8) and continuity properties of the operators involved, we canthen pass to the limit and thus identify the limiting equation. The limit process satisfies in the weak formanother stochastic evolution, which has a unique solution (proven in Section 9). The difficulty is to identifya rich enough space where tightness and uniqueness can be simultaneously proven. This is not trivial inour case, because the coefficients are not bounded and the equation degenerates (coefficients of the highestderivatives are not bounded away from zero). It turns out that the appopriate space in which one can proveboth tightness and uniqueness is a weighted Sobolev space, introduced in Purtukhia (1984) and furthergeneralized in Gy¨ongy & Krylov (1990) to study stochastic partial differential equations with unboundedcoefficients, and are briefly reviewed in Section 7.An application of Itˆo’s formula shows that for f ∈ C b ( ˆ P ), (cid:10) f, µ Nt (cid:11) E = (cid:10) f, µ N (cid:11) E + Z t n(cid:10) L f, µ Ns (cid:11) E + (cid:10) Q , µ Ns (cid:11) E (cid:10) L f, µ Ns (cid:11) E + D L X s f, µ Ns E E o ds + Z t ˆ A N [ f ]( X s , µ Ns ) ds + Z t D L X s f, µ Ns E E dV s + (cid:10) f, M Nt (cid:11) (18)where ˆ A N [ f ]( X s , µ Ns ) = N X n =1 λ N,ns J fN,n ( s ) m N,ns − (cid:8)(cid:10) Q , µ Ns (cid:11) E (cid:10) L f, µ Ns (cid:11) E − (cid:10) f, µ Ns (cid:11) E (cid:9) , fN,n ( s ) = 1 N N X n ′ =1 ( f p N,n ′ , λ N,n ′ s + β CN,n ′ N ! − f (cid:16) p N,n ′ , λ N,n ′ s (cid:17)) m N,n ′ s − N f (cid:0) p N,n , λ
N,ns (cid:1) for all s ≥ N ∈ N and n ∈ { , , . . . , N } , (cid:10) f, M Nt (cid:11) = 1 N N X n =1 Z t σ N,n q λ N,nt ∂f∂λ (ˆ p N,ns ) m N,ns dW ns + N X n =1 Z t J fN,n ( s ) d N N,ns and N N,ns = (1 − m N,ns ) − Z s λ N,nr m N,nr dr.
By subtracting (3) from (18) we find that Ξ Nt satisfies the equation (cid:10) f, Ξ Nt (cid:11) = (cid:10) f, Ξ N (cid:11) + Z t D L f + L X s f, Ξ Ns E ds + Z t (cid:2) hQ , ¯ µ s i (cid:10) L f, Ξ Ns (cid:11) + (cid:10) Q , Ξ Ns (cid:11) (cid:10) L f, µ Ns (cid:11)(cid:3) ds + Z t √ N ˆ A N [ f ]( X s , µ Ns ) ds + Z t D L X s f, Ξ Ns E dV s + D f, √ N M Nt E . In order to simplify the calculations later on, for each f ∈ C b ( ˆ P ), t ≥ N ∈ N and n ∈ { , , . . . , N } , wedefine the quantity˜ J fN,n ( t ) def = 1 N N X m =1 β CN,m ∂f∂λ (ˆ p N,mt ) m N,mt − f (cid:16) p N,n , λ
N,nt (cid:17) = (cid:10) L f, µ Nt (cid:11) E − f (ˆ p Nt ) . A simple computation shows that(19) (cid:12)(cid:12)(cid:12)(cid:12) J fN,n ( t ) − N ˜ J fN,n ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ K N (cid:13)(cid:13)(cid:13)(cid:13) ∂ f∂λ (cid:13)(cid:13)(cid:13)(cid:13) C . We can rewrite the stochastic evolution equation that Ξ N satisfies as follows: (cid:10) f, Ξ Nt (cid:11) = (cid:10) f, Ξ N (cid:11) + Z t D L f + L X t f, Ξ Ns E ds + Z t (cid:2) hQ , ¯ µ s i (cid:10) L f, Ξ Ns (cid:11) + (cid:10) Q , Ξ Ns (cid:11) (cid:10) L f, µ Ns (cid:11)(cid:3) ds + Z t D L X s f, Ξ Ns E dV s + D f, √ N ˜ M Nt E + R Nt, (20)where D f, √ N ˜ M Nt E = √ N N N X n =1 Z t σ N,n q λ N,nt ∂f∂λ (ˆ p N,ns ) m N,ns dW ns + 1 N N X n =1 Z t ˜ J fN,n ( s ) d N N,ns ! R Nt, = Z t √ N ˆ A N [ f ]( X s , µ Ns ) ds + N X n =1 Z t √ N ˆ B N,n [ f ]( s ) d N N,ns ˆ B N,n [ f ]( s ) = J fN,n ( s ) − N ˜ J fN,n ( s ) . The term R Nt, turns out to vanish in the limit as the following lemma shows. Lemma 6.1.
For any t ∈ [0 , T ] and any f ∈ C b ( ˆ P ) , there is a constant C , independent of n, N such that E (cid:20)Z t √ N (cid:12)(cid:12)(cid:12) ˆ A N [ f ]( X r , µ Nr ) (cid:12)(cid:12)(cid:12) dr (cid:21) ≤ C √ N t and sup ≤ t ≤ T E " N X n =1 Z t √ N ˆ B N,n [ f ]( s ) d N N,ns ≤ C N T. Moreover, we have that lim N →∞ E sup ≤ t ≤ T | R Nt, | = 0 . astly, regarding the conditional (on V t ) covariation of the martingale term D f, √ N ˜ M Nt E we have Cov hD f, √ N ˜ M Nt E , D g, √ N ˜ M Nt E (cid:12)(cid:12)(cid:12) V t ∨ t i = E (cid:20)Z t ∧ t (cid:2)(cid:10) L ( f, g ) , µ Ns (cid:11) + (cid:10) L ( f, g ) , µ Ns (cid:11) ++ (cid:10) L f, µ Ns (cid:11) (cid:10) L g, µ Ns (cid:11) (cid:10) Q , µ Ns (cid:11) −− (cid:10) L g, µ Ns (cid:11) (cid:10) L f, µ Ns (cid:11) − (cid:10) L f, µ Ns (cid:11) (cid:10) L g, µ Ns (cid:11)(cid:3) ds (cid:12)(cid:12)(cid:12) V t ∨ t i . (21)We continue with the proof of the theorem and defer the proof of Lemma 6.1 to the end of this section. Rel-ative compactness of the sequences n Ξ N,nt , t ∈ [0 , T ] o N ∈ N and n √ N ˜ M N,nt , t ∈ [0 , T ] o N ∈ N in D W − J ( w,ρ ) [0 , T ]follows by Lemmas 8.7 and 8.8. Relative compactness of the sequence (cid:8) µ Nt , t ∈ [0 , T ] (cid:9) N ∈ N in D E [0 , T ] followsby Lemma 7.1 in Giesecke et al. (2012). These imply that the sequence n(cid:16) µ N , √ N ˜ M N , Ξ N , V (cid:17) , N ∈ N o is relatively compact in D E × W − J ( w,ρ ) × W − J ( w,ρ ) × R [0 , T ]. Let us denote by (cid:8)(cid:0) ¯ µ, ¯ M , ¯Ξ , V (cid:1)(cid:9) a limit point of this sequence. We mention here that this is a limit in distribution, so the limit point may notbe defined on the same probability space as the prelimit sequence, but nevertheless V (and thus X ), havethe same distribution both in the limit and in the prelimit. Then, by (20), Lemma 6.1 and the continuityof the operators G and L we get that (cid:8)(cid:0) ¯ µ, ¯ M , ¯Ξ , V (cid:1)(cid:9) will satisfy, due to Theorem 5.5 in Kurtz & Protter(1996), the stochastic evolution equation (6). By Lemma 6.1, the conditional covariation of the distributionvalued martingale ¯ M is given by (7). Uniqueness follows by Theorem 9.7. This concludes the proof of thetheorem.We conclude this section with the proof of Lemma 6.1. Proof of Lemma 6.1.
We start by recalling that Lemma 3.4 in Giesecke et al. (2013) states that for each p ≥ T ≥ ≤ t ≤ TN ∈ N N N X n =1 E h | λ N,nt | p i ≤ C for some constant C >
N, n . Along with (19), this implies that E (cid:20)Z t √ N (cid:12)(cid:12)(cid:12) ˆ A N [ f ]( X r , µ Nr ) (cid:12)(cid:12)(cid:12) dr (cid:21) = E "Z t √ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 λ N,nr J fN,n ( r ) m N,nr − N N X n =1 λ N,nr ˜ J fN,n ( r ) m N,nr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dr ≤ K (cid:13)(cid:13)(cid:13)(cid:13) ∂ f∂λ (cid:13)(cid:13)(cid:13)(cid:13) C E "Z t √ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X n =1 λ N,ns (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dr ≤ C √ N t for some nonnegative constant C that is independent of N, n . Then, the first statement of the lemmafollows. The second statement follows similarly. In particular, using the martingale property, (19) and (22)we immediately obtain E " N X n =1 Z t √ N ˆ B N,n [ f ]( s ) d N N,ns ≤ C N t for some constant C >
0. The statement regarding R Nt, follows directly by the previous computations andDoob’s inequality. The statement involving the conditional covariation of the martingale term D f, √ N ˜ M Nt E follows directly by the definition of ˜ M N and the fact that jumps do not occur simultaneously. (cid:3) . The appropriate weighted Sobolev space
In this section we discuss the Sobolev space in which the fluctuations theorem is stated. Similar weightedSobolev spaces were introduced in Purtukhia (1984) and further generalized in Gy¨ongy & Krylov (1990) tostudy stochastic partial differential equations with unbounded coefficients. These weighted spaces turn outto be convenient and well adapted to our case. See Fernandez & M´el`eard (1997) and Kurtz & Xiong (2004)for the use of weighted Sobolev spaces in fluctuation analyses of other interacting particle systems.Let w and ρ be smooth functions in ˆ P with w ≥
0. Let J ≥ W J ( w, ρ ) theweighted Sobolev space which is the closure of C ∞ ( ˆ P ) in the norm k f k W J ( w,ρ ) = X k ≤ J Z ˆ p ∈ ˆ P w (ˆ p ) (cid:12)(cid:12) ρ k (ˆ p ) D k f (ˆ p ) (cid:12)(cid:12) d ˆ p / < ∞ For the weight functions w and ρ we impose the following condition, which guarantees that W J ( w, ρ ) willbe a Hilbert space, see Proposition 3.10 of Gy¨ongy & Krylov (1990). Condition 7.1.
For every k ≤ J , the functions ρ k − D k ρ and w − ρ k D k w are bounded. The inner product in this space is h f, g i W J ( w,ρ ) = X k ≤ J Z ˆ p ∈ ˆ P w (ˆ p ) ρ k (ˆ p ) D k f (ˆ p ) D k g (ˆ p ) d ˆ p Moreover, we denote by W − J ( w, ρ ), the dual space to W J ( w, ρ ) equipped with the norm k f k W − J ( w,ρ ) = sup g ∈ W J ( w,ρ ) | < f, g > |k g k W J ( w,ρ ) For notational convenience we will sometimes write k f k J and k f k − J in the place of k f k W J ( w,ρ ) and k f k W − J ( w,ρ ) respectively, if no confusion arises.The coefficients of our operators have linear growth in the first order terms and quadratic growth in thesecond order terms. For example, we easily see that the choices ρ (ˆ p ) = p | ˆ p | and w (ˆ p ) = (cid:0) | ˆ p | (cid:1) β with β < − J satisfy the assumptions of Condition 7.1. We refer the interested reader to Gy¨ongy & Krylov(1990) for more examples on possible choices for the weight functions ( w, ρ ). Lemma 7.2.
Let Condition 7.1 with J + 2 in place of J hold, and fix x ∈ R and µ ∈ E such that R ˆ P (cid:0) w (ˆ p ) ρ (ˆ p ) (cid:1) − µ ( d ˆ p ) < ∞ . Then the operator G x,µ is a linear map from W J +20 ( w, ρ ) into W J ( w, ρ ) and for all f ∈ W J +20 ( w, ρ ) , there exists a constant C > such that kG x,µ f k J ≤ C (cid:16) | b ( x ) | + | σ ( x ) | + hQ , µ i (cid:17) k f k J +2 Moreover, the operator L x is a linear map from W J +10 ( w, ρ ) into W J ( w, ρ ) and for all f ∈ W J +10 ( w, ρ ) kL x f k J ≤ C | σ ( x ) | k f k J +1 ≤ C | σ ( x ) | k f k J +2 Proof.
Recall the definition of G x,µ f and examine term by term. For the first term, Condition 7.1 gives us kL f k J = J X k =0 Z ˆ P w (ˆ p ) ρ k (ˆ p ) (cid:0) D k L f (ˆ p ) (cid:1) d ˆ p ≤ C J +2 X k =0 Z ˆ P w (ˆ p ) ρ k (ˆ p ) (cid:0) D k f (ˆ p ) (cid:1) d ˆ p ≤ C k f k J +2 For the second term again Condition 7.1 gives us kL x f k J = J X k =0 Z ˆ P w (ˆ p ) ρ k (ˆ p ) (cid:0) D k L x f (ˆ p ) (cid:1) d ˆ p C (cid:0) | b ( x ) | + | σ ( x ) | (cid:1) J +2 X k =0 Z ˆ P w (ˆ p ) ρ k (ˆ p ) (cid:0) D k f (ˆ p ) (cid:1) d ˆ p ≤ C (cid:0) | b ( x ) | + | σ ( x ) | (cid:1) k f k J +2 For the third term we have khQ , µ i L f k J = hQ , µ i J X k =0 Z ˆ P w (ˆ p ) ρ k (ˆ p ) (cid:0) D k L f (ˆ p ) (cid:1) d ˆ p ≤ C hQ , µ i k f k J +2 For the fourth term we have khL f, µ i Qk J ≤ C hL f, µ i J X k =0 Z ˆ P w (ˆ p ) ρ k (ˆ p )(1 + λ ) d ˆ p ≤ C (cid:18)Z ˆ P | Df (ˆ p ) | µ ( d ˆ p ) (cid:19) J X k =0 Z ˆ P w (ˆ p ) ρ k (ˆ p )(1 + | ˆ p | ) d ˆ p ≤ Z ˆ P w (ˆ p ) ρ (ˆ p ) | Df (ˆ p ) | d ˆ p Z ˆ P (cid:0) w (ˆ p ) ρ (ˆ p ) (cid:1) − µ ( d ˆ p ) J X k =0 Z ˆ P w (ˆ p ) ρ k (ˆ p )(1 + | ˆ p | ) d ˆ p ≤ C k f k J ≤ C k f k J +2 (23)The statement for L x follows analogously. This completes the proof of the lemma. (cid:3) Notice that if the weights w, ρ are chosen as above, then the condition on µ of Lemma 7.2 is equivalentto assuming that µ ∈ E has finite moments up to order 2 | β | − Tightness and continuity properties of the limiting process
We first recall three key preliminary results from Giesecke et al. (2013) and Giesecke et al. (2012) thatwill be useful in the sequel.
Lemma 8.1. [Lemma 3.4 in Giesecke et al. (2013).] For each p ≥ and T ≥ , there is a constant C > ,independent of N, n such that sup ≤ t ≤ TN ∈ N N N X n =1 E h | λ N,nt | p i ≤ C. Lemma 8.2. [Lemma 8.2 in Giesecke et al. (2012).] Let W ∗ be a reference Brownian motion. For each ˆ p = ( p , λ ◦ ) ∈ ˆ P where p = ( α, ¯ λ, σ, β C , β S ) , there is a unique pair { ( Q ( t ) , λ ∗ t (ˆ p )) : t ∈ [0 , T ] } taking values in R + × R + such that (24) Q ( t ) = Z ˆ P E ( V , ˆ p ) (cid:26) λ ∗ t (ˆ p ) exp (cid:20) − Z t λ ∗ s (ˆ p ) ds (cid:21)(cid:27) ν ( d ˆ p ) . and (25) λ ∗ t (ˆ p ) = λ ◦ − α Z t ( λ ∗ s (ˆ p ) − ¯ λ ) ds + σ Z t p λ ∗ s (ˆ p ) dW ∗ s + β C Z t Q ( s ) ds + β S Z t λ ∗ s (ˆ p ) dX s . Lemma 8.3. [Lemma 8.4 in Giesecke et al. (2012).] For all A ∈ B ( P ) and B ∈ B ( R + ) , ¯ µ satisfying (3)is given by ¯ µ t ( A × B ) = Z ˆ P χ A ( p ) E ( V t , ˆ p ) (cid:20) χ B ( λ ∗ t (ˆ p )) exp (cid:20) − Z t λ ∗ s (ˆ p ) ds (cid:21)(cid:21) ν ( d ˆ p ) , where λ ∗ t (ˆ p ) is defined via (24)-(25). Therefore, for any f ∈ C ( ˆ P ) , h f, ¯ µ t i E = Z ˆ P E ( V t , ˆ p ) (cid:20) f ( p , λ ∗ t (ˆ p )) exp (cid:20) − Z t λ ∗ s (ˆ p ) ds (cid:21)(cid:21) ν ( d ˆ p ) . et us now set G ( t ) def = Z ˆ P (cid:18) − E ( V , ˆ p ) (cid:20) exp (cid:20) − Z t λ ∗ s (ˆ p ) ds (cid:21)(cid:21)(cid:19) ν ( d ˆ p ) . Notice that ˙ G ( t ) = Q ( t ) . We prove tightness based on a coupling argument. The role of the coupled intensities will be played by λ N,n,Gt , defined as follows. Define λ N,n,Gt to be the solution to (1) when the loss process term β CN,n L Nt in theequation for the intensities has been replaced by G ( t ). Let us also denote by τ N,n,G the corresponding defaulttime and by L N,Gt the corresponding loss process. It is easy to see that conditional on the systematic process X , the λ N,n,G for n = 1 , . . . , N are independent. The related empirical distribution will be accordinglydenoted by µ N,Gt .Let κ > θ pN,κ = inf ( t : 1 N N X n =1 h | λ N,nt | p + | λ N,n,Gt | p i ≥ κ p ) . Notice that the estimate in Lemma 8.1 implies that for
T > p ≥ κ →∞ sup N ∈ N P ( θ pN,κ ≤ T ) = 0Thus, it is enough to prove tightness for { Ξ N,nt ∧ θ N,κ , t ∈ [0 , T ] } N ∈ N .The following lemma provides a key estimate for the tightness proof. Lemma 8.4.
For J large enough (in particular for J > D + 1 ) and weights ( w, ρ ) such that Condition 7.1holds and for every T > , there is a constant C independent of N such that sup N ∈ N sup ≤ t ≤ T E (cid:18) N (cid:13)(cid:13)(cid:13) µ Nt ∧ θ N,κ − µ N,Gt ∧ θ N,κ (cid:13)(cid:13)(cid:13) W − J ( w,ρ ) (cid:19) ≤ C. Proof.
For notational convenience we shall write θ N in place of θ N,κ . Let us denote by η Nt = µ Nt − µ N,Gt andlet f ∈ C c ( ˆ P ). For notational convenience we define the operator( G x,ν,µ f )(ˆ p ) = ( L f )(ˆ p ) + ( L x f )(ˆ p ) + hQ , ν i ( L f )(ˆ p ) + hL f, µ i Q (ˆ p )After some term rearrangement, Itˆo’s formula gives us, via the representations of Lemmas 8.2-8.3, (cid:10) f, η Nt (cid:11) = Z t D G X s ,µ N,Gs ,µ Ns f, η Ns E ds + Z t (cid:10) L f, µ N,Gs (cid:11) (cid:10) Q , µ N,Gs − ¯ µ s (cid:11) ds + Z t D L X s f, η Ns E dV s − N N X n =1 Z t (cid:10) L f, µ Ns (cid:11) d N N,ns + N X n =1 Z t ˆ B N,n [ f ]( s ) d N N,ns + Z t ˆ A N [ f ]( X s , µ Ns ) ds + 1 N N X n =1 Z t f ( λ N,ns ) d N N,ns − N N X n =1 Z t f ( λ N,n,Gs ) d N N,n,Gs + 1 N N X n =1 Z t (cid:18) σ N,n q λ N,ns L f ( λ N,ns ) − σ N,n q λ N,n,Gs L f ( λ N,n,Gs ) (cid:19) dW ns Using the bounds of Lemma 6.1, Itˆo’s formula and the fact that common jumps occur with probabilityzero, we have E (cid:10) f, η Nt ∧ θ N (cid:11) = Z t E (cid:20) (cid:10) f, η Ns (cid:11) D G X s ,µ N,Gs ,µ Ns f, η Ns E + (cid:12)(cid:12)(cid:12)D L X s f, η Ns E(cid:12)(cid:12)(cid:12) (cid:21) χ { θ N ≥ s } ds + E Z t (cid:10) f, η Ns (cid:11) (cid:10) L f, µ N,Gs (cid:11) (cid:10) Q , µ N,Gs − ¯ µ s (cid:11) χ { θ N ≥ s } ds + O (cid:18) N (cid:19) here the O (cid:0) N (cid:1) term originates from the martingales and from Lemma 6.1. Next we apply Young’sinequality to get E (cid:10) f, η Nt ∧ θ N (cid:11) ≤ Z t E (cid:20) (cid:10) f, η Ns (cid:11) D G X s ,µ N,Gs ,µ Ns f, η Ns E + (cid:12)(cid:12)(cid:12)D L X s f, η Ns E(cid:12)(cid:12)(cid:12) (cid:21) χ { θ N ≥ s } ds + E Z t (cid:20) (cid:10) f, η Ns (cid:11) + 12 (cid:10) L f, µ N,Gs (cid:11) (cid:10) Q , µ N,Gs − ¯ µ s (cid:11) (cid:21) χ { θ N ≥ s } ds + O (cid:18) N (cid:19) = Z t E (cid:20) (cid:10) f, η Ns (cid:11) D G X s ,µ N,Gs ,µ Ns f, η Ns E + (cid:12)(cid:12)(cid:12)D L X s f, η Ns E(cid:12)(cid:12)(cid:12) + 2 (cid:10) f, η Ns (cid:11) (cid:21) χ { θ N ≥ s } ds + E Z t (cid:10) L f, µ N,Gs (cid:11) (cid:10) Q , µ N,Gs − ¯ µ s (cid:11) χ { θ N ≥ s } ds + O (cid:18) N (cid:19) Let us next bound the term on the last line of the previous equation. It is easy to see that Y s def = N (cid:10) Q , µ N,Gs − ¯ µ s (cid:11) = N X n =1 (cid:0) λ N,n,Gs m N,n,Gs − hQ , ¯ µ s i (cid:1) is a discrete time martingale with respect to P ( ·|V ). Thus, by Theorem 3.2 of Burkholder (1973), we havethat E h | Y s | (cid:12)(cid:12)(cid:12) V i ≤ C E h | S ( Y s ) | (cid:12)(cid:12)(cid:12) V i where, maintaining the notation of Burkholder (1973), S ( Y s ) = vuut N X n =1 h λ N,n,Gs m N,n,Gs − hQ , ¯ µ s i i . Therefore, recalling the bound from Lemma 8.1 we obtain1 N E | Y s | = 1 N E h E h | Y s | (cid:12)(cid:12)(cid:12) V ii ≤ CN E h E h | S ( Y s ) | (cid:12)(cid:12)(cid:12) V ii = CN E N X n =1 (cid:2) λ N,n,Gs m N,n,Gs − hQ , ¯ µ s i (cid:3) ≤ C This bound implies that E Z t (cid:20) (cid:10) L f, µ N,Gs (cid:11) (cid:10) Q , µ N,Gs − ¯ µ s (cid:11) χ { θ N ≥ s } (cid:21) ds ≤ N C ( t, K, κ ) k f k W For J large enough ( J > D + 1), Proposition 3.15 of Gy¨ongy & Krylov (1990) and Theorem 6.53 of Adams(1978) immediately imply that the embedding W J ֒ → W is of Hilbert-Schmidt type. So, by Lemma 1 andTheorem 2 in Chapter 2.2 of Gel’fand & Vilenkin (1964), if { f a } a ≥ is a complete orthonormal basis for W J ,then P a ≥ k f a k W < ∞ .Hence, if { f a } a ≥ is a complete orthonormal basis for W J ( w, ρ ) we obtain that E (cid:10) f a , η Nt ∧ θ N (cid:11) ≤ Z t E (cid:20) (cid:10) f a , η Ns (cid:11) D G X s ,µ N,Gs ,µ Ns f a , η Ns E + (cid:12)(cid:12)(cid:12)D L X s f a , η Ns E(cid:12)(cid:12)(cid:12) + 2 (cid:10) f a , η Ns (cid:11) (cid:21) χ { θ N ≥ s } ds + 1 N C k f a k W Summing over a ≥ E (cid:13)(cid:13) η Nt ∧ θ N (cid:13)(cid:13) − J ≤ Z t E (cid:20) D η Ns , G ∗ X s ,µ N,Gs ,µ Ns η Ns E − J + (cid:13)(cid:13)(cid:13) L ∗ ,X s η Ns (cid:13)(cid:13)(cid:13) − J + 2 (cid:13)(cid:13) η Ns (cid:13)(cid:13) − J (cid:21) χ { θ N ≥ s } ds + 1 N C. hen as in Lemma 9.6 we get that E (cid:20) D η Ns , G ∗ X s ,µ N,Gs ,µ Ns η Ns E − J + (cid:13)(cid:13)(cid:13) L ∗ ,X s η Ns (cid:13)(cid:13)(cid:13) − J (cid:21) χ { θ N ≥ s } ≤ C E (cid:13)(cid:13) η Ns (cid:13)(cid:13) − J χ { θ N ≥ s } So, E (cid:13)(cid:13) η Nt ∧ θ N (cid:13)(cid:13) − J ≤ C (cid:20)Z t E (cid:13)(cid:13) η Ns ∧ θ N (cid:13)(cid:13) − J ds + 1 N (cid:21) Finally, an application of Gronwall’s lemma concludes the proof. (cid:3)
The following lemma provides a uniform bound for the fluctuation process.
Lemma 8.5.
Let
J > D + 1 and weights ( w, ρ ) such that Condition 7.1 holds. For every T > , there is aconstant C independent of N such that sup N ∈ N sup ≤ t ≤ T E (cid:18)(cid:13)(cid:13)(cid:13) Ξ Nt ∧ θ N,κ (cid:13)(cid:13)(cid:13) W − J ( w,ρ ) (cid:19) ≤ C. Proof.
Clearly Ξ Nt = √ N ( µ Nt − µ N,Gt ) + √ N ( µ N,Gt − ¯ µ t ) = √ N η Nt + √ N η
N,Gt . Notice now that by Lemmas8.2 and 8.3 N D f, η N,Gt E = N X n =1 h f ( λ N,n,Gt ) m N,n,Gt − h f, ¯ µ t i i is a P ( ·|V t ) discrete time martingale, which in turn implies via Theorem 3.2 of Burkholder (1973) andProposition 3.15 of Gy¨ongy & Krylov (1990) (similarly to the proof of Lemma 8.4) that E D f, η N,Gt E ≤ N C k f k W D +10 . Therefore, if { f a } a ≥ is a complete orthonormal basis for W J ( w, ρ ) with J > D + 1, Parseval’s identitygives (similarly to Lemma 8.4)(27) N E (cid:13)(cid:13)(cid:13) η N,Gt ∧ θ N (cid:13)(cid:13)(cid:13) − J ≤ C. Since, E (cid:10) f a , Ξ Nt ∧ θ N (cid:11) ≤ C (cid:20) N E (cid:10) f a , η Nt ∧ θ N (cid:11) + N E D f a , η N,Gt ∧ θ N E (cid:21) summing over a ≥ E (cid:13)(cid:13) Ξ Nt ∧ θ N (cid:13)(cid:13) − J ≤ C (cid:20) N E (cid:13)(cid:13) η Nt ∧ θ N (cid:13)(cid:13) − J + N E (cid:13)(cid:13)(cid:13) η N,Gt ∧ θ N (cid:13)(cid:13)(cid:13) − J (cid:21) which by Lemma 8.4 (obviously J > D + 1 > D + 1) and (27) yields the statement of the lemma. (cid:3) Next we discuss relative compactness for {√ N ˜ M Nt , t ∈ [0 , T ] } N ∈ N . In particular, we have the followinglemma. Lemma 8.6.
Let
T > and let J > D + 1 and weights ( w, ρ ) such that Condition 7.1 holds. The process {√ N ˜ M Nt , t ∈ [0 , T ] } N ∈ N is a W − J ( w, ρ ) − valued martingale such that sup N ∈ N E (cid:20) sup ≤ t ≤ T (cid:13)(cid:13)(cid:13) √ N ˜ M Nt (cid:13)(cid:13)(cid:13) W − J ( w,ρ ) (cid:21) ≤ C < ∞ . Furthermore, it is relatively compact in D W − ( J + D )0 ( w,ρ ) [0 , T ] .Proof. Clearly, it is enough to prove tightness for {√ N ˜ M Nt ∧ θ pN,κ , t ∈ [0 , T ] } N ∈ N . Let us defineΓ Ns [ f ] = (cid:10) L f, µ Ns (cid:11) + (cid:10) L f, µ Ns (cid:11) + (cid:10) L f, µ Ns (cid:11) (cid:10) Q , µ Ns (cid:11) − (cid:10) L f, µ Ns (cid:11) (cid:10) L f, µ Ns (cid:11) ach one of the terms in the last display can be bounded by above similarly to the derivation of the upperbound for the term hL f, µ i in (23). The bound from Lemma 8.1 is then used in order to treat the integralsof the weight functions with respect to µ Ns . It then follows that there exists a constant C such that E (cid:20)Z tr Γ s [ f ] χ { θ pN,κ ≥ s } ds (cid:12)(cid:12)(cid:12) F Nr (cid:21) ≤ C ( t, K, κ ) k f k W ( t − r )Therefore, if { f a } a ≥ is a complete orthonormal basis for W J ( w, ρ ) with J > D + 1, we obtain E (cid:20)(cid:13)(cid:13)(cid:13) √ N ˜ M Nt ∧ θ pN,κ − √ N ˜ M Nr ∧ θ pN,κ (cid:13)(cid:13)(cid:13) W − J ( w,ρ ) | F Nr (cid:21) ≤ X a ≥ E (cid:20)D f a , √ N ˜ M Nt ∧ θ pN,κ − √ N ˜ M Nr ∧ θ pN,κ E | F Nr (cid:21) ≤ C ( T, K, κ ) X a ≥ E (cid:20)Z tr Γ s [ f a ] χ { θ pN,κ ≥ s } ds | F Nr (cid:21) ≤ C ( T, K, κ ) X a ≥ k f a k W ( t − r )(28)As in the proof of Lemma 8.4, the restriction J > D + 1 implies that P a ≥ k f a k W < ∞ . Similarly, we canalso show that there exists a constant C such thatsup N ∈ N E (cid:20) sup ≤ t ≤ T (cid:13)(cid:13)(cid:13) √ N ˜ M Nt ∧ θ pN,κ (cid:13)(cid:13)(cid:13) W − J (cid:21) ≤ C Moreover, due to the inequality k·k W − ( J + D )0 ≤ C k·k W − J , the inequality (28) implies thatsup N ∈ N E (cid:20)(cid:13)(cid:13)(cid:13) √ N ˜ M Nt ∧ θ pN,κ − √ N ˜ M Nr ∧ θ pN,κ (cid:13)(cid:13)(cid:13) W − ( J + D )0 | F Nr (cid:21) ≤ C ( t − r )which obviously goes to zero as | t − r | ↓
0. The last two displays give relative compactness of {√ N ˜ M Nt , t ∈ [0 , T ] } N ∈ N in D W − ( J + D )0 [0 , T ] (Theorem 8.6 of Chapter 3 of Ethier & Kurtz (1986) and page 35 of Joffe &M´etivier (1986)).The uniform bound of the lemma follows by the fact that θ pN,κ → ∞ as κ → ∞ , see (26). (cid:3) Regarding the convergence of the martingale √ N ˜ M N we have the following lemma. Lemma 8.7.
Let
J > D +1 and weights ( w, ρ ) such that Condition 7.1 holds. The sequence n √ N ˜ M Nt , t ∈ [0 , T ] o N ∈ N is relatively compact in D W − J ( w,ρ ) [0 , T ] . Moreover, it converges towards a distribution valued martingale (cid:8) ¯ M t , t ∈ [0 , T ] (cid:9) with conditional (on the σ − algebra V ) covariance function, defined, for f, g ∈ W J ( w, ρ ) , by(7). The martingale (cid:8) ¯ M t , t ∈ [0 , T ] (cid:9) is conditionally on the σ − algebra V , Gaussian.Proof. Relative compactness follows by Lemma 8.6. Due to continuous dependence of (21) on µ N and onthe weak convergence of µ N · → ¯ µ · by Giesecke et al. (2012), we obtain that any limit point of √ N ˜ M Nt as N → ∞ , ¯ M , will satisfy (7). Conditionally on V , the limiting ¯ M is a continuous square integrablemartingale and its predictable variation is deterministic. Thus, by Theorem 7.1.4 in Ethier & Kurtz (1986),it is conditionally Gaussian. This concludes the proof. (cid:3) Next we discuss relative compactness of the process n Ξ N,nt , t ∈ [0 , T ] o N ∈ N . Lemma 8.8.
Let
T > , J > D + 1 and weights ( w, ρ ) such that Condition 7.1 holds. The process { Ξ N,nt , t ∈ [0 , T ] } N ∈ N is relatively compact in D W − J ( w,ρ ) [0 , T ] .Proof. It is enough to prove tightness for { Ξ N,nt ∧ θ N,κ , t ∈ [0 , T ] } N ∈ N . For J > D + 1, the bound from Lemma8.5 holds, i.e.,(29) sup N ∈ N sup ≤ t ≤ T E (cid:18)(cid:13)(cid:13)(cid:13) Ξ Nt ∧ θ N,κ (cid:13)(cid:13)(cid:13) W − J (cid:19) ≤ C. et { f a } a ≥ be a complete orthonormal basis for W J with J > J + D > D + 1. By (20) we have(30) (cid:10) f a , Ξ Nt (cid:11) = (cid:10) f a , Ξ Nr (cid:11) + Z tr (cid:10) G X s , ¯ µ s ,µ Ns f a , Ξ Ns (cid:11) ds + Z tr D L X s f a , Ξ Ns E dV s + D f a , √ N ˜ M Nt,r E + R Nt,r . Next, we consider the mapping H from W J ( w, ρ ) into R defined by H ( f ) = (cid:10) G X s , ¯ µ s ,µ Ns f, Ξ Ns (cid:11) and we notice that (cid:10) G X s , ¯ µ s ,µ Ns f, Ξ Ns (cid:11) ≤ (cid:13)(cid:13) G X s , ¯ µ s ,µ Ns f (cid:13)(cid:13) W J (cid:13)(cid:13) Ξ Ns (cid:13)(cid:13) W − J ≤ C k f k W J (cid:13)(cid:13) Ξ Ns (cid:13)(cid:13) W − J ≤ C k f k W J D (cid:13)(cid:13) Ξ Ns (cid:13)(cid:13) W − J where the second inequality is due to Lemma 7.2 and the third inequality because D >
2. Hence, byParseval’s identity we have k H k W − J ≤ C (cid:13)(cid:13) Ξ Ns (cid:13)(cid:13) W − J Thus, by (29) we get X a ≥ E (cid:20)Z tr (cid:10) G X s , ¯ µ s ,µ Ns f a , Ξ Ns (cid:11) χ { θ pN,κ ≥ s } ds | F Nr (cid:21) ≤ C E (cid:20)Z tr (cid:13)(cid:13) Ξ Ns (cid:13)(cid:13) W − J χ { θ pN,κ ≥ s } ds | F Nr (cid:21) ≤ C ( t − r )Similarly we also obtain that X a ≥ E (cid:20)Z tr D L X s f a , Ξ Ns E χ { θ pN,κ ≥ s } ds | F Nr (cid:21) ≤ C ( t − r )The last estimates, the uniform bound from Lemma 8.5, Lemma 8.6 for √ N ˜ M Nt and Lemma 6.1 for theremainder term R Nt,r imply the statement of the lemma. We follow the same steps as in the proof of Lemma8.6 and thus the details are omitted. (cid:3)
We end this section by proving a continuity result.
Lemma 8.9.
Any limit point of { Ξ Nt , t ∈ [0 , T ] } N ∈ N and {√ N ˜ N Nt , t ∈ [0 , T ] } N ∈ N is continuous, i.e., ittakes values in C W − J ( w,ρ ) [0 , T ] , with J > D + 1 .Proof. In order to prove that any limit point of (cid:8) Ξ Nt , t ∈ [0 , T ] (cid:9) N ∈ N takes values in C W − J ( w,ρ ) [0 , T ], it issufficient to show that lim N →∞ E (cid:20) sup t ≤ T (cid:13)(cid:13) Ξ Nt − Ξ Nt − (cid:13)(cid:13) W − J ( w,ρ ) (cid:21) = 0Let { f a } a ≥ be a complete orthonormal basis for W J ( w, ρ ). Then, by definition, we have (cid:10) f a , Ξ Nt − Ξ Nt − (cid:11) = √ N (cid:2)(cid:10) f a , µ Nt − ¯ µ t (cid:11) − (cid:10) f a , µ Nt − − ¯ µ t − (cid:11)(cid:3) = √ N (cid:2)(cid:10) f a , µ Nt − µ Nt − (cid:11) − h f a , ¯ µ t − ¯ µ t − i (cid:3) = √ N h J f a N,n ( t ) − h f a , ¯ µ t − ¯ µ t − i i = √ N (cid:20) J f a N,n ( t ) − N ˜ J f a N,n ( t ) (cid:21) + 1 √ N ˜ J f a N,n ( t ) − √ N [ h f a , ¯ µ t − ¯ µ t − i ] . Clearly, t ¯ µ t is continuous, so we only need to consider the first two terms. The first term is bounded by K N (cid:13)(cid:13)(cid:13) ∂ f a ∂λ (cid:13)(cid:13)(cid:13) C , see (19), whereas for the second we immediately get (cid:12)(cid:12)(cid:12) ˜ J f a N,n ( t ) (cid:12)(cid:12)(cid:12) ≤ K [ k f a k + k f ′ a k ] . o, as in Proposition 3.15 of Gy¨ongy & Krylov (1990), we get that E sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:10) f a , Ξ Nt − Ξ Nt − (cid:11)(cid:12)(cid:12) ≤ C (cid:20) N / + 1 √ N (cid:21) k f a k W D +30 . Thus, E (cid:20) sup t ≤ T (cid:13)(cid:13) Ξ Nt − Ξ Nt − (cid:13)(cid:13) W − J ( w,ρ ) (cid:21) ≤ X a ≥ E (cid:20) sup t ≤ T (cid:10) f a , Ξ Nt − Ξ Nt − (cid:11)(cid:21) ≤ C (cid:20) N / + 1 √ N (cid:21) X a ≥ k f a k W D +30 Since
J > D + 1, we certainly have W J ֒ → W D +30 , and then as in Lemma 8.4, P a ≥ k f a k W D +30 < ∞ ,which implies that after taking the limit as N → ∞ , the right hand side of the last display goes to zero.This concludes the proof of continuity of the limit point trajectories of { Ξ Nt , t ∈ [0 , T ] } N ∈ N .Next we consider continuity of the trajectories of the limit points of {√ N ˜ N Nt , t ∈ [0 , T ] } N ∈ N . It followsdirectly by (30) and Lemma 6.1 that Ξ Nt and √ N ˜ M Nt have the same discontinuities. Thus, the continuity ofany limit point of { Ξ Nt , t ∈ [0 , T ] } N ∈ N implies the continuity of any limit point of {√ N ˜ N Nt , t ∈ [0 , T ] } N ∈ N ,which concludes the proof of the lemma. (cid:3) Uniqueness
In this section we prove uniqueness of the stochastic evolution equation (6). Let ¯Ξ t , ¯Ξ t be two solutionsof (6) and let us define Φ t = ¯Ξ t − ¯Ξ t . Then, Φ t will satisfy the stochastic evolution equation h f, Φ t i = Z t D L f + L X s f, Φ s E ds + Z t [ hQ , ¯ µ s i hL f, Φ s i + hQ , Φ s i hL f, ¯ µ s i ] ds + Z t D L X s f, Φ s E dV s = Z t hG X s , ¯ µ s f, Φ s i ds + Z t D L X s f, Φ s E dV s a.s. Notice that this is a linear equation. In order to prove uniqueness, it is enough to show that E k Φ t k − J = 0Let { f a } be a complete orthonormal basis for W J ( w, ρ ). By Itˆo’s formula we get that |h f a , Φ t i| = Z t h f a , Φ s i hD L f a + L X s f a , Φ s E + hQ , ¯ µ s i hL f a , Φ s i + hQ , Φ s i hL f a , ¯ µ s i i ds ++ Z t (cid:12)(cid:12)(cid:12)D L X s f a , Φ s E(cid:12)(cid:12)(cid:12) ds + Z t h f a , Φ s i D L X s f a , Φ s E dV s = Z t h f a , Φ s i hG X s , ¯ µ s f a , Φ s i ds + Z t (cid:12)(cid:12)(cid:12)D L X s f a , Φ s E(cid:12)(cid:12)(cid:12) ds + Z t h f a , Φ s i D L X s f a , Φ s E dV s Then, summing over a and taking expected value (due to Lemma 7.2, the expected value of the stochasticintegral is zero) we have(31) E k Φ t k − J = E Z t h Φ s , G ∗ s Φ s i − J ds + E Z t (cid:13)(cid:13)(cid:13) L ∗ ,X s Φ s (cid:13)(cid:13)(cid:13) − J ds Hence, if we prove that there is a constant
C > h φ, G ∗ φ i − J + (cid:13)(cid:13) L ∗ ,x φ (cid:13)(cid:13) − J ≤ C k φ k − J then we can conclude by Gronwall inequality that Φ t = 0 a.s.Let us recall now that( G x,µ f )(ˆ p ) = ( L f )(ˆ p ) + ( L x f )(ˆ p ) + hQ , µ i ( L f )(ˆ p ) + hL f, µ i Q (ˆ p ) nd set ( G x,µ f )(ˆ p ) = ( L f )(ˆ p ) + ( L x f )(ˆ p ) + hQ , µ i ( L f )(ˆ p )( G x,µ f )(ˆ p ) = hL f, µ i Q (ˆ p )and ( G is f )(ˆ p ) = ( G iX s , ¯ µ s f )(ˆ p ) for i=1,2.Moreover, for simplicity in presentation and without loss of generality, we shall consider from now ononly the case of a homogeneous portfolio, i.e, set ν = δ ˆ p . This is done without loss of generality, since theoperators L i f involve differentiation only with respect to λ . The proof for the general heterogeneous case,is identical with heavier notation.Before a term by term examination, we gather some straightforward results in the following lemma. Lemma 9.1.
Let ψ ∈ C ∞ c ( R + ) and J ≥ and assume Condition 7.1. Then, up to a multiplicative constantfor the term O (cid:16) k ψ k J (cid:17) , that may be different from line to line, we have J X k =0 Z R + w ( λ ) ρ k ( λ ) λψ ( k +1) ( λ ) ψ ( k ) ( λ ) dλ = O (cid:16) k ψ k J (cid:17) J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k +1) ( λ ) ψ ( k ) ( λ ) dλ = O (cid:16) k ψ k J (cid:17) J X k =0 Z R + w ( λ ) ρ k ( λ ) λψ ( k +2) ( λ ) ψ ( k ) ( λ ) dλ = − Z R + w ( λ ) ρ k ( λ ) λ (cid:12)(cid:12)(cid:12) ψ ( k +1) ( λ ) (cid:12)(cid:12)(cid:12) dλ + O (cid:16) k ψ k J (cid:17) J X k =0 Z R + w ( λ ) ρ k ( λ ) λ ψ ( k +2) ( λ ) ψ ( k ) ( λ ) dλ = − Z R + w ( λ ) ρ k ( λ ) (cid:12)(cid:12)(cid:12) λψ ( k +1) ( λ ) (cid:12)(cid:12)(cid:12) dλ + O (cid:16) k ψ k J (cid:17) Proof.
It follows directly by integration by parts using Condition 7.1 and the assumption that ψ and itsderivatives are compactly supported. (cid:3) Then we bound each term on the right hand side of (31). First, we notice that, by Riesz representationtheorem, for φ ∈ W − J there exists a unique ψ = F ( φ ) ∈ W J such that h φ, f i = h ψ, f i J By a density argument we may assume that ψ = F ( φ ) ∈ W J +20 and obtain h φ, G ∗ φ i − J = h ψ, G ∗ φ i = hG ψ, φ i = hG ψ, ψ i J . which is true since by Lemma 7.2, G ψ ∈ W J . Lemma 9.2.
For φ ∈ W − J such that ψ = F ( φ ) ∈ W J +20 we have (cid:10) φ, G , ∗ φ (cid:11) − J = (cid:0) | b ( x ) | + | σ ( x ) | + hQ , µ i (cid:1) O (cid:16) k φ k − J (cid:17) − (cid:12)(cid:12) β S σ ( x ) (cid:12)(cid:12) J X k =0 Z R + w ( λ ) ρ k ( λ ) (cid:12)(cid:12)(cid:12) λψ ( k +1) ( λ ) (cid:12)(cid:12)(cid:12) dλ − σ J X k =0 Z R + w ( λ ) ρ k ( λ ) λ (cid:12)(cid:12)(cid:12) ψ ( k +1) ( λ ) (cid:12)(cid:12)(cid:12) dλ − J X k =0 Z R + w ( λ ) ρ k ( λ ) λ (cid:12)(cid:12)(cid:12) ψ ( k ) ( λ ) (cid:12)(cid:12)(cid:12) dλ Since ψ ∈ W J +20 , the integrals on the right hand side are well defined and bounded.Proof. We recall that (cid:10) φ, G , ∗ φ (cid:11) − J = (cid:10) G ψ, ψ (cid:11) J . and that ( G ψ )(ˆ p ) = ( L ψ )(ˆ p ) + ( L x ψ )(ˆ p ) + hQ , µ i ( L ψ )(ˆ p )Hence we bound each of the three terms separately. Using the statements of Lemma 9.1 we have for L ψ X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) ( L ψ ( λ )) ( k ) dλ == 12 σ J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) (cid:16) λψ (2) ( λ ) (cid:17) ( k ) dλ − α J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) (cid:16) λψ (1) ( λ ) (cid:17) ( k ) dλ + α ¯ λ J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) ψ ( k +1) ( λ ) dλ − J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) ( λψ ( λ )) ( k ) dλ = 12 σ J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) (cid:16) κψ ( k +1) ( λ ) + λψ ( k +2) ( λ ) (cid:17) dλ − J X k =0 α Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) (cid:16) κψ ( k ) ( λ ) + λψ ( k +1) ( λ ) (cid:17) dλ + J X k =0 α ¯ λ Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) ψ ( k +1) ( λ ) dλ − J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) (cid:16) κψ ( k − ( λ ) + λψ ( k ) ( λ ) (cid:17) dλ = O (cid:16) k ψ k J (cid:17) − σ J X k =0 Z R + w ( λ ) ρ k ( λ ) λ (cid:12)(cid:12)(cid:12) ψ ( k +1) ( λ ) (cid:12)(cid:12)(cid:12) dλ − J X k =0 Z R + w ( λ ) ρ k ( λ ) λ (cid:12)(cid:12)(cid:12) ψ ( k ) ( λ ) (cid:12)(cid:12)(cid:12) dλ = O (cid:16) k φ k − J (cid:17) − σ J X k =0 Z R + w ( λ ) ρ k ( λ ) λ (cid:12)(cid:12)(cid:12) ψ ( k +1) ( λ ) (cid:12)(cid:12)(cid:12) dλ − J X k =0 Z R + w ( λ ) ρ k ( λ ) λ (cid:12)(cid:12)(cid:12) ψ ( k ) ( λ ) (cid:12)(cid:12)(cid:12) dλ Similarly, for L x ψ we have J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) ( L x ψ ( λ )) ( k ) dλ == β S b ( x ) J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) (cid:16) λψ (1) ( λ ) (cid:17) ( k ) dλ + 12 (cid:12)(cid:12) β S σ ( x ) (cid:12)(cid:12) J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) (cid:16) λ ψ (2) ( λ ) (cid:17) ( k ) dλ = β S b ( x ) J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) (cid:16) kψ ( k ) ( λ ) + λψ ( k +1) ( λ ) (cid:17) dλ + 12 (cid:12)(cid:12) β S σ ( x ) (cid:12)(cid:12) J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) (cid:18) λ ψ ( k +2) ( λ ) + 2 kλψ ( k +1) + w ( λ ) ρ k ( λ ) k !( k − ψ ( k ) (cid:19) dλ = (cid:16) b ( x ) + (cid:12)(cid:12) β S σ ( x ) (cid:12)(cid:12) (cid:17) O (cid:16) k ψ k J (cid:17) − (cid:12)(cid:12) β S σ ( x ) (cid:12)(cid:12) J X k =0 Z R + w ( λ ) ρ k ( λ ) (cid:12)(cid:12)(cid:12) λψ ( k +1) ( λ ) (cid:12)(cid:12)(cid:12) dλ = (cid:16) b ( x ) + (cid:12)(cid:12) β S σ ( x ) (cid:12)(cid:12) (cid:17) O (cid:16) k φ k − J (cid:17) − (cid:12)(cid:12) β S σ ( x ) (cid:12)(cid:12) J X k =0 Z R + w ( λ ) ρ k ( λ ) (cid:12)(cid:12)(cid:12) λψ ( k +1) ( λ ) (cid:12)(cid:12)(cid:12) dλ Lastly, for the third term, hQ , µ i ( L ψ )(ˆ p ), we have hQ , µ i J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) ( L ψ ( λ )) ( k ) dλ = hQ , µ i J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) ψ ( k +1) ( λ ) dλ = hQ , µ i O (cid:16) k φ k − J (cid:17) ence, putting all the estimates together, we conclude the proof of the lemma. (cid:3) Lemma 9.3.
Let us assume that µ is such that R R + (cid:0) w ( λ ) ρ ( λ ) (cid:1) − µ ( dλ ) < ∞ . For φ ∈ W − J such that ψ ∈ W J we have (cid:10) φ, G , ∗ φ (cid:11) − J = O (cid:16) k φ k − J (cid:17) Proof.
By definition we have (cid:10) G ψ, ψ (cid:11) J = hL ψ, µ i J X k =0 Z R + w ( λ ) ρ k ( λ ) λ ( k ) ψ ( k ) ( λ ) dλ ≤ s k ψ k Z R + ( w ( λ ) ρ ( λ )) − µ ( dλ ) ×× Z R + w ( λ ) λ dλ ! / k ψ k + Z R + w ( λ ) ρ ( λ ) dλ ! / k ψ k ≤ C k ψ k J = C k φ k − J which concludes the proof of the lemma. (cid:3) Remark 9.4.
Notice that the condition on the integrability of µ that appears in the statement of Lemma9.3, R R + (cid:0) w ( λ ) ρ ( λ ) (cid:1) − µ ( dλ ) < ∞ , is equivalent to assuming that µ has finite moments at least up to order | β | − if the weights are chosen such that ρ ( λ ) = p | λ | and w ( λ ) = (cid:0) | λ | (cid:1) β and β < . Lemma 9.5.
For φ ∈ W − J such that ψ ∈ W J +10 we have (cid:13)(cid:13) L ∗ ,x φ (cid:13)(cid:13) − J = (cid:12)(cid:12) β S σ ( x ) (cid:12)(cid:12) O (cid:16) k φ k − J (cid:17) + (cid:12)(cid:12) β S σ ( x ) (cid:12)(cid:12) J X k =0 Z R + w ( λ ) ρ k ( λ ) (cid:12)(cid:12)(cid:12) λψ ( k +1) ( λ ) (cid:12)(cid:12)(cid:12) dλ Since ψ ∈ W J +10 the integral on the right hand side is well defined and bounded.Proof. By definition we have (cid:13)(cid:13) L ∗ ,x φ (cid:13)(cid:13) − J = sup ζ ∈ W J (cid:12)(cid:12)(cid:10) L ∗ ,x φ, ζ (cid:11)(cid:12)(cid:12) k ζ k J = sup ζ ∈ W J |h ψ, L x ζ i J |k ζ k J By the definition of the operator L x we have J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) ( L x ζ ) ( k ) dλ == β S σ ( x ) J X k =0 Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) (cid:16) λζ (1) ( λ ) (cid:17) ( k ) dλ = β S σ ( x ) " J X k =0 k Z R + w ( λ ) ρ k ( λ ) ψ ( k ) ( λ ) ζ ( k ) ( λ ) dλ + J X k =0 Z R + w ( λ ) ρ k ( λ ) λψ ( k ) ( λ ) ζ ( k +1) ( λ ) dλ ≤ (cid:12)(cid:12) β S σ ( x ) (cid:12)(cid:12) " J X k =0 Z R + w ( λ ) ρ k ( λ ) (cid:12)(cid:12)(cid:12) λψ ( k +1) ( λ ) + Cψ ( k ) ( λ ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ζ ( k ) ( λ ) (cid:12)(cid:12)(cid:12) dλ ≤ (cid:12)(cid:12) β S σ ( x ) (cid:12)(cid:12) Z R + vuut J X k =0 w ( λ ) ρ k ( λ ) (cid:12)(cid:12) λψ ( k +1) ( λ ) + Cψ ( k ) ( λ ) (cid:12)(cid:12) vuut J X k =0 w ( λ ) ρ k ( λ ) (cid:12)(cid:12) ζ ( k ) ( λ ) (cid:12)(cid:12) dλ (cid:12)(cid:12) β S σ ( x ) (cid:12)(cid:12) vuut J X k =0 Z R + w ( λ ) ρ k ( λ ) (cid:12)(cid:12) λψ ( k +1) ( λ ) + Cψ ( k ) ( λ ) (cid:12)(cid:12) dλ vuut J X k =0 Z R + w ( λ ) ρ k ( λ ) (cid:12)(cid:12) ζ ( k ) ( λ ) (cid:12)(cid:12) dλ = (cid:12)(cid:12) β S σ ( x ) (cid:12)(cid:12) vuut J X k =0 Z R + w ( λ ) ρ k ( λ ) h(cid:12)(cid:12) λψ ( k +1) ( λ ) (cid:12)(cid:12) + C (cid:12)(cid:12) ψ ( k ) ( λ ) (cid:12)(cid:12) + 2 Cλψ ( k +1) ( λ ) ψ ( k ) ( λ ) i dλ k ζ k J The first inequality follows from integration by parts on the second integral and the second and thirdinequality by Cauchy-Schwartz using Condition 7.1. Therefore, by canceling the term k ζ k J and then takingthe square of the resulting expression, the statement of the lemma follows. (cid:3) Collecting the statements of Lemma 9.2 and Lemma 9.3 we obtain the following lemma.
Lemma 9.6.
For φ ∈ W − J ( w, ρ ) we have h φ, G ∗ φ i − J + (cid:13)(cid:13) L ∗ ,x φ (cid:13)(cid:13) − J ≤ C (cid:16) | b ( x ) | + | σ ( x ) | + hQ , µ i (cid:17) k φ k − J . Then we are in position to prove uniqueness for the limiting stochastic evolution equation.
Theorem 9.7.
Let us assume that R R + (cid:0) w ( λ ) ρ ( λ ) (cid:1) − µ ( dλ ) < ∞ . Then, the solution to the stochasticevolution equation (6) is unique in W − J ( w, ρ ) .Proof. Equation (31) gives, via Lemma 9.6, E k Φ t k − J = E Z t (cid:18) (cid:10) Φ s , G ∗ X s , ¯ µ s Φ s (cid:11) − J + (cid:13)(cid:13)(cid:13) L ∗ ,X s Φ s (cid:13)(cid:13)(cid:13) − J (cid:19) ds ≤ C Z t E k Φ s k − J ds. Therefore, by applying Gronwall inequality we get that E k Φ t k − J = 0, which concludes the proof of thetheorem. (cid:3) Appendix A. Performance of Numerical Schemes
This appendix provides additional numerical results on the performance of the numerical schemes de-scribed in Sections 5.3.1 and 5.3.2.In Figure 6, we compare the standard error of Schemes 1 and 2 when estimating the expectation of f ( L NT ) = ( L NT − S ) + (i.e., a call option on the portfolio loss). Parameters are a time step of 0 . S = 0 .
12, and K = 6 as the truncation level. Scheme 1 is performed using an optimal choice of J and M given by (14). For small β S , Scheme 2 outperforms Scheme 1 since the conditionally Gaussian noisedominates. For large β S , the process X dominates the dynamics and the semi-analytic calculation reducesthe standard deviation an insufficient amount to justify the additional computational time.We also compare the standard error of the second-order approximation (calculated using Scheme 1) anddirect simulation of the original finite system (1) in Figure 7. We use a truncation level of K = 6 and a timestep of 0 .
005 for both asymptotic and finite systems. We report exact values in the following table for a totalcomputational time of 50 seconds. For N = 1 ,
000 and N = 2 , N = 1 , × − × − Standard Error ( N = 2 , × − × − igure 6. Comparison of standard error for Scheme 1 and Scheme 2, respectively describedin Sections 5.3.1 and 5.3.2. Parameter case is T = 0 . , N = 500 , σ = 0 . , α = 4 , λ = ¯ λ = 0 . , and β C = 1. The systematic risk X is an OU process with mean 1, reversion speed 2,volatility 1, and initial value 1. References
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Department of Mathematics & Statistics, Boston University, Boston, MA 02215
E-mail address : [email protected] Department of Management Science and Engineering, Stanford University, Stanford, CA 94305
E-mail address : [email protected] Department of Management Science and Engineering, Stanford University, Stanford, CA 94305
E-mail address : [email protected]@stanford.edu