Interacting Particle Systems for Systemic Risk
Interacting Particle Systems for Systemic Risk
Interacting Particle Systems for Systemic Risk
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
<strong>Systemic</strong> <strong>Risk</strong>: Motivation Stochastic Model Simulating <strong>Systemic</strong> Shocks Examples of Rare EventsIS ApproximationApproximate:E[f(X(T))] = E[f(X(T))= η T (˜f(X))T∏k=1G −1kT∏η k (G k ),k=1η k (g) = γ k (g)/γ k (1)(X)]T∏G k (X)k=1γ k (g) := E [ g(X)˜f(X) := f(X(T)) T ∏k=1k∏G l (X) ] .l=1G −1k(X)Obtain an unbiased estimator based onη (J)k(G k ) = 1 ∑ JJ j=1 G k(X (j) ).Typical ( potential: G k (X (j) (t 0 ), X (j) (t 1 ),... ) X (j) (t k )) =exp α(min l≤k−1 N(t l )−min l≤k N(t l )) (multiplicative).Preference to particles where N(t) is setting new lows.15Ludkovski<strong>Systemic</strong> <strong>Risk</strong>