Uncertainty quantification and systemic risk Josselin Garnier (Université Paris 7) with George Papanicolaou and Tzu-Wei Yang (Stanford University) April 9, 2013
Modeling Systemic Risk We consider evolving systems with a large number of inter-connected components, each of which can be in a normal state or in a failed state. We want to study the probability of overall failure of the system, that is, its systemic risk. There are three effects that we want to model and that contribute to the behavior of systemic risk: The intrinsic stability of each component The strength of external random perturbations to the system The degree of inter-connectedness or cooperation between components
Possible Applications Engineering systems with a large number of interacting parts. Components can fail but the system fails only when a large number of components fail simultaneously. Power distribution systems. Individual components of the system are calibrated to withstand fluctuations in demand by sharing loads, but sharing also increases the probability of an overall failure. Banking systems. Banks cooperate and by spreading the risk of credit shocks between them can operate with less restrictive individual risk policies. However, this increases the risk that they may all fail, that is, the systemic risk. We want to propose a simple model to explain that individual risk does not affect the systemic risk in an obvious way. In fact, it is possible to simultaneously reduce individual risk and increase the systemic risk.
A Basic, Bistable Mean-Field Model The (banking) system has N components (banks). The risk variable X j (t) of each component j = 1,...,N, satisfies the SDE dx j (t) = hv (X j (t))dt +σdw j (t)+θ ( X(t) Xj (t) ) dt. V(x) = x 4 /4 x 2 /2 is a potential with two stable states ±1. With θ = σ = 0, X j (t) converges to one of the states. We define 1 the normal state and +1 the failed state. h 0 is the intrinsic stability. X(t) := 1 N N i=1 X i(t) is the risk variable of the system. θ 0 is the attractive interaction. {W j (t),j = 1,...,N} are independent Brownian motions. σ 0 is the noise strength.
Why this Model? h, σ and θ control the three effects we want to study: intrinsic stability, random perturbations, and degree of cooperation. Why mean field interaction? Because it is the simplest interaction that models cooperative behavior. It can be generalized to include diversity as well as other more complex interactions such as hierarchical ones. Connection with UQ (Uncertainty Quantification): UQ quantifies the variance of the quantity of interest. The variance quantifies the normal fluctuations. It may not characterize a rare event. Connection with QMU (Quantification of Margins of Uncertainty): In an inter-connected system, individual components can be operating closer to their margin of failure, as they can benefit from the stability of the rest of the system. This, however, reduces the overall margin of uncertainty.
Schematic for the Individual Risk and the Systemic Risk -1 0 +1 Status
The probability-measure-valued process µ N (t) First idea: to analyze X(t) = 1 N N i=1 X i(t). Unfortunately, no closed equation for X(t), so we need to generalize this problem into a larger space. We define a probability-measure-valued process µ N (t), µ N (t,dx) := 1 N N j=1 δ X j (t)(dx). Then X(t) = yµ N (t,dy). (Dawson, 1983) µ N (t) converges in law as N to a deterministic process u(t). For t > 0, u(t) has a pdf u(t,x) solution of the nonlinear Fokker-Planck equation: t u = h [ V (x)u ] θ {[ ] 12 2 yu(t,y)dy x u }+ σ2 x x x 2u, recalling that dx j (t) = hv (X j (t))dt +θ( X(t) X j (t))dt +σdw j (t).
Existence of two stable equilibria u(t) converges to an equilibrium uξ e with a pdf of the form: { } uξ e (x) = 1 (x ξ)2 exp 2h Z ξ σ 2V(x), 2 σ2 2θ with the compatibility condition: ξ = m(ξ) := xue ξ (x)dx. Given θ and h, there exists a critical value σ c such that ξ = m(ξ) has two stable solutions ±ξ b if and only if σ < σ c. Simplification: for small h, we have σc 2 = 2θ/3+O(h) and ξ b = 1 3σ2 2θ +O(h). Let us say that u ξ e b is the normal state of the system, u+ξ e b is the failed state. We have µ N (t) N u(t). If u(0) = u ξ e b, then u(t) = u ξ e b for all time.
Simulation of X: impact of N 1.5 N=120, h=0.1, σ=1, θ=10 dm(0)/dξ=1.0086, 3σ 2 /2θ=0.15 1 Systemic Risk 0.5 0 0.5 1 1.5 N=100, h=0.1, σ=1, θ=10 dm(0)/dξ=1.0086, 3σ 2 /2θ=0.15 1.5 0 2000 4000 6000 8000 10000 t 1.5 N=80, h=0.1, σ=1, θ=10 dm(0)/dξ=1.0086, 3σ 2 /2θ=0.15 1 1 Systemic Risk 0.5 0 0.5 Systemic Risk 0.5 0 0.5 1 1 1.5 0 2000 4000 6000 8000 10000 t 1.5 0 2000 4000 6000 8000 10000 t X n+1 j = Xj n hv (Xj n ) t +σ Wj n+1 +θ( 1 N N Xk n Xn j ) t. k=1
Simulation of X: impact of θ 1.5 N=100, h=0.1, σ=1, θ=10 dm(0)/dξ=1.0086, 3σ 2 /2θ=0.15 1 Systemic Risk 0.5 0 0.5 1 1.5 N=100, h=0.1, σ=1, θ=6 dm(0)/dξ=1.0126, 3σ 2 /2θ=0.25 1.5 0 2000 4000 6000 8000 10000 t 1.5 N=100, h=0.1, σ=1, θ=5 dm(0)/dξ=1.0141, 3σ 2 /2θ=0.3 1 1 Systemic Risk 0.5 0 0.5 Systemic Risk 0.5 0 0.5 1 1 1.5 0 2000 4000 6000 8000 10000 t 1.5 0 2000 4000 6000 8000 10000 t X n+1 j = Xj n hv (Xj n ) t +σ Wj n+1 +θ( 1 N N Xk n Xn j ) t. k=1
Probability of system failures Assume that σ < σ c so the limit of µ N (t) has two stable equilibria u e ξ b (resp. u e ξ b ), the system s normal (resp. failed) state. For large N, let the empirical density µ N (0) u e ξ b. Then we expect that µ N (t) u e ξ b for all t > 0. However, as long as N is finite, the system collapse: µ N (0) u e ξ b, µ N (T) u e +ξ b happens in the time interval [0,T] with small but nonzero probability. We use large deviations to compute this small probability.
Large deviation principle (Dawson & Gärtner, 1987) Given an event A in the suitable space (smaller than C([0,T],M 1 (R))) Detail, then µ N (t,dx) = 1 N N i=1 δ X j (t)(dx) satisfies the large deviation principle with the rate function I h : Detail ( ) P(µ N A) N 1 exp N inf I h(φ). φ A The rate function I h has the following variational form: Detail I h (φ) = 1 T 2σ 2 0 sup f: φ,f 2 L h φ = 1 2 {[ σ2 2 x 2φ+θ x x φ,f (t) = φ t L h φ,f 2 x (t) 0 φ,fx 2 dt, ] } yφ(t,dy) φ +h [ V (x)φ ], x φ(t, dx)f(x).
How to interpret the rate function? I h (φ) = 1 T 2σ 2 0 sup f: φ,fx 2 0 φ t L h φ,f 2 / φ,fx 2 dt. Recall that the limit u(t) µ e ξ b of µ N (t) satisfies u t L h u = 0, and I h(u) = 0. So P(µ N A) N 1 exp( Ninf φ A I h (u)) = 1 if u A. Given a smooth probability-measure valued process φ u with φ(0) = µ e ξ b, let g satisfy φ t L h φ = (φg) x. (1) g is the external force/randomness making φ deviate from u. Note that by (1), sup f: φ,f 2 y 0 φ t L h φ,f 2 φ,f 2 x = sup f: φ,f 2 x 0 φ,f x g 2 φ,f 2 x = φ,g 2. Then I h (φ) = 1 0 φ,g2 dt, which is the L 2 -norm of g (the randomness) and indicates the price to pay to have µ N φ. This expression (1) exists only if (very unfortunately) φ is a density function. 2σ 2 T
Probability of system failures and small h analysis P(µ N A) N 1 exp( Ninf φ A I h (φ)). The rare event A of system failures is the set of all possible paths starting from u e ξ b (the normal state) to u e +ξ b (the failed state): Detail A = { φ : φ(0) = u e ξ b, φ(t) = u e +ξ b }. The major work is to compute inf φ A I h (φ), which is a nonlinear and infinite-dimensional problem. Here we assume that the intrinsic stability, h, is small so that we can make this problem tractable. Two good reasons to consider a small intrinsic stability h: 1. P(µ N A) is extremely small for large h, which is not the regime we have in mind. 2. The case h = 0 is analytically solvable and the cases h small are perturbations of it.
The case h = 0 In this case, inf φ A I 0 (φ) can be solved without approximation. Result: For h = 0, inf φ A I 0 (φ) has the unique minimizer p e (t,x)dx, a path of Gaussian measures: { } p e 1 (t,x) = exp (x ae (t)) 2, 2π σ2 2 σ2 2θ 2θ where a e (t) = 2ξ b t/t ξ b and ξ b = 1 3σ2 2θ. Therefore, Recall that A = inf I 0(φ) = I 0 (p e ) = 2ξ2 b φ A σ 2 T = 2 σ 2 T {φ : φ(0) = u e ξb, φ(t) = u e +ξb }. ) (1 3σ2. 2θ
The case h small When h is small, good candidates for the transition path of empirical densities are Gaussian with small perturbations: { [ 1 (x a(t)) 2 φ = p +hq : p(t,x) = 2πb2 (t) exp 2b 2 (t) ],a(0) = ξ b,a(t) = ξ b } For h small, the large deviation problem is solvable approximately, and the transition probability is P(µ N A) ( [ exp N ) 2 (1 3σ2 + 6h σ 2 T 2θ σ 2 Comments: ( σ 2 θ ) 2 ) ]) (1 σ2 +O(h 2 ). θ A large system is more stable than a small system. In the long run (T large), a transition will happen. Increase of the intrinsic stabilization parameter h reduces systemic risk. Mean transition times are simply related to transition probabilities in this approximation (Williams 82)
What about the individual risk? The risk variable X j (t) of component j satisfies dx j = h(x 3 j X j )dt +θ( X X j )dt +σdw j. Assume X j (0) = 1 (the normal state) and linearize X j around 1: X j (t) = 1+δX j (t), X(t) = 1+δ X(t) and δ X(t) = 1 N N j=1 δx j(t). δx j (t) and δ X(t) satisfy linear SDEs: dδx j = (θ+2h)δx j dt+θδ Xdt+σdW j, dδ X = 2hδ Xdt+ σ N δx j (t) is a Gaussian process with the stationary distribution σ N(0, 2 2(2h+θ) ) as N. Qualitatively speaking, the individual risk is external risk(σ 2 ) intrinsic stability(h)+risk diversification(θ), and note that σ 2 2(2h+θ) σ2 2θ for small h. N dw j. j=1
Why and when is risk diversification undesirable? For h small: Systemic Risk exp ( 2N σ 2 T )) (1 3σ2, Individual Risk σ2 2θ 2θ. If a banker wants to take more external risk (to earn more profits) or the economy is more uncertain (increase σ 2 ), in order to keep the individual risk low, he can diversify the risk to the other banks (increase θ). Let us assume that everyone simultaneously increase σ 2 and θ but carefully keep the ratio σ2 2θ at a low level. - The individual risk is kept low. - The systemic risk increases, although this cannot be observed by the banks (until the catastrophic transition happens).
Extensions Diversity : the values of the parameters θ,σ,h are component-dependent. Hierarchical model : the system is stabilized by a central component.
Modeling of Diversity in Cooperative Behavior t u1 t u K The cooperative behavior of components can be different across groups: dx j (t) = hv (X j (t))dt +σdw j (t)+θ j ( X (t) X j (t) ) dt. The components are partitioned into K groups. In group k, the components have cooperative parameter Θ k. In the limit N the empirical densities of each group converge to the solution of the joint Fokker-Planck equations: = h [ V ] 1 2 (x)u 1 + x 2 σ2 x 2u1 Θ1 x. = h [ V ] 1 2 (x)u K + x 2 σ2 x 2u K Θ K x where ρ k N is the size of group k. {[ y {[ y } K ρ k u k (t,y)dy x ]u 1 k=1 } K ρ k u k (t,y)dy x ]u K k=1
Impact of Component Diversity on the Systemic Risk Why is the diversity interesting? The model is more realistic and more widely applicable. Diversity significantly affects the system stability by reducing it. Impact from the diversity: Analytical and numerical studies show that even with the same parameters and with {θ j } whose average equals θ the system still changes significantly.
Simulation 2 - Impact of Diversity, Change of Θ k and ρ k N=100, h=0.1, σ=1, Θ=[10;10;10] ρ=[0.33;0.34;0.33], stdev(θ)/mean(θ)=0 dm(0)/dξ=1.0086, Σ (ρ /Θ )(3σ 2 /2Θ 1)= 0.085 i i i i N=100, h=0.1, σ=1, Θ=[6;10;14] ρ=[0.33;0.34;0.33], stdev(θ)/mean(θ)=0.3266 dm(0)/dξ=1.0092, Σ (ρ /Θ )(3σ 2 /2Θ 1)= 0.091196 i i i i N=100, h=0.1, σ=1, Θ=[2;10;18] ρ=[0.33;0.34;0.33], stdev(θ)/mean(θ)=0.6532 dm(0)/dξ=1.0089, Σ (ρ /Θ )(3σ 2 /2Θ 1)= 0.086956 i i i i 1.5 1.5 1.5 1 1 1 Systemic Risk 0.5 0 0.5 Systemic Risk 0.5 0 0.5 Systemic Risk 0.5 0 0.5 1 1 1 1.5 0 2000 4000 6000 8000 10000 t 1.5 0 2000 4000 6000 8000 10000 t 1.5 0 2000 4000 6000 8000 10000 t N=100, h=0.1, σ=1, Θ=[5;10;15] ρ=[0.1;0.8;0.1], stdev(θ)/mean(θ)=0.22473 dm(0)/dξ=1.0089, Σ i (ρ i /Θ i )(3σ 2 /2Θ i 1)= 0.088 N=100, h=0.1, σ=1, Θ=[5;10;15] ρ=[0.3;0.4;0.3], stdev(θ)/mean(θ)=0.38925 dm(0)/dξ=1.0095, Σ i (ρ i /Θ i )(3σ 2 /2Θ i 1)= 0.094 N=100, h=0.1, σ=1, Θ=[5;10;15] ρ=[0.45;0.1;0.45], stdev(θ)/mean(θ)=0.47673 dm(0)/dξ=1.0099, Σ i (ρ i /Θ i )(3σ 2 /2Θ i 1)= 0.0985 1.5 1.5 1.5 1 1 1 Systemic Risk 0.5 0 0.5 1 Systemic Risk 0.5 0 0.5 1 Systemic Risk 0.5 0 0.5 1.5 1.5 1 2 0 2000 4000 6000 8000 10000 t 2 0 2000 4000 6000 8000 10000 t 1.5 0 2000 4000 6000 8000 10000 t
Analysis in the Diversity Case System with diversity have larger transition probabilities: When h is zero, the average of θ j is θ, and the diversity of θ j is small, then the system has a higher transition probability. Mathematically, if h = 0, Θ k = θ(1+δα k ) with δ 1, and K k=1 ρ kα k = 0, then ( ) P µ div N A { [ exp N ) ( K ) 2 (1 3σ2 (3σ 2δ 2 ρ σ 2 k α 2 2 k T 2θ 2θ + 1 T k=1 T 0 ( 1 e θs) 2 ds ) ]}
A Hierarchical Model of Systemic Risk Here we consider a hierarchical model with a central component: dx 0 = h 0 V 0 (X 0)dt θ 0 ( X0 1 N dx j = σdw j hv (X j )dt θ ( X j X 0 ) dt, N ) X j dt j=1 j = 1,...,N X 0 models the central stable component. It is intrinsically stable (h 0 > 0), and not subjected to external fluctuations. It interacts with the other components through a mean field interaction. X j, j = 1,...,N model individual components that are subjected to external fluctuations. They are (h > 0) or are not (h = 0) intrinsically stable. They interact with the central component X 0.
A Hierarchical Model of Systemic Risk - Analysis Nonlinear Fokker-Planck equations: In the limit N the pair ( X 0 (t), 1 N N j=1 δ X j (t)(dx) ) converges to (x 0 (t),u(t,x)dx) solution of the nonlinear Fokker-Planck equation t u = σ2 2 2 xx u + [ x hv (x)+θ(x x 0 (t))u ], with dx 0 dt = h 0V 0(x ( 0 ) θ 0 x0 xu(t,x)dx ). Existence of two equilibrium states (x 0 (t),u(t,x)) (x e,u e (x)) when σ is below a critical level: u e (x) = 1 exp ( 2hV(x)+θ(x x e) 2 ) Z e σ 2 with the compatibility equation xu e (x)dx = x e + h 0 θ 0 V 0 (x e). Large deviations principle to compute the probability of transition.
A Hierarchical Model of Systemic Risk - Results Exact results for h = 0 and expansions for small h (for the optimal paths and for the probability of transition). For h = 0: Resolution of a fourth-order ODE for the optimal x 0 (t) with boundary conditions. For the optimal path the mean of the individual components x(t) is ahead of x 0 (t): the individual components drive the transition. Stability increases with θ and decreases with θ 0.
On-going and future work Study models with diversities everywhere: The most general case (when the limit exists) is that V, σ and θ in each group can be different, i.e. if X j is in the group k, then the SDE is dx j (t) = hv k (X j (t))dt+σ k dw j (t)+θ k ( X (t) X j (t) ) dt. Using the analysis as a guide, we design importance sampling algorithms for computing efficiently (very) small systemic failure probabilities.
Topological spaces for the mean field model M 1 (R) is the space of probability measures on R with the Prohorov metric ρ, associated with the weak convergence. C([0,T],M 1 (R)) is the space of continuous functions from [0,T] to M 1 (R) with the metric sup 0 t T ρ(µ 1 (t),µ 2 (t)). M (R) = {µ M 1 (R), ϕ(y)µ(dy) < }, where ϕ(y) = y 4, serves as a Lyapunov function. M (R) is endowed with the inductive topology: µ n µ in M (R) if and only if µ n µ in M 1 (R) and sup n ϕ(y)µn (dy) <. C([0,T],M (R)) is the space of continuous functions from [0,T] to M (R) endowed with the topology: φ n ( ) φ( ) in C([0,T],M (R)) if and only if φ n ( ) φ( ) in C([0,T],M 1 (R)) and sup 0 t T sup n ϕ(y)φn (t,dy) <. Go Back
Large Deviations Principle Exact statement: inf I h (φ) liminf φ A N limsup N 1 N logp(µ N A) 1 N logp(µ N A) inf I h (φ) φ A Go Back
The rare event A = { φ : φ(0) = u e ξ b, φ(t) = u e +ξ b }. Definition of an enlarged rare event (with non-empty interior): A δ = { φ : φ(0) = u e ξ b, ρ(φ(t),u e +ξ b ) δ }. We have lim inf I h (φ) = inf I h(φ) δ 0φ A δ φ A Go Back
The classical Freidlin-Wentzell formula Let dx N = b(x N )dt + 1 N σdw t (X N (t)) t [0,T] satisfies a large deviation principle in C([0,T],R) with the rate function I((x(t)) t [0,T] ) = { 1 T 2σ 2 0 +, ( t x(t) b(x(t)) ) 2 dt, (x(t))t [0,T] H 1 otherwise. Go Back