# Dynamic Credit Quality Evaluation with Social Network Data

Authors: Stanley Sewe; Philip Ngare; Patrick Weke
Journal: Journal of Applied Mathematics
 (2019)

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## Abstract

We investigate the filtering problem where the borrower’s time varying credit quality process is estimated using continuous time observation process and her (in this paper we refer to the borrower as female and the lender as male) ego-network data. The hidden credit quality is modeled as a hidden Gaussian mean-reverting process whilst the social network is modeled as a continuous time latent space network model. At discrete times, the network data provides unbiased estimates of the current credit state of the borrower and her ego-network. Combining the continuous time observed behavioral data and network information, we provide filter equations for the hidden credit quality and show how the network information reduces information asymmetry between the borrower and the lender. Further, we consider the case when the network information arrival times are random and solve stochastic optimal control problem for a lender having linear quadratic utility function.


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## Body

## 1. Introduction

In this study we consider the problem of stochastic filtering in the presence of network generated information. Consider a continuous time credit quality processXt which is hidden and is only partially observed through its randomized function Yt. Additionally, at discrete time points the observer has access to unbiased signals of the process Xt and of nodes directly linked to the Xt node. Thus the standard stochastic filtering is carried out before the arrival of the network information. The unbiased signals from the network are used to improve the estimates of the hidden true credit quality.We assume that the nodes are individual (potential) borrowers in a dynamic social network with the processXt being borrower’s true credit quality modeled as an Ornstein-Ulehnbeck process. The hidden process Xt is partially observed through its function Yt in continuous time. Thus Yt models the typical borrower’s behavior and financial information available to the lender (e.g., account turnover, periodic account balance). Xt and Yt are modeled as scalar processes for simplicity, though the results apply to the vector case too. The borrower knows her true credit quality, and this is also visible to her direct social contacts. A social link is formed upon mutual consent between the parties. The probability of a link forming is influenced by the distance between the borrowers’ credit quality. Thus links are formed by credit type homophily. Homophily ([1]) is the idea that individuals with similar characteristics are more likely to have a network tie than individuals with different characteristics. The network is thus modeled as an undirected continuous time latent space model. The lender is able to observe at discrete time points the borrower’s ego-network. Hence at time points 0=t0<t1<⋯<tN-1<T the lender is able to update his estimate of the borrower’s credit quality using the network data.Kalman-Bucy filtering techniques provide the estimates for the hidden processXt and the conditional variance based on the continuous time observations Yt. At time tk,k=0,1, N∈N the estimates and the conditional variances are updated using the vector Zk of network signals. The dimension of the vector is depended on the borrower’s degree (number of friends) ηt,k at time tk. Thus we combine Kalman-Bucy filtering and Bayesian updating techniques to improve the lender’s estimate of the borrower’s credit quality. Further, we prove that the network data leads to a lower variance for the estimated credit quality X^t. In addition, we consider the case whereby the network information arrives at random times with intensity of a Poisson process. The dynamics of the filtered estimates and conditional variances are the same as in the case of discrete time information arrivals, though we have the conditional variance being a piecewise deterministic stochastic process. We solve a stochastic optimization problem for a lender using the linear quadratic utility. Due to the random times tk, the problem is reduced to a linear quadratic Gaussian utility with jumps. Our credit model can be seen as continuous time version of the static credit scoring model of [2]. Also related is the work of [3] who derived filtered estimates and conditional variance of a hidden drift process in the presence of continuous time observation and discrete time scalar expert opinion. Our stochastic optimal control problem is close to the optimal credit limit problem of [4] whereby the lender optimizes the credit limit available to a corporate entity based on the status of the entity’s surplus process. Reference [5] solved a singular optimization problem within a random time horizon to obtain the optimal interest rate payable by a corporate entity. Reference [6] provides a good account of stochastic optimization for jump diffusion processes whilst [7] is an excellent treatment of discrete and stochastic optimization.Application of social network analysis and social network data in consumer credit has become ubiquitous; see [8] for a review. With the availability of social network data, digital credit and mobile phone based lending have emerged as key areas of study in finance. Some previous studies of dynamic social network include the coevolution model of [9] where the node’s attributes and network ties influence each other. The nodal attributes and network ties are modeled as continuous time Markov chains. The static latent distance models of [10] and discrete time model of [11] are able to capture key network properties like homophily and transitivity (two nodes connected to a common third node are likely to be connected) and group structures. Reference [12] provides a Bayesian dynamic model of relational structure on a latent variable continuous time social network. Reference [13] gives a good review of the static and dynamic latent distance and latent space network models.There exist two approaches in modelling credit risk: the structural and reduced form approach, respectively. In the structural approach, default occurs when the observed process related to the value of the firm (or firm cash flow) passes below a deterministic default threshold. The threshold is chosen by the managers of the firm, with the default time being a predictable stopping time. In the reduced form approach, default is modeled as an inaccessible stopping time, with the default threshold modelled as a random variable. Reference [14] provides a good introduction to credit risk modelling. There have been attempts to apply the corporate credit risk models in modelling unsecured consumer credit. The challenge therein lies in the fact that it is hard to measure a borrower’s assets. In addition, default in consumer loans is related more to cash flow issues than to the borrower’s debts exceeding her assets. Reference [15] modeled the consumer credit risk using the structural approach of credit risk modeling. Treating the consumer’s behavioral score as a continuous time jump diffusion process, the authors were able to apply the option pricing theory to model the probability of default. Our model is related to the structural approach of credit risk modeling in that the credit worthiness score is seen as an asset to the borrower. However, in the model the lender manages credit risk through continuous updating of the credit limit and borrower’s credit worthiness. The credit limit is affected by the borrower’s credit worthiness, which is related to the probability of default. In turn, the sanctioned credit limit (credit exposure) affects the borrower’s credit worthiness.Some existing studies on consumer credit scoring include [16] whereby the borrower’s credit rating is modelled as a discrete time Markov chain process upon incorporating a latent variable driven by economic conditions. Reference [2] used social network links and signals from the borrower’s ego-network to estimate her credit quality. The model assumed that the credit quality which is normally distributed is hidden and can only be estimated using the noisy signals from the borrower and her ego-network. Thus the model was based on homophily by preference. Reference [17] proposed the SEN-HMM-CSD (Social Economic Network-Hidden Markov Model-Credit Scores and Default) model whereby economic and social network variables including trust, distrust, and reputation were used to estimate the hidden credit quality of a borrower in a social network. In the SEN-HMM-CSD model, the true credit quality was modeled as a discrete time, discrete state hidden Markov model. Assuming a fully connected social network on a population with no drift, the network based variables were used to generate the credit risk analysis factors, which in turn were used to estimate the hidden Markov parameters for each individual borrower. Our work generalizes the SEN-HMM-CSD model in the continuous time continuous state Hidden Markov model direction through the following: inclusion of neighbouring nodes signals in the estimation of credit worthiness, credit limit variability tied to individual’s credit worthiness, and the network ties based on credit type homophily; thus there is no assumption on a fully connected network model.The objective of stochastic filtering is to find the best estimate of a hidden processX partially seen through the observation process Y. Excellent textbook treatment of stochastic filtering includes [18–20]. Recent studies on stochastic filtering and information modeling include [3] who studied a market model whereby the unobserved drift process (modeled as an Ornstein-Uhlenbeck process) is filtered conditioned on a continuous time stock return observations and discrete time expert opinion. By incorporating expert opinion in their estimation, the model was a continuous time version of the Black-Litterman model [21]. Reference [22] considered a market model whereby the unobserved drift parameter is estimated on an observation filtration initially enlarged with the terminal value of the stock price perturbed by some constant variance noise. Reference [23] considered an enlargement of filtration problem with a random variable combined with partial information, with an application in linear stochastic control.The paper is organized as follows. In Section2, we present our credit model, dynamic social network model, and the information setup. Stochastic filtering and Bayesian updating results are presented in Section 3. Within this section, we also present the properties of the conditional variance. Credit limit management optimization problem is solved in Section 4. Brief numerical results are presented in Section 5, whilst Section 6 concludes.


## 2. The Credit Model

Consider a filtered probability space(Ω,A,A=Att≥0,P) satisfying the usual conditions of right continuity and completeness. All processes are assumed to be A adapted. We are interested in the following filtering problem.Borrower’s Behavioral Dynamics. We model the borrower’s observed behavioral dynamics as a linear diffusion process defined as(1)dYt=αXtdt+σdWtThe parametersα,σ>0 are assumed to be constants and Wt is a A adapted one dimensional Brownian motion. The hidden credit quality process Xt driving the drift is modeled as a mean-reverting Ornstein Ulehnbeck process defined as(2)dXt=μδ-Xtdt+γdBtX0=xwhere δ∈R,μ>0,γ>0 are constants and Bt is a Brownian motion. We assume that Bt and Wt are independent. x is a A measurable normally distributed random variable independent of B and W with mean m0∈R and variance v0>0. Xt is a Gaussian process; its mean and variance are given by(3)δ+e-μtm0-δγ22μ+e-2μtv0-γ22μNetwork Dynamics. We consider a society with a population Z. Each individual in the population has a normally distributed time varying credit quality Xit. The credit qualities Xit are assumed to be independent across the individuals. At each time t>0, with no other additional borrower information, an individual’s true credit quality Xt is assumed to be normally distributed with mean 0 and variance q-1. Individuals i and j interact and a network tie is formed by mutual consent. Affecting equations (4) and (5)-the highlighted section should read: Assuming an undirected network i.e. Yij(t)=Yji(t), then we let(4)Yijt∣πijt~Bernπijt for every i≠j and t⩾0 with(5)πijt=e-1/2Xit-Xjt2Thus the network ties Yij(t)∈{0,1} are conditionally independent Bernoulli random variables given the corresponding probabilities of tie formation πij(t)∈(0,1). πij(t) is the survival function of a standard Raleigh distribution. Existence of a network tie between individual i and j is dependent on the distance between their credit types, and shorter distance leads to higher probability of network tie. Thus the network model captures homophily, whereby individuals with similar characteristics are more likely to have social network ties than individuals with different characteristics. We assume there is no cost incurred in network tie formation or destruction.LetGt denote the set of friendship ties in the society at time t. The set of borrower i’s friends at time t, known as her ego-network, is defined as gt1=ij∣ij∈Gt. For a particular borrower i, we consider her true credit quality process Xt and behavioral score Yt.Lender’s information. As a result of the information asymmetry between the lender and the borrowers, the lender is unable to directly observe the true credit quality process Xt for the individual i. He however gets noisy observation of the hidden process via the behavioural process Yt. Additionally, the lender is able to receive at discrete time points 0=t0<t1<⋯<tN-1<T,N∈N noisy signals about the current state of Xi and neighbouring nodes within her ego-network. For each k∈0,1,…,N, let the set Zk=Zjk∣ij∈gt1 be the set of signals of the borrower’s ego-network such that Zjk=Xjtk+Λkεjk. The variable εjk~N(0,1) is i.i.d across individuals and E(εjsεlu)=0 for j≠ls≠u. Thus at time point tk the lender observes borrower i′s ego-network and the vector Zk comprising of the noisy observation of her current credit state Zk=Xtk+εik and the credit states of her immediate social contacts Zkj,ij∈gtk1. The variance Λk is a measure of the reliability of the network information received by the lender.The information available to the lender can thus be represented by the following filtrations:(6)FY=FtYt∈0,TwithFtYgeneratedbyYs,s⩽tFZ=FtZt∈0,TwithFtZgeneratedbyYs,s⩽t,Zk,tk⩽tFO=FtOt∈0,TwithFtOgeneratedbyZk,tk⩽tFY corresponds to the continuous time behavioral score only and FO consists of the network information received at discrete times whilst FZ is the combination of behavioral score and the network information. We assume that the σ-algebras FtY and FtZ are augmented by the P null sets.Assumption 1.
It is obvious thatFtY⊂FtZ for all t>0. FtY is generated by the innovation process dMt=dWt+ασ-1Xt-X^tdt. We assume that E(Zk,Mt-Mt,k)=0 for all t⩾tk. Thus FtY is immersed in FtZ; hence every FY square integrable martingale is a FZ square integrable martingale.


## 3. Stochastic Filtering

Stochastic filtering entails obtaining estimate of the hidden processXt conditioned on the observed process. Let X^tH be the projection of the process Xt onto the observed filtration FH,H∈O,Y,Z, i.e., X^tH=EXt∣FtH. X^tH is the optimal estimator of Xt in the mean square sense. In this section, we derive the recursive equations for the filtered estimates conditioned on the different information settings.


### 3.1. Behavioural Observations

In the case whereby the lender observes the borrower’s behavioral information only, i.e., where the lender’s observation filtration isFtY, the state and observation processes constitute a linear system of equations. Besides, the bivariate process (Xt,Yt) is Gaussian. Thus the usual Kalman-Bucy filtering technique, see, e.g., [19, 20], can be used to obtain the conditional mean X^tY=EXt∣FtY and the conditional variance λtY=EXt-X^tY2∣FtY.The dynamics ofX^tY is given by the following SDE:(7)dX^tY=μδ-X^tY-α2σ-2λtYX^tYdt+ασ-2λtYdYt,X^0Y=m0 whilst the dynamics of λtY is given by the deterministic ODE(8)dλtYdt=-2μλtY+γ2-α2σ-2λtY2,λ0Y=v0Equation (8) is the wellknown Riccati equation. With initial value λ0Y=v0, the unique solution for the equation can be given as(9)λtY=-μσ2α2+CoC1+C2e-2α2/σ2CotC1-C2e-2α2/σ2Cotwith C0=σ/αμ2σ2/α2+γ2, C1=v0+C0+μσ2/α2, and C2=v0-C0+μσ2/α2 (see, e.g., [24]).


### 3.2. Behavioral Observations and Network Information

This is the case whereby the lender uses both the observed behavioral score and the network information to obtain an estimate of the borrower’s credit quality. The following lemma shows that the expected number of friends for individuali at any time t⩾0 can be treated as a constant.Lemma 2.
Letηt be the borrower’s degree (number of friends) at time t. At each time t⩾0, conditioned on the borrower’s true credit quality Xt, the expected degree Eηt∣Xt is a constant denoted as η.Proof.
Conditioned on the borrower’s true credit typeXt, the probability of having a network tie with any other individual is(10)∫-∞∞e-Xt-s2/2q2πe-qs2/2ds=qq+1e-q/2q+1Xt2Thus the conditional expected number of friends Eηt∣Xt is given by(11)Eηt∣Xt=Zqq+1e-q/2q+1Xt2As q→0, Z→∞ we make the simplifying assumption that Zq/q+1 is a constant denoted by η. Thus in the limit Eηt∣Xt=η.The simplifying assumption is justified by the consideration that in a typical human social network; types are diffuse and the population size is large. Besides, in a small social network with less diffuse individual types, the benefits of social network scoring would be limited.Proposition 3.
For anyk∈0,1,…,N and t>0, let pk=Λk-1 be the precision of the network information at time tk. Further define the parameter(12)θkZ=pk+1λtk-Zpkpk+1+ηk+pk+1Then it holds that(i)
For anyt∈[tk,tk+1), X^tZ is Gaussian and its dynamics satisfies the following:(13)dX^tZ=μδ-X^tZ-α2σ-2λtZX^tZdt+ασ-2λtZdYtwhilst the conditional variance is given by (14)λtZ=-μσ2α2+CoC1k+C2ke-2α2/σ2CotC1k-C2ke-2α2/σ2Cotwith initial values X^tkZ,λtkZ. C0 is same as in (9) whilst C1k=λtkZ+C0+μσ2/α2 and C2k=λtkZ-C0+μσ2/α2(ii)
At information datetk, X^t,kZ is Gaussian with the conditional mean and variance X^t,kZ,λt,kZ updated from their respective values at tk- (that is, before the arrival of the network information) to be mean(15)X^t,kZ=θkZλtk-ZX^t,k-Zλtk-Z+pkZik+pkpk+1∑j∈ZZjkand variance(16)θkZλtk-Z=λtk-Zpk+1λtk-Zpkpk+1+ηk+pk+1Proof.
Part (i). Between two information dates, t∈[tk,tk+1), there is no arrival of new network information. The only observed process is the continuous time returns Yt with the lender’s information set being FtZ=Ft,kZ∨σYs,tk<s⩽t. Thus, we have the standard Kalman-Bucy filtering case, with the initial values for conditional mean and conditional variance being X^tkZ and λtkZ, respectively. The result thus follows from (8) and (9).
Part (ii). At the information date, tk, the estimates for the mean and variance are updated from their values at time tk- using the Bayesian method. The borrower’s ego network at time tk comprises all her direct network ties. We assume that the lender is able to observe the borrower’s complete ego-network at time tk. The posterior probability of the borrower’s credit type PXtk∣Zk is obtained by (17)PXtk∣Zk∝PXtk,Zk=∫-∞∞PZk∣Xtk,Xj,tkPXtkPXj,tk∣XtkdXj,tk=∫-∞∞PZk∣Xtk,Xj,tkPXtkPXj,tkdXj,tkThe last equality is as a result of the assumption of independence for the Xjtk. We have PXj,tk=∏ij∈gk1e-q/2Xj,tk2 being the assumed density of any individual Xj for j≠i. The lender does not update his knowledge of individual j′s signal. The remaining part of the integrand is(18)PZk∣Xtk,Xj,tkPXtk=e-Xtk-Xtk-Z2/2λtk-Z×e-pk/2Zik-Xtk2×∏j∈Zke-pk/2Zjk-Xjtk2×∏ij∈gk1e-1/2Xtk-Xjtk2︸a×∏j∈gk0j∉Zk1-Eηtk∣XtkZ︸bwhere(i)
The first term denotes the product of the prior density (before the arrival of network information) and the likelihood function for the observationZit,k.(ii)
(a) denotes the probability that at timetk borrower i is friends with the individuals within her ego-network (in gtk1) whose signals are in Zk and that these friends have the signals as collected in Zk(iii)
(b) denotes the probability that at timetk borrower i is not friends with anyone outside gtk1.
Asq→0, Z→∞, by the monotone convergence theorem and applying Lemma 2 above then(19)∏j∈gk0j∉Zk1-Eηtk∣XtkZ⟶e-ηwhich has no Xtk term. Hence we have(20)PXtk∣Zk,gk∝∫-∞∞e-Xtk-Xtk-Z2/2λtk-Z×e-pk/2Zik-Xtk2×∏j∈Zke-pk/2Zjk-Xjtk2×∏ij∈gk1e-1/2Xtk-Xjtk2dXj,tkwhereby the integrand is a product of Gaussian densities. Upon integrating out Xjtk and matching the terms of Xtk and Xtk2 we obtain the posterior density which is Gaussian with the given expectation and variance.For the case whereby the lender observes only the network information, i.e.,H=O, we have that between information dates t∈[tk,tk+1), there is no arrival of new information. We thus get the following corollary.Corollary 4.
When the lender’s information set isFtO we have the following:(i)
Fort∈[tk,tk+1), the estimates for the mean and variance are given by(21)X^tO=δ+e-μt-tkX^tkO-δ(22)λtO=e-2μt-tkλtkO+γ22μ1-e-2μt-tk(ii)
At information datetk, it holds that XtO is Gaussian with mean and variance(23)X^t,k0=θk0λtk-0X^t,k-0λtk-0+pkZik+pkpk+1∑j∈ZZjk(24)θk0λtk-0=λtk-0pk+1λtk-0pkpk+1+ηk+pk+1respectively.Proof.
Between information dates, it follows thatFtO=FtkO for t∈[tk,tk+1). Thus X^tO=EXt∣FtkO and λtO=EXt-X^tO2∣FtkO. From the state/observation model, we get that Xt has the form(25)Xt=δ+e-μt-tkXtk+γ∫tkteμsdBsTherefore(26)X^tO=δ+e-μt-tkEXtk-δ∣FtkO=δ+e-μt-tkX^tkO-δ
The conditional variance is given by(27)EXt-X^tO2∣FtkO=Ee-μt-tkXtk-X^tkO+e-μt-tkγ∫tkteμs-tkdBs2∣FtkO=e-2μt-tkEXtk-X^tkO2∣FtkO+e-2μt-tkγ2∫tkte2μs-tkds=e-2μt-tkλtkO+γ22μ1-e-2μt-tk
Here we have employed the martingale property of the Brownian motion process and used the Ito isometry to obtain the variance. At information datetk, the estimates are updated by the information received from the random vector Zk. With a Gaussian prior for Xtk, we use the Bayesian update similar to Proposition 3 part (ii) to get the posterior expectation and variance.Remark 5.
(i)
For the case where the individual has no network ties (isolated node), i.e., whenηtk=0, then θk0 reduces to Λk/(Λk+λtk-0). Considering the behavior of the estimates with limiting values of Λk, when Λk=0 we have that θk0=0 and X^tO=Xtk whilst when Λk=∞ we have θk0=1 and X^tO=Xtk-. For Λk=∞ the implication is that no additional information on the hidden process Xt can be obtained by observing the network data. (ii)
Further, the estimatesX^tO and λtO can be obtained from X^tZ and λtZ as limiting cases as σ→∞. Thus between information dates, the dynamics for the conditional mean and conditional variance are dX^t0=μδ-X^t0dt and dλtO=-2μλt0+γ2dt, respectively. These are deterministic O.D.E equations which can be solved to yield the expressions in Corollary 4. The implication is that on account of σ→∞ no additional information relating to the hidden process Xt can be obtained by observing the continuous time process Yt. (iii)
We may consider a case whereby the information arrival timestk,k=0,1,…,N-1 are random times with intensity φ akin to the jump times of a Poisson process. The random times are independent of the processes X,Y,Z and the Brownian motions W,B; thus the timing of jumps does not carry any information. We deal with this case in Section 4 on optimal credit limit management.In the following proposition, we show how the additional network informationZk improves the lender’s estimate of the borrower’s true credit quality. This is captured by the lower variance associated with the estimates from the information set FtZ compared to the other two alternative information settings. Thus a lender incorporating signals from the borrower’s ego-network in his analysis is bound to have a better estimate of the borrower’s true credit history.Proposition 6 (properties of the conditional variance).
ForH∈{Y,O}, t∈[0,T] and k=1,2,…,N(28)λtZ⩽λtHProof.
Fort∈[tk,tk+1), the dynamics of the conditional variance is given by the function (29)dλtHdt=fHλtHfor H∈{Y,O,Z}. From Proposition 3 (i), we have(30)fZr=fYr=-2μr+γ2-α2σ-2r2whilst following Remark 5(31)fOr=-2μr+γ2whichisfOr=fZr+α2σ-2r2
Comparingλ0Z and λ0Y, we note that, at time t=0, λ0Z=θ00v0⩽v0=λ0Y since θ0∈[0,1] for all values of η0. Since fZ(r)=fY(r) and a unique solution for the ODE fZ(r) exists, we note that the inequality will persist for all t∈[t0,t1). At time t=t1, we again have the inequality λtkZ⩽λt1Y. Extending the argument for all t∈[0,T) yields that that λtZ⩽λtY.
To prove thatλ0Z⩽λ0O, we note that, at time t=0, λ0Z=λ0O. For t∈[0,t1), we have fZ(r)⩽fO(r). This means that the two deterministic functions starting at the same initial value will have the relation λtZ⩽λtO. At time t=t1, since the map x→x(p+1)/xpkpk+1+ηk+pk+1 is nondecreasing, we shall have λt1Z⩽λt1O. Iterating this argument for all t∈[0,T] we conclude that λ0Z⩽λ0O.The implication of this result is that the network information improves the accuracy of the lender’s belief about the borrower’s true credit quality. Periodically observing the borrower’s ego-network at discrete times without even observing the entire network leads to a better estimate of the hidden true credit quality. This is as a result of the network ties being based on borrower type homophily. This is quite in line with industry’s practice in digital and mobile phone based lending whereby an individual’s social network data can be used to improve the estimate of her credit quality.


## 3.1. Behavioural Observations

In the case whereby the lender observes the borrower’s behavioral information only, i.e., where the lender’s observation filtration isFtY, the state and observation processes constitute a linear system of equations. Besides, the bivariate process (Xt,Yt) is Gaussian. Thus the usual Kalman-Bucy filtering technique, see, e.g., [19, 20], can be used to obtain the conditional mean X^tY=EXt∣FtY and the conditional variance λtY=EXt-X^tY2∣FtY.The dynamics ofX^tY is given by the following SDE:(7)dX^tY=μδ-X^tY-α2σ-2λtYX^tYdt+ασ-2λtYdYt,X^0Y=m0 whilst the dynamics of λtY is given by the deterministic ODE(8)dλtYdt=-2μλtY+γ2-α2σ-2λtY2,λ0Y=v0Equation (8) is the wellknown Riccati equation. With initial value λ0Y=v0, the unique solution for the equation can be given as(9)λtY=-μσ2α2+CoC1+C2e-2α2/σ2CotC1-C2e-2α2/σ2Cotwith C0=σ/αμ2σ2/α2+γ2, C1=v0+C0+μσ2/α2, and C2=v0-C0+μσ2/α2 (see, e.g., [24]).


## 3.2. Behavioral Observations and Network Information

This is the case whereby the lender uses both the observed behavioral score and the network information to obtain an estimate of the borrower’s credit quality. The following lemma shows that the expected number of friends for individuali at any time t⩾0 can be treated as a constant.Lemma 2.
Letηt be the borrower’s degree (number of friends) at time t. At each time t⩾0, conditioned on the borrower’s true credit quality Xt, the expected degree Eηt∣Xt is a constant denoted as η.Proof.
Conditioned on the borrower’s true credit typeXt, the probability of having a network tie with any other individual is(10)∫-∞∞e-Xt-s2/2q2πe-qs2/2ds=qq+1e-q/2q+1Xt2Thus the conditional expected number of friends Eηt∣Xt is given by(11)Eηt∣Xt=Zqq+1e-q/2q+1Xt2As q→0, Z→∞ we make the simplifying assumption that Zq/q+1 is a constant denoted by η. Thus in the limit Eηt∣Xt=η.The simplifying assumption is justified by the consideration that in a typical human social network; types are diffuse and the population size is large. Besides, in a small social network with less diffuse individual types, the benefits of social network scoring would be limited.Proposition 3.
For anyk∈0,1,…,N and t>0, let pk=Λk-1 be the precision of the network information at time tk. Further define the parameter(12)θkZ=pk+1λtk-Zpkpk+1+ηk+pk+1Then it holds that(i)
For anyt∈[tk,tk+1), X^tZ is Gaussian and its dynamics satisfies the following:(13)dX^tZ=μδ-X^tZ-α2σ-2λtZX^tZdt+ασ-2λtZdYtwhilst the conditional variance is given by (14)λtZ=-μσ2α2+CoC1k+C2ke-2α2/σ2CotC1k-C2ke-2α2/σ2Cotwith initial values X^tkZ,λtkZ. C0 is same as in (9) whilst C1k=λtkZ+C0+μσ2/α2 and C2k=λtkZ-C0+μσ2/α2(ii)
At information datetk, X^t,kZ is Gaussian with the conditional mean and variance X^t,kZ,λt,kZ updated from their respective values at tk- (that is, before the arrival of the network information) to be mean(15)X^t,kZ=θkZλtk-ZX^t,k-Zλtk-Z+pkZik+pkpk+1∑j∈ZZjkand variance(16)θkZλtk-Z=λtk-Zpk+1λtk-Zpkpk+1+ηk+pk+1Proof.
Part (i). Between two information dates, t∈[tk,tk+1), there is no arrival of new network information. The only observed process is the continuous time returns Yt with the lender’s information set being FtZ=Ft,kZ∨σYs,tk<s⩽t. Thus, we have the standard Kalman-Bucy filtering case, with the initial values for conditional mean and conditional variance being X^tkZ and λtkZ, respectively. The result thus follows from (8) and (9).
Part (ii). At the information date, tk, the estimates for the mean and variance are updated from their values at time tk- using the Bayesian method. The borrower’s ego network at time tk comprises all her direct network ties. We assume that the lender is able to observe the borrower’s complete ego-network at time tk. The posterior probability of the borrower’s credit type PXtk∣Zk is obtained by (17)PXtk∣Zk∝PXtk,Zk=∫-∞∞PZk∣Xtk,Xj,tkPXtkPXj,tk∣XtkdXj,tk=∫-∞∞PZk∣Xtk,Xj,tkPXtkPXj,tkdXj,tkThe last equality is as a result of the assumption of independence for the Xjtk. We have PXj,tk=∏ij∈gk1e-q/2Xj,tk2 being the assumed density of any individual Xj for j≠i. The lender does not update his knowledge of individual j′s signal. The remaining part of the integrand is(18)PZk∣Xtk,Xj,tkPXtk=e-Xtk-Xtk-Z2/2λtk-Z×e-pk/2Zik-Xtk2×∏j∈Zke-pk/2Zjk-Xjtk2×∏ij∈gk1e-1/2Xtk-Xjtk2︸a×∏j∈gk0j∉Zk1-Eηtk∣XtkZ︸bwhere(i)
The first term denotes the product of the prior density (before the arrival of network information) and the likelihood function for the observationZit,k.(ii)
(a) denotes the probability that at timetk borrower i is friends with the individuals within her ego-network (in gtk1) whose signals are in Zk and that these friends have the signals as collected in Zk(iii)
(b) denotes the probability that at timetk borrower i is not friends with anyone outside gtk1.
Asq→0, Z→∞, by the monotone convergence theorem and applying Lemma 2 above then(19)∏j∈gk0j∉Zk1-Eηtk∣XtkZ⟶e-ηwhich has no Xtk term. Hence we have(20)PXtk∣Zk,gk∝∫-∞∞e-Xtk-Xtk-Z2/2λtk-Z×e-pk/2Zik-Xtk2×∏j∈Zke-pk/2Zjk-Xjtk2×∏ij∈gk1e-1/2Xtk-Xjtk2dXj,tkwhereby the integrand is a product of Gaussian densities. Upon integrating out Xjtk and matching the terms of Xtk and Xtk2 we obtain the posterior density which is Gaussian with the given expectation and variance.For the case whereby the lender observes only the network information, i.e.,H=O, we have that between information dates t∈[tk,tk+1), there is no arrival of new information. We thus get the following corollary.Corollary 4.
When the lender’s information set isFtO we have the following:(i)
Fort∈[tk,tk+1), the estimates for the mean and variance are given by(21)X^tO=δ+e-μt-tkX^tkO-δ(22)λtO=e-2μt-tkλtkO+γ22μ1-e-2μt-tk(ii)
At information datetk, it holds that XtO is Gaussian with mean and variance(23)X^t,k0=θk0λtk-0X^t,k-0λtk-0+pkZik+pkpk+1∑j∈ZZjk(24)θk0λtk-0=λtk-0pk+1λtk-0pkpk+1+ηk+pk+1respectively.Proof.
Between information dates, it follows thatFtO=FtkO for t∈[tk,tk+1). Thus X^tO=EXt∣FtkO and λtO=EXt-X^tO2∣FtkO. From the state/observation model, we get that Xt has the form(25)Xt=δ+e-μt-tkXtk+γ∫tkteμsdBsTherefore(26)X^tO=δ+e-μt-tkEXtk-δ∣FtkO=δ+e-μt-tkX^tkO-δ
The conditional variance is given by(27)EXt-X^tO2∣FtkO=Ee-μt-tkXtk-X^tkO+e-μt-tkγ∫tkteμs-tkdBs2∣FtkO=e-2μt-tkEXtk-X^tkO2∣FtkO+e-2μt-tkγ2∫tkte2μs-tkds=e-2μt-tkλtkO+γ22μ1-e-2μt-tk
Here we have employed the martingale property of the Brownian motion process and used the Ito isometry to obtain the variance. At information datetk, the estimates are updated by the information received from the random vector Zk. With a Gaussian prior for Xtk, we use the Bayesian update similar to Proposition 3 part (ii) to get the posterior expectation and variance.Remark 5.
(i)
For the case where the individual has no network ties (isolated node), i.e., whenηtk=0, then θk0 reduces to Λk/(Λk+λtk-0). Considering the behavior of the estimates with limiting values of Λk, when Λk=0 we have that θk0=0 and X^tO=Xtk whilst when Λk=∞ we have θk0=1 and X^tO=Xtk-. For Λk=∞ the implication is that no additional information on the hidden process Xt can be obtained by observing the network data. (ii)
Further, the estimatesX^tO and λtO can be obtained from X^tZ and λtZ as limiting cases as σ→∞. Thus between information dates, the dynamics for the conditional mean and conditional variance are dX^t0=μδ-X^t0dt and dλtO=-2μλt0+γ2dt, respectively. These are deterministic O.D.E equations which can be solved to yield the expressions in Corollary 4. The implication is that on account of σ→∞ no additional information relating to the hidden process Xt can be obtained by observing the continuous time process Yt. (iii)
We may consider a case whereby the information arrival timestk,k=0,1,…,N-1 are random times with intensity φ akin to the jump times of a Poisson process. The random times are independent of the processes X,Y,Z and the Brownian motions W,B; thus the timing of jumps does not carry any information. We deal with this case in Section 4 on optimal credit limit management.In the following proposition, we show how the additional network informationZk improves the lender’s estimate of the borrower’s true credit quality. This is captured by the lower variance associated with the estimates from the information set FtZ compared to the other two alternative information settings. Thus a lender incorporating signals from the borrower’s ego-network in his analysis is bound to have a better estimate of the borrower’s true credit history.Proposition 6 (properties of the conditional variance).
ForH∈{Y,O}, t∈[0,T] and k=1,2,…,N(28)λtZ⩽λtHProof.
Fort∈[tk,tk+1), the dynamics of the conditional variance is given by the function (29)dλtHdt=fHλtHfor H∈{Y,O,Z}. From Proposition 3 (i), we have(30)fZr=fYr=-2μr+γ2-α2σ-2r2whilst following Remark 5(31)fOr=-2μr+γ2whichisfOr=fZr+α2σ-2r2
Comparingλ0Z and λ0Y, we note that, at time t=0, λ0Z=θ00v0⩽v0=λ0Y since θ0∈[0,1] for all values of η0. Since fZ(r)=fY(r) and a unique solution for the ODE fZ(r) exists, we note that the inequality will persist for all t∈[t0,t1). At time t=t1, we again have the inequality λtkZ⩽λt1Y. Extending the argument for all t∈[0,T) yields that that λtZ⩽λtY.
To prove thatλ0Z⩽λ0O, we note that, at time t=0, λ0Z=λ0O. For t∈[0,t1), we have fZ(r)⩽fO(r). This means that the two deterministic functions starting at the same initial value will have the relation λtZ⩽λtO. At time t=t1, since the map x→x(p+1)/xpkpk+1+ηk+pk+1 is nondecreasing, we shall have λt1Z⩽λt1O. Iterating this argument for all t∈[0,T] we conclude that λ0Z⩽λ0O.The implication of this result is that the network information improves the accuracy of the lender’s belief about the borrower’s true credit quality. Periodically observing the borrower’s ego-network at discrete times without even observing the entire network leads to a better estimate of the hidden true credit quality. This is as a result of the network ties being based on borrower type homophily. This is quite in line with industry’s practice in digital and mobile phone based lending whereby an individual’s social network data can be used to improve the estimate of her credit quality.


## 4. Optimal Credit Limit Management

Following Remark5 (iii), in this section, we consider the case whereby the network information arrival times tk are modeled as Poisson jump times, with jump intensity φ. No information arrives at time t0. As such the sequence (tk,Zk)k∈I is a marked point process, though the dimension of the vector Zk is governed by the borrower’s degree ηtk at time tk. In this case, the dynamics of the filtered process X^tZ and λtZ will be same as in Proposition 3, save for the fact that λtZ will be a piecewise deterministic process with stochastic jump times. Denote the jumps of X^t,kZ at time tk as (32)Rt,k=X^t,kZ-X^t,k-Z From the properties of the filtered process X^t,kZ, Rt,k has a Gaussian distribution. Let f(X^t,kZ,rt,k) denote the density of Rt,k with first and second moments μt,k1=E(Rt,k) and μt,k2=E(Rt,k2), respectively. We note that the jump sizes are not i.i.d like in the compound Poisson case, since the density f(X^t,kZ,rt,k) is dependent on the state of the system at time tk.Consider a financial market model with terminal timeT whereby a lender has access to the continuous time information FY and the network information generated by the marked point process (tk,Zk)k∈I. The lender desires to obtain an estimate of the borrower true credit quality Xt based on the information set FZ. Besides, the lender desires to optimize his credit decision viz the credit limit lt sanctioned to the borrower from time to time. Thus we are faced with a stochastic optimal control problem. Due to mathematical tractability, we assume a nondefaulting borrower who is credit constrained and thus will accept and fully utilize the credit limit as and when availed. Thus, the lender avails a credit line facility on revolving fund basis, and the borrower is expected to continuously utilize and repay any outstanding amounts. The lender seeks to adjust the sanctioned credit limit in line with the borrower’s time varying credit quality. Different from the consumer credit card limits, the borrower is expected to fully repay any outstanding amounts before any new borrowing. Such a credit arrangement is applicable in unsecured mobile and digital banking products especially within the emerging market economies; see, e.g., [25, 26].Define the controlled credit quality process as(33)dXt=μδ-Xt+α~-pltdt+γdBt When the borrower has no outstanding credit balance, lt=0. As soon as the borrower obtains credit within the sanctioned limit, the finances are channeled towards short-term working capital/consumption with return of α~ (different from the drift before borrowing) which we net the price per unit of funds p. We assume that borrowing impacts the drift of the borrower’s credit quality without affecting her volatility. Let ϕ=(α~-p)>0 and (34)Jt,x=Et,x∫tTe-ωslsp-κ-12βls2ds be the lender’s objective function, where ω is the lender’s discounting factor, p is the price per unit of funds, κ is the lender’s cost per unit of funds, and β is the marginal lending cost. ω,κ,p, and β are all positive. The lender’s optimization problem is (35)supl∈UJt,x where U, the admissible set of lending strategies, is defined below. Since the controlled state process X is not observable, this is a situation of stochastic optimization with partially observed process. From Proposition 3 and Remark 5 and applying Ito’s formula for jump diffusion process, we condition on the information set FZ to obtain the filtered controlled state process as(36)dX^tZ=μδ-X^tZ+ϕldt+RtdNs+ασλtZdMtZwhere MtZ is the FtZ innovation process and N is the Poisson process with jump intensity φ. Let φdtf(X^t,kZ,rt,k)dR be the FZ compensator for the jump process. The complete information optimization problem is given as(37)supl∈UJ^t,x^=-infl∈U-J^t,x^=-infl∈UEt,x^∫tTe-ωs12βls2-lsp-κdsThus the lender seeks to obtain the value functionV(t,x^)=supl∈UJ^(t,x^) and the optimal loan limit l∗ such that V(t,x^)=Vl∗(t,x^)=supl∈UJ^(t,x^). (37) with dynamics (36) is a linear quadratic Gaussian regulator with jump diffusion problem. We solve the optimization problem using dynamic programming approach.Definition 7.
The admissible setU is defined as(38)U=l=ltt∈0,T,lisFZadapted,lt∈0,L~,X^tZ>0,E∫0Tls<∞,E∫0T∫-∞∞ħXul+R-ħXulfXul,rdRdu<∞,∀ħx∈C2where C2 is the family of functions ħ(x) such that ħ and its derivatives ħx,ħxx are continuous on R. The upper bound on the loan amount L~ is established exogenously. The upper bound may be in respect to the credit product specification or the lender’s risk management criteria.


### 4.1. The Dynamic Programming Method

We know that if the value functionV(x^)∈C2, then it satisfies the Hamilton-Jacobi-Bellman (HJB) equation(39)infl∈H-ωV+12βl2-lp-κ+μδ-x+ϕlVx+12α2λ2σ2Vxx+φEVx+R-Vx=0,∀t,x∈0,T×RWe proceed to solve the optimization problem in the following proposition.Proposition 8.
The function(40)νx=-Jx2+Kx+L solves the HJB equation with the constants J,K, and L given as (41)J=βω+2μ2ϕ2K=-ω+2μβμδ+βφμt1+ϕp-κμϕ2andL=ω+2μ2ωβϕ2μ2ωβμδ+βφμt1+ϕp-κ2+2β2μ2α2λ2σ2+φμt2+ϕ2μ2p-κ2ω+2μProof.
By solving the minimization problem in the HJB equation, we obtain the candidate optimal loan amount as(42)l=1βp-κ-ϕνx Plugging this into the HJB equation yields the following second-order ODE equation: (43)-ων+μδ-xνx+12α2λ2σ2νxx+φEνt,x+r-νt,x-12βp-κ-ϕνx2=0
Given that the value function is a twice continuously differentiable function ofx, we make the following quadraticansatz: (44)νx=-Jx2+Kx+Lwhereby J,K, and L are constants to be determined. The derivatives of ν are given by νx=-2Jx-K and νxx=-2J. Further, we have(45)φEνt,x+r-νt,x=φ-EJx+R2+Kx+R+L+Jx2+Kx+L=φ-EJx2+JR2+2JRx+Kx+R+L+Jx2+Kx+L=-φJμt2+2Jxμt1+Kμt1Inserting the derivatives of ν together with (45) and rearranging yields the following:(46)x2ωJ+2Jμ-2J2ϕ2β+xωK-2Jμδ+Kμ-2Jφμt1-2Jϕ2Kβ-2ϕJp-κβ+ωL-μδK-α2λ2Jσ2-Jφμt2-Kφμt2-p-κ22β-ϕ2K22β-ϕKp-κβ=0which holds for all (t,x). The terms in the large parentheses must equal zero. As such solving for the constants J,K, and L we conclude the proof. The case J=0 is trivial since then, with the optimal control l(t,x)=p-κ/β being a constant.Proposition 9.
(i)
The value function is given by(47)Vx=-Jx2+Kx+Lwhere (48)J=βω+2μ2ϕ2K=-ω+2μβμδ+βφμt1+ϕp-κμϕ2andL=-ω+2μ2ωβϕ2μ2ωβμδ+βφμt1+ϕp-κ2+2β2μ2α2λ2σ2+φμt2+ϕ2μ2p-κ2ω+2μ (ii)
The optimal credit limit process is given by(49)lt∗=ω+2μϕX^tZ-1μβϕϕp-κω+μ+βω+2μμδ+φμt1Proof.
(i)
Given that the solutionν(x) of the HJB equation is a function in C2, it follows by verification theorem for jump diffusion processes that the value function V(x)=ν(x). (ii)
In addition, the candidate optimal loan limit processlt∗ obtained from optimization problem with the HJB is indeed the optimal loan limit process. Reference [6] provides an excellent review of the verification theorem for jump diffusion processesThe optimal loan limit is a straight line with y-intercept depended on the expectation of jump sizeRt. Since the density of Rt is depended on the state of the system at time t, the intercept will change in line with the time varying values of Xt,kZ. The positive gradient of the line is independent of the jump sizes.It may be optimal for the lender to decline to avail any credit line to the borrower, i.e., to havelt∗=0. This occurs when (50)X^tZ=ϕp-κω+μ+βω+2μμδ+φμt1μβω+2μ In this case, for X^tZ to be positive we must have μt1>-1/φμδ+ϕ(p-κ)(ω+μ)/(ω+2μ).Corollary 10.
For the case when there is no network data, i.e., whenF=FY, the value function is given by(51)V0x=-Jx2+Kx+Lwhere (52)J=βω+2μ2ϕ2,K=-ω+2μβμδ+ϕp-κμϕ2andL=ω+2μ2ωβϕ2μ2ωβμδ+ϕp-κ2+2β2μ2α2λ2σ2+ϕ2μ2p-κ2ω+2μ and the optimal credit limit process is(53)l0,t∗=ω+2μϕX^tY-1μβϕϕp-κω+μ+μδβω+2μProof.
To obtain the optimal value function when the observation filtration isFY, simply replace the jump intensity parameter φ=0 in Proposition 9 and hence the proof.Whenφ=0, the lender’s optimization problem reduces to the normal linear Gaussian quadratic regulator and the objective function J(t,x)=J^(t,x^); see, e.g., [7].Remark 11.
The impact of the network information on the lender’s optimal decision process is seen by noting that wheneverμt1<0 for all t then lt∗≥l0,t∗. This, coupled with the variance results in Proposition 6, captures the gains made by incorporating the borrower’s ego-network information in the lender’s credit risk strategy.


## 4.1. The Dynamic Programming Method

We know that if the value functionV(x^)∈C2, then it satisfies the Hamilton-Jacobi-Bellman (HJB) equation(39)infl∈H-ωV+12βl2-lp-κ+μδ-x+ϕlVx+12α2λ2σ2Vxx+φEVx+R-Vx=0,∀t,x∈0,T×RWe proceed to solve the optimization problem in the following proposition.Proposition 8.
The function(40)νx=-Jx2+Kx+L solves the HJB equation with the constants J,K, and L given as (41)J=βω+2μ2ϕ2K=-ω+2μβμδ+βφμt1+ϕp-κμϕ2andL=ω+2μ2ωβϕ2μ2ωβμδ+βφμt1+ϕp-κ2+2β2μ2α2λ2σ2+φμt2+ϕ2μ2p-κ2ω+2μProof.
By solving the minimization problem in the HJB equation, we obtain the candidate optimal loan amount as(42)l=1βp-κ-ϕνx Plugging this into the HJB equation yields the following second-order ODE equation: (43)-ων+μδ-xνx+12α2λ2σ2νxx+φEνt,x+r-νt,x-12βp-κ-ϕνx2=0
Given that the value function is a twice continuously differentiable function ofx, we make the following quadraticansatz: (44)νx=-Jx2+Kx+Lwhereby J,K, and L are constants to be determined. The derivatives of ν are given by νx=-2Jx-K and νxx=-2J. Further, we have(45)φEνt,x+r-νt,x=φ-EJx+R2+Kx+R+L+Jx2+Kx+L=φ-EJx2+JR2+2JRx+Kx+R+L+Jx2+Kx+L=-φJμt2+2Jxμt1+Kμt1Inserting the derivatives of ν together with (45) and rearranging yields the following:(46)x2ωJ+2Jμ-2J2ϕ2β+xωK-2Jμδ+Kμ-2Jφμt1-2Jϕ2Kβ-2ϕJp-κβ+ωL-μδK-α2λ2Jσ2-Jφμt2-Kφμt2-p-κ22β-ϕ2K22β-ϕKp-κβ=0which holds for all (t,x). The terms in the large parentheses must equal zero. As such solving for the constants J,K, and L we conclude the proof. The case J=0 is trivial since then, with the optimal control l(t,x)=p-κ/β being a constant.Proposition 9.
(i)
The value function is given by(47)Vx=-Jx2+Kx+Lwhere (48)J=βω+2μ2ϕ2K=-ω+2μβμδ+βφμt1+ϕp-κμϕ2andL=-ω+2μ2ωβϕ2μ2ωβμδ+βφμt1+ϕp-κ2+2β2μ2α2λ2σ2+φμt2+ϕ2μ2p-κ2ω+2μ (ii)
The optimal credit limit process is given by(49)lt∗=ω+2μϕX^tZ-1μβϕϕp-κω+μ+βω+2μμδ+φμt1Proof.
(i)
Given that the solutionν(x) of the HJB equation is a function in C2, it follows by verification theorem for jump diffusion processes that the value function V(x)=ν(x). (ii)
In addition, the candidate optimal loan limit processlt∗ obtained from optimization problem with the HJB is indeed the optimal loan limit process. Reference [6] provides an excellent review of the verification theorem for jump diffusion processesThe optimal loan limit is a straight line with y-intercept depended on the expectation of jump sizeRt. Since the density of Rt is depended on the state of the system at time t, the intercept will change in line with the time varying values of Xt,kZ. The positive gradient of the line is independent of the jump sizes.It may be optimal for the lender to decline to avail any credit line to the borrower, i.e., to havelt∗=0. This occurs when (50)X^tZ=ϕp-κω+μ+βω+2μμδ+φμt1μβω+2μ In this case, for X^tZ to be positive we must have μt1>-1/φμδ+ϕ(p-κ)(ω+μ)/(ω+2μ).Corollary 10.
For the case when there is no network data, i.e., whenF=FY, the value function is given by(51)V0x=-Jx2+Kx+Lwhere (52)J=βω+2μ2ϕ2,K=-ω+2μβμδ+ϕp-κμϕ2andL=ω+2μ2ωβϕ2μ2ωβμδ+ϕp-κ2+2β2μ2α2λ2σ2+ϕ2μ2p-κ2ω+2μ and the optimal credit limit process is(53)l0,t∗=ω+2μϕX^tY-1μβϕϕp-κω+μ+μδβω+2μProof.
To obtain the optimal value function when the observation filtration isFY, simply replace the jump intensity parameter φ=0 in Proposition 9 and hence the proof.Whenφ=0, the lender’s optimization problem reduces to the normal linear Gaussian quadratic regulator and the objective function J(t,x)=J^(t,x^); see, e.g., [7].Remark 11.
The impact of the network information on the lender’s optimal decision process is seen by noting that wheneverμt1<0 for all t then lt∗≥l0,t∗. This, coupled with the variance results in Proposition 6, captures the gains made by incorporating the borrower’s ego-network information in the lender’s credit risk strategy.


## 5. Numerical Results

In this section we illustrate our findings in the preceding sections. We model the hidden credit quality process as an Ornstein Ulehnbeck process and the observed returns process as a linear diffusion process. We assume that the network informationZk arrives at equidistant time points tk,k=0,1,…,N. We simulate the processes using the parameter values as found in Table 1.Table 1
Model parameter values.
Mean reversion speedδ 0.05 Driftα 2  Return’s volatilityσ 0.3 Volatilityγ 1  Mean reversion levelμ 2.0 Network varianceΛ 0.04In order to capture how the network information causes jumps on the filtered estimates and variances, Figure1 shows the drift Xt, filters X^tH for H∈{Y,O,Z} and variances λtH for the case when the borrower’s degree ηt is fixed with values 0,2. The borrower’s own network information Zitk is indicated with red cross. We choose the number of network information times N=6. In panels one and two, we note that on information dates the filters X^tO and X^tZ jumps towards Zitk whilst between information dates the filters are driven back to the stationary mean of the drift δ. In panels 3 and 4, we note that λtY moves towards its asymptotic value C0-μσ2/α2 as t→∞. λtO and λtZ rise between information dates but jump down on information dates.Figure 1
Panel 1: The driftXt and filtered estimates XtH for the case when the borrower with degree η=2. Panel 2: The drift Xt and filtered estimates XtH for the case when the borrower with degree η=0. Panel 3 and 4: The conditional variances λtH when the degrees ηt=2 and ηt=0.To capture well the effect of number of direct friends on the conditional variancesλtO and λtZ Figure 2 plots a comparison between the variances of λtO and λtZ with and with no friends. The left panel plots a comparison of the variances for the case when the number of friends is zero and five. For the right panel, we model the random number of friends at information dates tk as a Poisson random variable with parameter ρ=5. The conditional variances for the case when there exists friends’ data in Zk is lower as compared to the case with zero friends. The right panel captures the perturbation of conditional variance as the number of friends varies.Figure 2
Left: Plot of variancesλtH when no. of friends ηt=0 and ηt=5 Right: Plot of variances λtH when no. of friends ηt=0 and ηt=ρ.


## 6. Conclusion

In this paper, we have presented the detailed results for the estimates and properties of conditional variance for a hidden Ornstein-Ulehnbeck equation driving the drift process of a borrower’s observed credit score. The network information obtained from the borrower’s evolving ego-network has been found to improve the quality of the lender’s estimate. These results offer a theoretical basis for the emerging area of unsecured digital and mobile phone based lending prevalent in emerging market economies. We have also solved a stochastic optimization problem whereby the lender is able to obtain the optimal credit limit for a borrower, the optimal limit being a linear function of the borrower’s credit quality estimate. The optimization problem is reduced to a linear quadratic Gaussian problem with jump diffusion. The jumps caused by the network information arriving at random times have been found to impact on the credit limit amount availed to the borrower from time to time.



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