A Queueing Network Approach

to Portfolio Credit Risk

Mark Davis and Juan C. EsparragozaDepartment of MathematicsImperial College, London

31st October 2004

1 Introduction

In the modelling of credit risk it has been always important to consider aportfolio as a whole. Moreover, both risk management and valuation hasbeen challenged by the development of the credit risk market with instru-ments based on baskets such as first to default swaps and CDOs.Modelling the credit risk of a portfolio involve:

• Credible Modelling of Interaction Effects: The literature distinguishbetween two main interaction effects. First, common factors that in-fluence defaults, such as the economic cycle. Second, the contagioneffect where the default of an obligor may affect the probability ofdefault of others obligors.

• Efficient Computational Methods: Many models that are convenientfor a single obligor or a few of them may be computationally expen-sive to use for large portfolios. For example, the use of reduced formmodels with copula dependence structure can be cumbersome whenthe number of obligors increases. Algorithms of simulation/pricingmust show little sensitivity in computational terms to the number ofelements.

• Ease of Calibration: Together with the computational issue, a desir-able property of a model is an easy calibration. It is reflected in theexistence f a methodology that allows for simplicity . A parsimonywith the number of parameters must be reached. While a large num-ber of parameters may be cumbersome and redundant, too few mayprove insufficient to reflect all the market data.

In the case of large portfolios it makes sense to consider the use oflarge-sample approximations in order to achieve efficient computational al-gorithms and keep easy calibrations.

1

Maybe the best known work that illustrates the modelling of large port-folios by using an homogeneity assumption assumption is by Vasicek (1987).We briefly present below this model to introduce our proposal. More recentwork by Hull and White (2004) extends this to non-homogeneous portfolios.The authors present an algorithm to simplify the computational burden ofthe model. Unfortunately, sometimes this proves to be unstable and somemodifications are required.Another model that is widely used and exploits the use of approxima-

tions is CreditRisk+ (CSFB 1997). Recently, Gordy (2001) proposed theuse of saddlepoint approximations to calculate quantiles of the loss distribu-tion under CreditRisk+ outperforming the commonly used Panjer recursionmethod.A different approach to the loss distribution estimation is given by Giesecke

and Weber (2003) applying a gaussian approximation to the loss distributionin a voter model.All models mentioned above try to estimate the loss distribution of a

portfolio in a fixed point in the time, this is, they are static. Recently,Frey and Backhaus (2003) presented a model similar to the one presentedhere and considering a mean field interaction approach. They derive strongconvergence results conditional on a general exogenous process.This paper is structured as follows. In order to introduce the idea of

large portfolio modelling we review briefly the Vasicek model and commenton some of its advantages and critical points. In section 3 we introduceheuristically the ideas and assumptions of the proposed model, in particularwe intend to present the queuing network as a natural structure for thecredit risk modelling. The general model is presented formally in section4 together with the results that allow approximate the distribution of theprocess. Section 5 presents the application to a simple credit risk model andexplicit solutions to the approximated processes are presented. Finally wepresent some numerical results in section 6 and conclude.

2 The Vasicek Large Portfolio Model

The single factor gaussian copula model proposed by Vasicek (1987) is maybethe most popular framework used to price and quote portfolio basket relatedderivatives. The use of the model is wide in the CDO tranche pricing dueto its simplicity.Given a portfolio with n obligors labeled i = 1, .., n, the obligor i defaults

ifXi < Ki whereXi is a standard normal random variable andKi = N−1(pi)

is a barrier defined to match pi the marginal default probability of the obligori. Represent Xi as

Xi = ρiX +√1− ρ2i εi,

2

where X, ε1, ε2, . . . are independent standard normal random variables. Us-ing the homogeneity assumptions pi = p and ρi = ρ, i = 1, . . . , n, theconditional marginal probability of default for each obligor i is

P [Obligor i defaults|X] = N

(K − ρX√1− ρ2

)

≡ p(X),

where K = N−1(p). From the construction the defaults are independentconditional on X. The conditional distribution of proportion π of oblig-ors defaulting is then binomial with mean p(X) and standard deviation√p(X)(1− p(X)/n where n is the portfolio size.By numerical integration it is possible to calculate the unconditional loss

distribution

P [π = a] =

(n

k

)∫ ∞

−∞pa(x)(1− p(x))n−aφ(x)dx

where φ(x) denotes the normal density.Moreover, for large n the standard deviation is small and we have ap-

proximately(π > α)⇔ p(X) > α

giving the unconditional distribution

P [π > α] ∼ N

(K1 −

√1− ρ2N−1(α)ρ

)

.

The model as described is static, useful only for valuations in a singlepoint of the future. Extensions of the copula approach to different copulastructures and multi-stage valuation points for CDOs and baskets defaultsderivatives have been developed, for a survey consult Laurent and Gregory(2004). However, there is no interpretation to the variable X and the pa-rameter ρ is unobservable in the market.Other criticism to the model is the inability to reflect the quoted prices

for CDOs tranches producing the “correlation skew”. Using a single cor-relation parameters seems to undervalue equity and senior tranches whileproducing the opposite effect to the mezzanine tranches.Our objective is create a modelling framework in a dynamic context

keeping the simplicity that makes the Vasicek model appealing. By intro-ducing a larger number of parameters, we expect to be able to reflect thecorrelation skew observed in the market.

3 Using Queuing Networks

Rating based models have been widely used for pricing and risk manage-ment purposes. Maybe the best known of all of them is the Jarrow, Lando

3

and Turnbull model (1997). Most recent application of rating models inthe context of Markov chains can be found in, for example, Frydman andSchuermann (2004) and Lando and Christensen (2004).Even if there are some criticisms of the use of credit ratings for pricing

purposes, it is appealing to consider having buckets of obligors with similarcharacteristics to simplify the mathematics while modelling the likelihood ofobligors to modify their credit worthiness. This is particularly useful whenhandling large portfolios.The use of latent variables and common factors has been common in

practice when dealing with large portfolios. Examples well known are theCreditMetrics model (Gupton et al. 1987) and PortfolioRisk+ (CSFB 1997).Moreover, empirical evidence (Crowder, Giampieri and Davis 2003) suggeststhat the pattern of realized defaults is well represented by a latent variablemodel where the latent process Xt is a 2-state (good times / bad times)economic variable.Finally, taking an approach of ratings and latent variables related with

economic interpretation, it is sensible to assume that obligors move aroundrating categories at a faster time scale than the economic cycle.These facts suggest a model in which obligors move around the rating

categories at rates depending on the latent process and occasionally default.Moreover, there is an obvious analogy with queueing networks in which ‘jobs’move around ‘service stations’ for processing.Recent work by Choudhury, Mandelbaum et al. (1998) studies fluid

and diffusion limits for queueing networks under random environments. Wegive some modifications of their results in this paper, in a form suitable forapplication to credit portfolios.

3.1 Rating Transitions, No Latent Variable and the LeakyBucket Analysis

For a portfolio with n obligors and K possible ratings k = 1, ...,K we definea vector process Qn(t) taking values in Nk, with each component Qnk(t)containing the number of elements in each rating category at time t. Then,∑Qnk(t) is the number of non defaulted obligors at time t. When n = 5,K =

2, the state space of Qn(t) is as shown in figure 1. The figure shows all thepossible movements given the current credit ratings: transitions (move alongthe diagonal) and defaults (move to the next diagonal).

4

Figure 1. The Sample Space in the Simplest Case

We can analyse heuristically the model by using a stationary argumentapplied to the case of two credit ratings, 0 and 1, as in the figure 2. Thetransition rate rate from 0 to 1 and viceversa are μ0 and μ1 respectively,while the default rate in state 0 (1) is ν0 (ν1). We call this the leaky bucketanalysis. Two buckets (0 and 1) are leaking at rating νi, i = 0, 1. Addi-tionally, liquid goes form one bucket to the other at a rate μi. If there isno leaking in the system, we expect that πt, the proportion of obigors incategory 0 (bucket 0) will reach a stationary state

π =μ1

μ0 + μ1

solution to the mass balance equation

μ0π − μ1(1− π) = 0.

Assuming that ν0, ν1 << μ0, μ1 we anticipate an average leakage (default)rate

ν = ν0π + ν1(1− π) =ν0μ1 + ν1μ0μ0 + μ1

Moreover, since the obligors are independent in this model, the standarddeviation of cumulative defaults in [0, t] is just

√d

nt

5

π 1−π

μ1

μ0

ν1ν0

0 1

Default

Figure 2. The Leaky Bucket System

0.00

0.10

0.20

0.30

0.40

0.50

0.0 0.5 1.0 1.5 2.0 2.5 3.0

exact mean exact std dev exact default rate est default rate est std dev

parametersμ0 4.0 μ1 1.0ν0 0.3 ν1 0.2n 20 π0 0.75

Figure 3. Results of the Leaky Bucket Analysis

As illustrated in figure 3, this simple analysis is surprisingly succesful inpredicting the mean an variance of the exact distribution obtained by solvingthe forward equation for the finite-state Markov process, if an adjustmentfor mean defaults is made.However the leaky bucket analysis does not depend on the initial distri-

bution of the obligors’ ratings and therefore does not capture the short-termdefault behavior. As seen in figure 3, the success of the analysis will dependon reaching the stationary state (assuming it exists) and on the relation ofleaking and transition rates.

3.2 Random Environments and Fluid and Diffusion Limits

As argued above, the economic cycle is relevant in the default rates of theobligors. We propose a finite state environment process Xt to reflect thiseffect.

6

X=0

X=1

Figure 4. Multiple layer setting

The finite state environment process defines different “layers” in whichtransition parameters are different. These multiple layers are copies of theinitial state space shown in Figure 1. Now the process jumps across thelayers anytime the random environment process Xt jumps to a differentstate as illustrated in figure 4. In this model the movements across thegrid are expected to be more frequent than the movements across layers,particularly when the number of elements in the system increases. Thisis why in some literature this structure is referred as a slowly changingenvironment (Mandelbaum, Massey et al. 1988). All we can observe howeveris the second coordinate of the pair process (Xt, Q

n(t)).A more accurate analysis of the model that keeps some of the tractability

of the leaky bucket analysis can be done using fluid and diffusion limits.Conditional in the realisation of the random environment, the process Qn(t)may be approximated by two processes

Qn(t) ' Q(0)(t) +√nQ(1)(t)

where Q(0) is a deterministic process called the “fluid limit” and Q(1)(t) isa diffusion called the “diffusion limit” of the sequence Qn(t). This is, condi-tional on the random environment, the distribution of the process Qn(t), atany time t > 0, may be approximated by a normal distribution. Fluid andDiffusion limits are sometimes referred as Functional Law of Large Num-bers and Functional Central Limit Theorem respectively. Now we state ourgeneral model.

7

4 The General Model

The random environment process X(t) is a finite state process in continuoustime, having at most a finite number of jumps in any bounded interval of[0,∞).To construct the process Q(t) we consider a collection of mutually inde-

pendent Poisson processes {Ai}i∈I={1,...,n} and a collection of vectors {vi}in RK , K ∈ N, and a collection of non-negative functions of the formαi(∙, ∙, x) : [0,∞) × RK → [0,∞) for all i ∈ I and x ∈ X . We assumeeach αi(t, ∙, x) is Lipschitz bounded with respect to the second argument,this is, exist a locally integrable function βt : [0,∞)→ [0,∞) such that

‖αi(t, ∙, x)‖ ≤ βt (1)

where ‖ ∙ ‖ is the Lipschitz norm defined as

‖f‖ = supx 6= yx, y ∈ D1

|f(x)− f(y)|D2|x− y|D1

∨ |f(0)|D2

We define the probability space (Ω1,F1, P 1) associated to the randomenvironment X(t), where Ω1 = D([0,∞),X ) the space of Right Continuouswith Left Limits (RCLL) functions from [0,∞) into X , F1 is a σ-algebra inΩ1 and P 1 is some probability measure. The measure space associated tothe collection {Ai}is defined by the state space Ω2 = Ω1 × ... × Ωn whereΩi = D([0,∞),Rm) and the minimum sigma-algebra F2 generated by Ω2.The probability measure P 2 is defined as the product measure. Finally, weassume independence between the environment process and each Poissonprocess Ai, therefore the probability space is defined as (Ω,F , P ) with Ω =Ω1 × Ω2, F = σ(Ω) and P = P 1 × P 2. We will denote by ω = (ω1, ω2) theelements of Ω being ω1 ∈ Ω1 and ω2 ∈ Ω2.We define a mapping Q from Ω into D([0,∞),RK) by (ω1, ω2) 7→ Q,

where Q is the process solution to the equation

Q(t) = Q(0) +n∑

i=1

Ai

(∫ t

0αi(s,Q(s), X(s))ds

)

vi (2)

The law of Q(t) is PQ(∙) = P (Q−1(∙)). The probability conditional on theenvironment process X(t) denoted by Pω1Q is then defined by the conditionalprobability under the inverse mapping.We are concerned with the convergence of sequences Qη : (ω1, ω2)→ RK

of the form

Qη(t)

η=Qη(0)

η+1

η

n∑

i=1

Ai

(

η

∫ t

0αi(s,Q

η(s), X(s))ds

)

vi

for η > 0.

8

Theorem 1 Assume a collection of functions of the form αi : [0,∞)×RK×X → [0,∞) for i ∈ I and such that

|αi(t, y, x)| ≤ C(1 + |y|) (3)

for some constant C < ∞, s ≥ 0 and y ∈ RK . Assume the processes X(t)and Ai(t) defined as above and vi a collection of vectors in R

K .Then the process Q(t) defined as the solution of the equation

Q(t) = Q(0) +n∑

i=1

Ai

(∫ t

0αi(s,Q(s), X(s))ds

)

vi (4)

has unique solution for a.e. ω.

We want to infer the behaviour of the state of the network when thenumber of arrivals and the number of serves per node increases while the rateof service remains unchanged. We will present some quenched convergenceresults for the process (Xt, Qt). The results obtained are two: Firstly, aquenched strong approximation limit for accelerated sequences of the form(X(t), Q(ηt)/η), η → ∞. Secondly, a quenched weak convergence resultstates the convergence of accelerated sequences of the form

1√η(Q(ηt)− ηQ0(t))

to a diffusion for a given realisation of the random process. Here Q(0) is thelimit derived from the strong approximation.

4.1 Strong Approximations

We define a sequence of network processes {(X(t), Qη(t)/η); η > 0} asso-ciated to (X(t), Q(t)) as the set of network processes where Qη(t) is thesolution to the system

Qη(t) = Qη(0) +N∑

i=1

Ai

(∫ t

0αηi (s,Q

η(s), X(s))) ds

)

vi (5)

where {αηi (s, ∙, x)}, with x ∈ X and i ∈ I, is a collection of functions satis-fying

‖αηi (t, ∙, x)‖ ≤ ηβt (6)

with βt a locally integrable function.The interpretation of the pair (X(t), 1/ηQη(t)) for some η > 0 is a

process with the same characteristics of Q(t) under the same environmentX(t) but where the number of servers and rates of arrivals have increased ηtimes.A quenched law of large numbers for the sequence (X(t), 1/ηQη(t)):

9

Theorem 2 If {αηi |i ∈ I, η > 0} are Lipschitz bounded and

limη→∞

N∑

i=0

∫ t

0

∥∥∥∥αηi (s, ∙, x)η

− α(0)i (s, ∙, x)

∥∥∥∥ ds = 0 (7)

For ω1 ∈ Ω1 where the process Qη(t) is that

Qη(0)

η→ Q(0)(0) as η →∞ (8)

then

Qη(t)

η→ Q(0)(t) as η →∞ (9)

a.s. in Pω1Q , where Q(0)(t) defined in RK is the solution to the equation

Q(0)(t) = Q(0)(0) +

∫ t

0α(0)(s,Q(0)(s), X(s))ds (10)

where

α(0)(t, ∙, x) =N∑

i=1

α(0)i (t, ∙, x)vi (11)

and the integral equation is deterministic conditional on the random envi-ronment process X(t).

The process Q(0)(t) is referred as the fluid limit of the sequence {Qη(t)}η≥0.

4.2 Weak convergence

It is possible to derive a quenched functional version of the central limittheorem for (X(t), (1/η)Qη(t)) conditional on Fω1t .To present this result, we require to define the scalable Lipschitz deriva-

tive of a function f : D1 → D2 at x ∈ D1, D1 and D2 two Banach spaces,as the function Λfx(y) : D1 → D2 such that

limy→0

|f(x+ y)− f(x)− Λfx(y)|D2|y|D1

= 0

whenever such function exists and it is Lipschitz bounded and homogeneous,this is,

‖Λfx(∙)‖ <∞,

and for all λ ≥ 0

λΛfx(y) = Λfx(λy).

10

We note that in the case of f : Rd1 → Rd2 , d1, d2 ∈ N, the operatorΛfx(y) generalises the notion of directional derivative, and whenever thedifferential operator exists we have

Λfx(y) = Df(x)y

where Df(x) is the Jacobian matrix valuated at x. When the Jacobianmatrix is not defined Λfx(y) may be not unique.For a sequence of r.v.’s defined in a measurable space (Ω,F) we will

say that a sequence of r.v.’s {Yn} converges in distribution of Y , denotedby limn→∞ Yn =

d Y , w.r.t. P some probability measure on (Ω,F) if forall f ∈ C(Ω), the set of bounded continuous real functions defined in Ω,limn→∞EP {Yn} = EP {Y }.Some weak convergence results in queuing theory assume a heavy traffic

condition, this is, the arrival and total service rates are nearly the same.Here we have no arrivals and eventually Qη(t) = 0, so no stationarity maybe assumed.

Theorem 3 Assume∑

i∈I

∫ t

0limη→∞

√η

∥∥∥∥αηi (s, ∙, x)η

− α(0)i (s, ∙, x)

∥∥∥∥ ds <∞ (12)

and

limη→∞

∑

i∈I

∫ t

0

∥∥∥∥√η

[αηi (s, ∙, x)η

− α(0)i (s, ∙, x)

]

− α(1)i (s, ∙, x)

∥∥∥∥ ds = 0 (13)

and for all x ∈ X and i ∈ I. Assume the function α(0)(s, ∙, x) has scalableLipschitz derivative for any values X(t) and Q(0)(t). For P 1 a.e. ω1 ∈ Ω1,if

limη→∞

√η

[Qη(0)

η−Q(0)(0)

]

=d Q(1)(0) (14)

w.r.t Pω1Q , then

limη→∞

√η

[Qη(t)

η−Q(0)(t)

]

=d Q(1)(t) (15)

w.r.t Pω1Q , where the process Q(1)(t) takes values in Ω2 and it is the solution

to the stochastic integral equation

Q(1)(t) = Q(1)(0) +

∫ t

0Λα(0)i (s,Xs, Q

(0);Q(1)(s)) + α(1)(s,Q(0)(s), Xs)ds

+∑

i∈I

Bi

(∫ t

0α(0)i (s,Q

(0)(s), Xs)ds

)

vi (16)

where {Bi(t)} is a collection of independent standard Brownian motions in(Ω2,F2, Pω1).

11

5 Application to Correlated Defaults

We propose a model for the default/rating process of a set of obligors.There is a finite-state random environment process {X(t)|t ≥ 0} repre-senting some macroeconomic (or sector associated) process that influencesthe default/transition rates of the obligors. The obligor credit events areindependent conditional on the realisation of the environment process andfollow a Markov chain with rates being function of the environment process.Assume a portfolio with n obligors and K possible ratings 1, . . . ,K.

The initial rating composition of the portfolio is represented by the ratingdistribution vector Qn(0) ∈ Rk. Qn(t) will represent the random ratingdistribution of the portfolio at a later time t.We define the index set of transition events I = {(i, j)|i, j = 1, ..., k}

and denote the transition rate from rating i to rate j, i, j = 1, ..., k, byμ(i,j)(x) = μij(x); default rates are denoted by μ(i,i)(x) ≡ μi(x). Associatedto these credit events we define the set of vectors {v(i,j) ∈ R

k|(i, j) ∈ I}that define the changes in the rating distribution vector in case of a creditevent. This is

v(i,j) =

{ej − ei i 6= i−ei i = j

where ei is the i-th canonical vector in Rk.

Under this assumptions the credit events of each obligors are identicallydistributed and occur according to the first jump of a Poisson process withrates

μi(x) =k∑

j=1

μij(x)

Once a credit event occurs at time t, the obligor defaults with probabilityμi(Xt)/μi(Xt) while a transition to rate j 6= i has probability μij(Xt)/μi(Xt).As stated above, the occurrence of a credit event for an obligor in the

credit rate i is given by the first jump of a Poisson process. The rate ofoccurrence depends on the environment random process but is independentof time. This is, for a set of positive real numbers {μxij |x ∈ X , (i, j) ∈ I} wedefine the transition default rates as

μ(i,j)(t,Xt) =∑

x∈X

μxijI{Xt=x}

for (i, j) ∈ I.

5.1 The Fluid Limit

According to the notation used above, we have the set of rate functions

αn(i,j)(t,y, x) = yiμ(i,j)(t, x)

12

and since the rate function does not depend on n is obvious that

α(0)(i,j)(t,y, x) = yiμ(i,j)(t, x)

satisfy the conditions of theorem 2. Using vector notation

α(0)(t,y, x) = At(x)y

where A is the infinitesimal generator of the process.We can verify that by assuming

lim1

nQn(0) = Q(0)(0)

for all n ≥ 0 all conditions of therorem 2 hold and the fluid limit processQ(0)(t) is the solution to the deterministic PDE system

d

dtQ(0)(t) = AtQ

(0)(t) (17)

and whenever A = At is time independent (no environment influence) thesolution is given by

Q(0)(t) = etAQ(0)(t)

otherwise, since Xt has at most finite jumps in any bounded interval of time,we can define a countable set of jump times of Xt t0 = 0, t1 < ... and defineQ(0)(t) recursively

Q(0)(t) = e(t−ti)A(xti )Q(0)(t) (18)

for ti < ti+1.

5.2 The diffusion limit

Since αn(i,j) = α(0)i,j for all n ≥ 0 and (i, j) ∈ I we can verify that by defining

α(1)(i,j) = 0 for all (i, j) ∈ I the conditions in theorem 3 hold. Thereforeassuming

limn→∞

√n

[Qn(0)

n−Q(0)(0)

]

=d Q(1)(0) (19)

implies

limn→∞

√n

[Qn(t)

n−Q(0)(t)

]

=d Q(1)(t) (20)

13

where Q(1)(t) satisfies a SDE which may be expressed as the following inte-gral equation

Q(1)(t) = Q(1)(0) +

∫ t

0AtQ

(1)(t) +k∑

l=1

∫ t

0

(Q(0)l (t)

)1/2BldW

(l)t

where W l is a k-dimensional vector of independent standard Brownian mo-tions for l = 1, ..., k. The matrices Bl have components

(Bl(t))ij =

−μ1/2(i,j) if i = j

μ1/2(i,j) if l = i 6= j

0 otherwise

It is always possible rewrite (21) as

Q(1)(t) = Q(1)(0) +

∫ t

0AtQ

(1)(t) +

∫ t

0B(s)dW

(l)t

where Wt is a k-dimensional vector of independent standard Brownian mo-tions. In the case k = 2 the SDE is

dQ(1)1 (t) = −Q(1)1 (t)(μ1(t) + μ12(t))dt+Q

(1)2 (t)μ12(t)dt

−(Q(0)1 (t))1/2(μ

1/21 (t)dW

(1)1,t + μ

1/212 (t)dW

(1)2,t ) + (Q

(0)2 (t)μ21(t))

1/2dW(2)1,t

dQ(1)2 (t) = −Q(1)2 (t)(μ2(t) + μ21(t))dt+Q

(1)1 (t)μ21(t)dt

−(Q(0)2 (t))1/2(μ

1/22 (t)dW

(2)2,t + μ

1/221 (t)dW

(2)1,t ) + (Q

(0)1 (t)μ12(t))

1/2dW(1)(2,t

That is equivalent to the the following SDE system

dQ(1)(t) = AtQ(1)dt+B(t)dWt

where

B(t) =

(σ1(t) 0

ρ(t)σ2(t)√1− ρ2(t)σ2(t)

)

σ21(t) = Q(0)1 (t)(μ1(t) + μ12(t)) +Q

(0)2 (t)μ21(t)

σ22(t) = Q(0)2 (t)(μ2(t) + μ21(t)) +Q

(0)1 (t)μ12(t)

ρ(t)σ1(t)σ2(t) = −Q(0)1 (t)μ12(t)−Q(0)2 (t)μ21(t)

and

Wt = (W(1)t , W

(2)t )

t

14

is a bi dimensional standard Brownian motion with independent compo-nents.Assuming At = A time independent, the solution of the SDE is given by

Q(1)(t) = etAQ(1)(0) +

∫ t

0e(t−s)AB(s)dWs (21)

with Q(1)(0) = 0. The process is a stable Gaussian system with covariancematrix given by the integral

Cov[Q(1)(t), Q(1)(t)] =

∫ t

0e(t−s)AB(s)B(s)tr(e(t−s)A)trds (22)

that can be calculated numerically by solving a matrix ordinary differentialequation (the Lyapunov equation).In the general case of Xt taking values in X the process is defined simi-

larly to the case of the fluid limit, this is

Q(1)(t) = e(t−ti)AtiQ(1)(ti) +

∫ t

ti

e(t−s)AtiB(s)dWt (23)

for ti < t < ti+1 where ti is the time of the i-th jump of Xt.

6 Some numerical results

We assume a two state (two credit rates) system and 20, 50 and 100 ele-ments. We assume a two state external random environment where jumpsoccur according to a standard Poisson process (rate 1). By sampling 1000times the random environment we construct both the diffusion approxima-tion and the exact distribution. The latter is obtained by integrating theKolmogorov forward equation associated to the process using Runge-Kutta.The parameters are shown in the table.

X(t) = 0 X(t) = 1

μ1 0.1 0.2μ2 0.2 0.4μ12 0.3 0.3μ21 0.2 0.2

The exact and approximated distributions for the case of 20 elementsand initial distribution 50/50 is shown in figure 5. We can observe thatthe fitting of the marginal distributions is outstanding despite the relativelysmall number of elements.

15

Figure 5. Exact and approximated distributions with 20 elements

We consider also the approximation to the number of survivors for dif-ferent number of elements and initial distribution. The results are shownin figure 6. We can observe the effect of the random process X(t) in theskewed distribution obtained.

16

Figure 6. Approximation to the number of survivors

7 Concluding Remarks

In this paper we present a method for credit risk modelling that seemseffective in predicting default performance with low computational cost.We present encouraging results that suggest the convenience of using limitapproximations to the distribution of the rating distribution in a credit riskportfolio.Further research involve investigating pricing applications. By now, we

have no methodology to calibrate to the market information. Particularly,we are interested in the calibration to iTraxx tranche quotes as a method toestimate implied basket correlation.Other applications may include the use of empirical change-of-rating

data for risk management purposes.

A Appendix

We present the proof of the main theorems shown in this paper. Most of theproofs are taken from Choudhury, Mandelbaum et al. (1998). First we showthe existence of the sequence of processes Qη and the tightness property. Inorder to do that, lemma A.1 constructs a bound for the sequence.

17

Lemma A.1: The process defined as solution of the equation

Zη(t) = Xη(∫ t

0β(s)Zη(s)ds

)

(24)

with

Xη(t) = X(0) +∑

i∈I

1

ηAi (ηt)vi (25)

is a pure jump process (no explosions) and the compensator is

β(s)∑

i∈I

|vi|

Proof. It can be seen that for any t > 0

E(Xη(t)) ≤ E(Xη(0)) +∑

i∈I

1

η(ηt)|vi| <∞ a.s.

Now, if (24) has a solution we can define τ(t) as

τ(t) =

∫ t

0β(s)Z(s)ds =

∫ t

0β(s)X(τ(s))ds

We need to prove we can define τ(t) uniquely and that τ(t) < ∞ if t < ∞a.s. Let us consider the one to one mapping τ(t) from t such that

∫ t

0β(s)ds =

∫ τ(t)

0

1

X(u)du (26)

and since by definition X(u) > 0 and β(s) > 0 then the mapping is uniquelydefined, moreover since X(u) has no explosions τ(t) <∞ as long as t <∞a.s. Finally, τ(t) is a stopping time with respect to the natural filtration ofX(t) from the definition of the mapping.Now by differentiating both sides in (26)

β(t) =1

X(τ(t))

dτ

dt

and then we define

Z(t) =τ ′(t)

β(t)

that satisfies

τ(t) =

∫ t

0β(s)Z(s)ds

18

and has the properties we required.Now, we know that

x(t)− t∑

i∈I

|vi|

is a martingale, since τ(t) is a stopping time

M(t) = Z(t)− τ(t)∑

i∈I

|vi|

is a martingale with respect to the natural filtration of M(t) and then thecompensator property is proved.Now we show the existence and properties of the sequences Qη and from

this result is automatic theorem 1.Theorem A.2: For a set of indices I let {Ai|i ∈ I} be a collection ofstandard Poisson processes in the space (Ω, F, P ) and X(s) a environmentprocess as specified above. Let {vi|i ∈ I} a collection of vectors in Rk.Consider the class of processes Qη(t), η > 0 of the form

Qη(t) = Qη(0) +∑

i∈I

Ai

(∫ s

0αηi

(

s,1

ηQη(s), X(s)

)

ds

)

vi (27)

where {αηi |η > 0, i ∈ I} is a family of measurable functions such that

|αηi (s, y, x)| ≤ ηC(1 + |y|) (28)

and {Qη(0)|η > 0} a family of rv’s independent of the collection {Ai}.For each η > 0, the process Qη is uniquely defined (solution exists and

it is unique) for almost ω in Ω.Moreover, if

limη→∞

|Qη(0)|η

<∞ a.s.

then

limη→∞

|Qη(t)|η

<∞ a.s.

and therefore the sequence is tight.Proof: First we prove that Qη(t) can be defined (unique solution exists)for all ω1 ∈ Ω1. We use and induction argument to construct Qη. For η > 0define Qη0(t) = Q

η(0), and

Qηn(t) = Qη(0) +

∑

i∈I

1/ηAi

(∫ t∧Tn

0αηi

(s, 1/ηQηn−1(s), X(s)

)ds

)

vi (29)

19

where

Tn = inf

{

s :∑

i∈I

Ai

(∫ t

0αi(t, 1/ηQ

ηn−1(s), X(s))ds

)

= n

}

(30)

where Tn = ∞ if the set is empty. By construction Qη0(t) = Q

η1(t) for

0 ≤ t < T1. Using the induction hypothesis Qn(t) = Qn−1(t) for t < Tnimplies

∫ t

0αηi (s, 1/ηQ

ηn(s), X(s))ds =

∫ t

0αηi (s, 1/ηQ

ηn−1(s), X(s))ds

for t < Tn and i ∈ I. By continuity of the integral, the same equalityholds for t = Tn. So Q

ηn(t) = Q

ηn+1(t) for t ≤ Tn. Now by the properties

of the Poisson process (pure jump process) Qηn+1(t) = Qηn(t) = Q

ηn(Tn) for

Tn ≤ t < Tn+1. Then Qηn(t) = Q

ηn−1(t) for 0 ≤ t < Tn and Qη(t) = Q

ηn(t)

for Tn ≤ t < Tn+1 and Qη(t) is defined in unique way.Now consider the process defined by the SDE

Z(t) = 1 +1

η|Qη(0)|+

∑ 1

ηAi

(

ηC

∫ t

0Z(s)ds

)

|vi|

such process is well defined (Lemma A.1). Notice that if

1 +1

η|Qηn−1(s)| < Z

η(s)

for t < Tn almost surely, then

1 +1

η|Qηn(t)| ≤ 1 +

1

η|Qη(0)|+

∑

i∈I

1

ηAi

(∫ t

0αηi (s, 1/ηQ

ηn−1(s), X(s))ds

)

|vi|

≤ 1 +1

η|Qη(0)|+

∑

i∈I

1

ηAi

(

ηC

∫ t

0(1 + 1/η|Qn−1(s)|)ds

)

|vi|

≤ 1 +1

η|Qη(0)|+

∑

i∈I

1

ηAi

(

ηC

∫ t

0Zη(s))ds

)

|vi|

almost surely, therefore we can conclude that

sup0≤s≤t

1 + |Qηn(t)| < Zη(s) (31)

for t > 0 a.s.Notice that for any i ∈ I and n ≥ 0

Ai

(∫ t

0αηi (s, 1/ηQ

ηn(s), X(s)ds)

)

≤ Ai

(

C

∫ t

01 + 1/η|Qη(t)|ds

)

≤ Ai

(

C

∫ t

0Zη(t)ds

)

20

then

∑

i∈I

Ai

(∫ t

0αηi (s, 1/ηQ

ηn(s), X(s)ds)

)

≤∑

i∈I

Ai

(

C

∫ t

0Zη(t)ds

)

and since {Ai} is a collection of independent standard Poisson process, thesum can be represented as a Poisson process A∗, this is

∑

i∈I

Ai

(

C

∫ t

0Zη(t)ds

)

= A∗(

ηCη

∫ t

0Zη(t)

)

that is a well behaved pure jump process, therefore we can define for τn thetime of the n-th jump in A∗ a sequence t(τn) such that

τn = ηC

∫ t(τn)

0Zη(t)ds

Clearly t(τ) → ∞ a.s. as η → ∞. Additionally t(τn) ≤ Tn, otherwise theinequality is broken, therefore Tn →∞ a.s.Finally, assume

limη→∞

|Qη(0)|η

≤ ∞ a.s.

then recall (31) , this is

sup0≤s≤t

1 + |Qηn(t)| < Zη(s)

and lemma 1 that implies

Zη(t) exp

{

−Ct∑

i∈I

|vi|

}

is a martingale, therefore

E{Zη(t)} =

(

1 +1

η|Qη(0)|

)

exp

{

Ct∑

i∈I

|vi|

}

and using the Tchebychev inequality se can prove the sequence Zη(t) istight. Therefore limZη(t) exists and is uniformly integrable, then

limη→∞Zη(t) <∞ a.s.

and consequently

limη→∞Qη(t) <∞ a.s. �

21

The strong approximation result is derived from a theorem appeared inEthier and Kurtz (1986) and previously in Kurtz (1978). We present theenunciation as a lemma without proof.Lemma A.3:[Kurtz (1986)] A standard Poisson process {A(t)|t ≥ 0} canbe realized in the same space as a standard Brownian motion {B(t)|t ≥ 0}in such way that the positive r.v. X given by

X =|A(t)− t−B(t)|ln(2 ∨ t

<∞ (32)

has a finite moment generating function in a neighborhood of the origin, inparticular has finite mean.With this result we show the convergence of the series 1/ηQη to after-

wards show the situations where the limit is indeed Q(0).Lemma A.4: If {αηi |η > 0, i ∈ I} is a family of measurable functions suchthat (28) holds and {Ai}, X(t), {Qη(0)} and {vi} are as defined above, then

lim sup1

ln η

∣∣∣∣Qη(t)−Qη(0)−

[∫ t

0αη (s, 1/ηQη(s), X(s)) ds

+∑

i∈I

Bi

(∫ t

0αηi (s, 1/ηQ(s), X(s)) ds

)

|vi|

]∣∣∣∣∣= C1 <∞ (33)

as η →∞ for almost all ω in Ω.Proof: The term inside the norm in the left hand side of equation (33) isequal to∣∣∣∣∣

∑

i∈I

Ai

(∫ t

0αηi (s, 1/ηQ(s), X(s)) ds

)

|vi| −∫ t

0αη (s, 1/ηQη(s), X(s)) ds

−∑

i∈I

Bi

(∫ t

0αηi (s, 1/ηQ(s), X(s)) ds

)

|vi|

∣∣∣∣∣

and applying lemma A.3 we can bound the term by

≤∑

i∈I

Xi ln

(

2 ∨∫ t

0αηi (s, 1/ηQ

η(s), X(s)) ds

)

|vi|

≤∑

i∈I

Xi ln

(

2 ∨ ηC∫ t

01 +1

η|Qη(s)|ds

)

ln

(

2 ∨ ηC∫ t

0Zη(t)ds

)

|vi|

≤∑

i∈I

Xi|vi|

where Zη(t) is as in the proof of theorem A.2. By lemma A.3

E[∑

Xi|vi|]<∞

22

and

ln(2 ∨ ηC

∫ t0 Zη(t)ds

)

ln(η)→ C2 <∞

as η →∞ and the lemma is proved �Once the strong convergence is guaranteed, we derive the strong limit

when the default rates are Lipschitz bounded, for this we require a general-isation of the Gronwall inequality we present without proof.Lemma A.5: Let x, y and z be measurable, non-negative functions on thereals. If y is bounded and z is integrable on [0, T ] and for all 0 ≤ t ≤ T

x(t) ≤ z(t) +∫ t

0s(x)y(s)ds

then

x(t) ≤ z(t) +∫ t

0z(s)y(s) exp

[∫ t

0y(r)dr

]

and

sup0≤t≤T

x(t) ≤ sup0≤t≤T

z(t) exp

[∫ T

0y(s)ds

]

Theorem A.6: (Theorem 2) If {αηi |i ∈ I, η > 0} are Lipschitz bounded and

limη→∞

N∑

i=0

∫ t

0

∥∥∥∥∥

αηi,x(s, ∙)

η− α(0)i,x (s, ∙)

∥∥∥∥∥ds = 0 (34)

For P 1 a.e. ω1 ∈ Ω1 the process Qη(t) is that

Qη(0)

η→ Q(0)(0) as η →∞ (35)

then

Qη(t)

η→ Q(0)(t) as η →∞ (36)

a.s. in P 1, where Q(0)(t) defined in RK is the solution to the equation

Q(0)(t) = Q(0)(0) +

∫ t

0α(0)(s,Q(0)(s), X(s))ds (37)

where

α(0)(t, ∙, x) =N∑

i=1

α(0)i (t, ∙, x)vi (38)

and the integral equation is deterministic conditional on the random envi-ronment process X(t).Proof: We need to prove that:

23

1. The function α(0) is well defined, this is, the function exists and it isLipschitz.

2. For all ω in Ω

limη→∞

sup0≤s≤t

∣∣∣∣1

ηQη(s)−Q(0)(s)

∣∣∣∣ = 0 (39)

almost surely under Pω1 .First, notice that

∫ t

0‖α(0)‖ds ≤

∫ t

0‖αη(s, y,X(s)‖

+

∫ t

0

∥∥∥∥1

ηαη(s, y,X(s))− α(0)(s, y,X(s))

∥∥∥∥ ds

≤ sup |vi|∑

i∈I

[∫ t

0β(s)ds

+

∫ t

0

∥∥∥∥1

ηαηi (s, y,X(s))− α

(0)i (s, y,X(s))

∥∥∥∥ ds

]

<∞

and since I is finite it is clear that

limη→∞

∫ t

0

∥∥∥∥1

ηαη(s, y,X(s))− α(0)(s, y,X(s))

∥∥∥∥ ds = 0

otherwise it can be derived form the inequality

∫ t

0

∥∥∥∥1

ηαη(s, y,X(s))− α(0)(s, y,X(s))

∥∥∥∥ ds

≤ sup0≤s≤t

∑

i∈I

∫ t

0

∥∥∥∥1

ηαηi (s, y,X(s))− α

(0)i (s, y,X(s)

∥∥∥∥

where the right hand side converges to zero as η → 0.Now as a consequence of lemma A.4 we have

Qη(t)

η−Q(0)(t) =

∫ t

0

1

ηαη(s,

1

ηQη(s), X(s))− α(0)(s,Q(0), X(s))ds

+∑

i∈I

1

ηBi (α

ηi (s, 1/ηQ

η(s), X(s)) ds)vi +1

ηO(ln η)

where O(ln η) is a function such that O(ln η)/ ln(η)→ C <∞ as η →∞.

24

Now

sup0≤s≤t

∣∣∣∣Qη(s)

η−Q(0)(s)

∣∣∣∣ ≤

∫ t

0

∣∣∣∣1

ηαη (s, 1/ηQη(s), X(s))− α(0) (s, 1/ηQη(s), X(s))

∣∣∣∣ ds

+

∫ t

0

∣∣∣α(0) (s, 1/ηQη(s), X(s))− α(0)

(s,Q0(s), X(s)

)∣∣∣ ds

+∑

i∈I

sup0≤s≤t

1

η||Bi

(∫ s

0αηi (r, 1/ηQ

η(u), X(u)) du

)

|vi|+1

ηO(ln η)

By lemma A.3 and the Lipschitz property of αη

≤∫ t

0

∥∥∥∥αη(s, y,X(s))

η− α(0)(s, y,X(s))

∥∥∥∥Z

η(t)

+

∫ t

0‖αη(s, y,X(s))‖ sup

0≤r≤s

∣∣∣∣Qη(r)

η−Q(0)(r)

∣∣∣∣ ds

+∑

i∈I

sup0≤s≤t

1

η

∣∣∣∣Bi

(

C

∫ t

0Zη(s)ds

)∣∣∣∣ |vi|+

1

ηO(ln η)

and by lemma A.5 (generalisation of Gronwall inequality)

sup0≤s≤t

∣∣∣∣Qη(s)

η−Q(0)(s)

∣∣∣∣ ≤ exp

(∫ t

0‖α(0)(s, y,X(s))‖ds

)

(∫ t

0

∥∥∥∥αη(s, y,X(s))

η− α(0)(s, y,X(s))

∥∥∥∥

)

+∑

i∈I

sup0≤s≤t

1

η|Bi(ηCsZ

η(s))|vi|+1

ηO(ln η)

and since ‖α(0)‖ ≤ ∞ and∫‖αη − α(0)‖ → 0 as η →∞, we conclude

limη→∞

sup0≤s≤t

∣∣∣∣Qη(s)

η−Q(0)(s)

∣∣∣∣ = 0 �

Finally, we prove the weak convergene result. Before proving theorem 3we need to prove that the sequence

1

η|Qη(s)− ηQ(0)|

has a subsequence that is tight and then the weak convergence is possible.Lemma A.7: For all t > 0 if {αηi (s, y, x)|η > 0, i ∈ I} is a family of positiveLipschitz bounded functions as in theorem 2 assumptions, and

∑

i∈I

limη→∞

sup0≤s≤t

1√η‖αη(s, y,X(s))− ηα(0)(s, y,X(s))‖ds <∞ (40)

25

P 1 a.s. then the sequence

1

η|Qη(s)− ηQ(0)(s)|

is such that

P

{1

η|Qη(s)− ηQ(0)(s)| <∞

}

= 1

Proof: The proof follows the same arguments as in the proof of theorem2.From condition (40)

∑

i∈I

limη→∞

sup0≤s≤t

1

η‖αη(s, y,X(s))− ηα(0)(s, y,X(s))‖ds = 0

and then condition (34) holds and theorem 2 is valid. Then ‖α(0)‖ is aLipschitz map. Moreover, from the proof of theorem 2 we recall

∫ t

0

∥∥∥∥1

ηαη(s, y,X(s))− α(0)(s, y,X(s))

∥∥∥∥ ds

≤ sup0≤s≤t

∑

i∈I

∫ t

0

∥∥∥∥1

ηαηi (s, y,X(s))− α

(0)i (s, y,X(s)

∥∥∥∥

and multiplying by√η

∫ t

0

1√η‖αη(s, y,X(s))− ηα(0)(s, y,X(s))‖ <∞

P a.s. for all t > 0By Lemma A.4

Qη(t) = Qη(0) +

∫ t

0αη (s, 1/ηQη(s), X(s)) ds

+∑

i∈I

Bi

(∫ t

0αη (s, 1/ηQη(s), X(s)) ds

)

vi +O(ln η)

=d Qη(0) +

∫ t

0αη (s, 1/ηQη(s), X(s)) ds

+∑

i∈I

√ηB′i

(1

η

∫ t

0αη (s, 1/ηQη(s), X(s)) ds

)

vi +O(ln η)

where {B′i} is a collection of independent Brownian motions different to thefamily {Bi} that can be constructed in the same probability space by theself-similarity properties of the Bm’s.

26

Then

1√η

[Qη(t)− ηQ(0)(t)

]=d

1√η

∫ t

0αη (s, 1/ηQ(s), X(s))− ηα(0)(s,Q(0)(s), X(s))ds

+∑

i∈I

B′i

(1

η

∫ t

0αη (s, 1/ηQη(s), X(s)) ds

)

vi +1√ηO(ln η)

=1√ηΘ(t)

We then work with Θ(t), it can be seen that

sup0≤s≤t

1√ηΘ(t) ≤

1√η

∫ t

0

∣∣∣αη (s, 1/ηQη(s), X(s))− ηα(0) (s, 1/ηQη(s), X(s))

∣∣∣ ds

+1√η

∫ t

0η∣∣∣α(0) (s, 1/ηQη(s), X(s))− α(0)

(s, 1/ηQ(0)(s), X(s)

)∣∣∣ ds

+∑

i∈I

sup0≤s≤t

∣∣∣∣B′i

(1

η

∫ s

0αη (u, 1/ηQη(u), X(u)) du

)∣∣∣∣ |vi|+

1√ηO(ln η)

By the Lipschitz continuity of α(0) and αη

≤∫ t

0

1√η‖αη(s, y,X(s))− ηα(0)(s, y,X(s))‖dsZη(t)

+

∫ t

0‖α(0)(s, y,X(s)‖

1

ηsup0≤r≤s

∣∣∣Qη(r)− ηQ(0)(r)

∣∣∣ ds

+∑

i∈I

sup0≤s≤t

∣∣B′i(sCZ

η(s))∣∣ |vi|+

1√ηO(ln η)

and by lemma A.5

sup0≤s≤t

1√η≤ exp

[∫ t

0‖α(0)(s, y,X(s))‖ds

]

(∫ t

0

1√η‖αη(s, y,X(s))− ηα(0)(s, y,X(s))‖dsZη(t)

+∑

i∈I

sup0≤s≤t

∣∣B′i(sCZ

η(s))∣∣ |vi|+

1√ηO(ln η)

)

By taking limits in both sides

limη→∞

1√ηQη(t) ≤ exp

[∫ t

0‖α(0)(s, y,X(s)‖ds

]∑

i∈I

sup0≤s≤t

∣∣B′i(sCZ

η(s))∣∣ <∞

27

and then

P

[1

η|Qη(t)− ηQ(0)(t)| <∞

]

= 1 �

Finally, we show the weak approximation Q(1) to the sequence.Theorem A.8: (Theorem 5) Assume

∑

i∈I

∫ t

0limη→∞

√η

∥∥∥∥αηi (s, ∙, x)η

− α(0)i (s, ∙, x)

∥∥∥∥ ds <∞ (41)

and

limη→∞

∑

i∈I

∫ t

0

∥∥∥∥√η

[αηi (s, ∙, x)η

− α(0)i (s, ∙, x)

]

− α(1)i (s, ∙, x)

∥∥∥∥ ds = 0 (42)

and for all x ∈ X and i ∈ I. Assume the function α(0)(s, ∙, x) has scalableLipschitz derivative for any values X(t) and Q(0)(t). For P 1 a.e. ω1 ∈ Ω1,If

limη→∞

√η

[Qη(0)

η−Q(0)(0)

]

=d Q(1)(0)

w.r.t Pω1Q , then

limη→∞

√η

[Qη(t)

η−Q(0)(t)

]

=d Q(1)(t) (43)

w.r.t Pω1Q , where the process Q(1)(t) takes values in Ω2 and it is the solution

to the stochastic integral equation

Q(1)(t) = Q(1)(0) +

∫ t

0Λα(0)i (s,Xs, Q

(0);Q(1)(s)) + α(1)(s,Q(0)(s), Xs)ds

+∑

i∈I

Bi

(∫ t

0α(0)i (s,Q

(0)(s), Xs)ds

)

vi (44)

where {Bi(t)} is a collection of independent standard Brownian motions in(Ω2,F2, Pω1).Proof: We need to prove the following:

1. The limit rates α(0) and α(1) are Lipschitz bounded and their Lipschitznorm locally integrable as functions of time.

2. That

limη→∞

sup0≤s≤t

∣∣∣∣∣Qη(s)− ηQ(0)(s)

√η

−Q(1)(s)

∣∣∣∣∣= 0 (45)

for almost all ω ∈ Ω.

28

If

∑

i∈I

∫ t

0limη→∞

√η

∥∥∥∥αηi (s, ∙, x)η

− α(0)i (s, ∙, x)

∥∥∥∥ ds <∞

then

∑

i∈I

∫ t

0limη→∞

∥∥∥∥αηi (s, ∙, x)η

− α(0)i (s, ∙, x)

∥∥∥∥ ds = 0

and then condition (34) of theorem 2 holds and α(0) is Lipschitz boundedand ‖α(0)‖ is locally integrable.By a similar argument to the used in the proof of theorem 2 from eq (42)

∫ t

0

∥∥∥α(1)

∥∥∥ ds ≤

∫ t

0

∥∥∥∥∥

[αη − ηα(0)√η

]∥∥∥∥∥ds+

∫ t

0

∥∥∥∥∥

[αη − ηα(0)√η

]

− α(1)∥∥∥∥∥ds

∫ t

0

∥∥∥∥∥

[αη − ηα(0)√η

]∥∥∥∥∥ds+ sup

i∈Ivi∑

i∈I

∫ t

0

∥∥∥∥∥

[αηi − ηα

(0)i√

η

]

− α(1)i

∥∥∥∥∥ds <∞

and therefore ‖α(1)‖ is locally integrable. Moreover, since

∫ t

0

∥∥∥∥∥

[αη − ηα(0)√η

]

− α(1)∥∥∥∥∥ds < sup

i∈Ivi∑

i∈I

∫ t

0

∥∥∥∥∥

[αηi − ηα

(0)i√

η

]

− α(1)i

∥∥∥∥∥ds

and since supi∈I vi <∞ then

limη→∞

∫ t

0

∥∥∥∥∥

[αη − ηα(0)√η

]

− α(1)∥∥∥∥∥ds = 0

and the first point is proved.Now we proof the second point, we bound the term in left hand side of

(45) and show each term goes to zero as η →∞.

1√η

[Qη(t)−Q(0)(t)

]

=d∫ t

0

1

η

(αη (s, 1/ηQη(s), X(s))− ηα(0)

(s,Q(0)(s), X(s)

))ds

−∫ t

0Λα(0)

(s,Q(0)(s), X(s);Q(1)(s)

)+ α(1)(s,Q(0)(s), X(s))ds

+∑

i∈I

[

B′i

(1

η

∫ t

0αη (s, 1/ηQη(s), X(s)) ds

)

−B′i

(∫ t

0α(0)

(s,Q(0)(s), X(s)

)ds

)]

vi

+1√ηO(ln η)

29

and

=d∫ t

0

1

η

(αη (s, 1/ηQη(s), X(s))− ηα(0) (s, 1/ηQη(s), X(s))

)

−α(1)(s,Q(0)(s), X(s)

)ds

+

∫ t

0

(ηα(0) (s, 1/ηQη(s), X(s))− ηα(0)

(s,Q(0)(s), X(s)

))

−Λα(0)(s,Q(0)(s), X(s);Q(1)(s)

)ds

+

∫ t

0α(1)

(s,Q(0)(s), X(s)

)− α(1)(s,Q(0)(s), X(s))ds

+∑

i∈I

[

B′i

(1

η

∫ t

0αη (s, 1/ηQη(s), X(s)) ds

)

−B′i

(∫ t

0α(0)

(s,Q(0)(s), X(s)

)ds

)]

vi

+1√ηO(ln η)

and again

=d∫ t

0

1

η

(αη (s, 1/ηQη(s), X(s))− ηα(0) (s, 1/ηQη(s), X(s))

)

−α(1)(s,Q(0)(s), X(s)

)ds (46a)

+

∫ t

0

(ηα(0) (s, 1/ηQη(s), X(s))− ηα(0)

(s,Q(0)(s), X(s)

)

−Λα(0)(s,Q(0)(s), X(s); 1/ηQη(s)−Q(0)(s)

))ds (46b)

∫ t

0

√ηΛα(0)

(

s,Q(0)(s), X(s);1

ηQη(s)−Q(0)(s)

)

−Λα(0)(s,Q(0)(s), X(s);Q(1)(s)

)ds (46c)

+

∫ t

0α(1)

(s,Q(0)(s), X(s)

)− α(1)(s,Q(0)(s), X(s))ds (46d)

+∑

i∈I

[

B′i

(1

η

∫ t

0αη (s, 1/ηQη(s), X(s)) ds

)

−B′i

(∫ t

0α(0)

(s,Q(0)(s), X(s)

)ds

)]

vi (46e)

+1√ηO(ln η)

Amazingly all terms tend to zero as η → 0. Now here is why. Using thesame arguments as in the proof of lemma A.4, the first term (46a) can be

30

bounded by the pure process sequence∫ t

0

∥∥∥∥∥αη(s, y,X(s))− ηα(0)(s, y,X(s))

√η

− α(1)(s, y,X(s))

∥∥∥∥∥dsZη(t)

The second term (46b) can be rewritten as

√η

∣∣∣∣1

ηQη(s)−Q(0)(s)

∣∣∣∣

11ηQη(s)−Q(0)(s)

∥∥∥α(0)(s, 1/ηQη(s), X(s))− α(0)(s,Q(0)(s), X(s))

−Λα(0)(s,Q(0), X(s); 1/ηQη(s)−Q(0)(s))∥∥∥

The right hand (first line and the rest) may be analysed using the propertiesof the scalable Lipschitz derivative while the left term is always bounded bytheorem 2.For the third term we use the fact that

‖Λf(x; y)‖ ≤ ‖f‖

therefore the term (46c) can be bounded by

∫ t

0‖α(0)(s, y,X(s))‖ lim

η→∞sup0≤u≤s

∣∣∣∣∣Qη(s)− ηQ(0)(u)

√η

−Q(1)(u)

∣∣∣∣∣

(47)

The term (46d) is bounded by the Lipschitz condition

‖α(1)(s, y,X(s))‖ds sup0≤s≤t

∣∣∣∣1

ηQη(s)−Q(0)(s)

∣∣∣∣

and by theorem 2 the term vanishes as η →∞.For the final term (46e) we recall the paths of a Brownian motion are

continuous with probability 1. Therefore if

limη→∞

sup0≤s≤t

∣∣∣∣

∫ s

0αη(

u,1

ηQη(u), X(u)

)

du−∫ s

0α(0)

(u,Q(0)(u), X(u)

)du

∣∣∣∣ = 0

holding in compact sets, the term (46e) will vanish as well. Notice that

sup0≤s≤t

∣∣∣∣1

η

∫ s

0αηi

(

u,1

ηQη(u), X(u)

)

du−∫ s

0α(0)i

(u,Q(0)(u), X(u)

)∣∣∣∣

≤∫ t

0

∣∣∣∣1

ηαηi (s, 1/ηQ

η(s), X(s))− α(0)i (s, 1/ηQη(s), X(s))

∣∣∣∣ ds

+

∫ t

0

∣∣∣α(0)i (s, 1/ηQ

η(s), X(s))− α(0)i

(s,Q(0)(s), X(s)

)∣∣∣ ds

≤∫ t

0

∥∥∥∥1

ηαη(s, y,X(s))− α(0)(s, y,X(s))

∥∥∥∥ dsZ

η(t) +

∫ t

0‖α(0)(s, y,X(s))‖ds sup

0≤s≤t

∣∣∣∣1

ηQη(s)−Q(0)(s)

∣∣∣∣

31

that vanishes as in proof of theorem 2. Then

limη→∞

sup0≤s≤t

∣∣∣∣∣

∑

i∈I

[

B′i

(1

η

∫ s

0αηi (u, 1/ηQ

η(u), X(u)) du

)

−B′i

(∫ s

0α(0)(u,Q(0)(u), X(u))du

)]

vi

∣∣∣∣ = 0

Therefore in the limit the whole expression may be bounded by (47), this is

limη→∞

sup0≤s≤t

∣∣∣∣∣Qη(s)− ηQ(0)(s)

√η

−Q(1)(s)

∣∣∣∣∣

≤∫ t

0‖α(0)(s, y,X(s))‖ lim

η→∞sup0≤u≤s

∣∣∣∣∣Qη(s)− ηQ(0)(s)

√η

−Q(1)(s)

∣∣∣∣∣

and applying the lemma A.5,the first inequality with z(t) = 0, we obtain

limη→∞

sup0≤s≤t

∣∣∣∣∣Qη(s)− ηQ(0)(s)

√η

−Q(1)(s)

∣∣∣∣∣= 0

and the proof is complete. �

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