Importance sampling for indicator Markov chains

Abstract

We consider a continuous-time, inhomogeneous Markov chain M taking values in {0, 1} n . Processes of this type arise in finance as models of correlated default timing in a portfolio of firms, in reliability as models of failure timing in a system of interdependent components, and in many other areas. We develop a logarithmically efficient importance sampling scheme for estimating the tail of the distribution of the total transition count of M at a fixed time horizon.

Sequential Importance Sampling and Resampling for Dynamic Portfolio Credit Risk

“…even worse than those of the IS scheme of Giesecke and Shkolnik (2010), reported in Table 3. Note that˜ raises the intensity of firm 1 by increasing 1 = 0 001 to˜ 1 = 1, while keeping all other parameters under unchanged.…”

“…The performance of SISR˜ is robust with respect to the choice of˜ 1 , yielding similar results when we vary˜ 1 from 0 3 to 2. The IS scheme of Giesecke and Shkolnik (2010) performs much better for relatively homogeneous portfolios, as in the parameter configuration treated in § §6.2 and 6.3. Using an appropriate number of simulation runs to roughly match the CPU time of the SISR Algorithm 2 reported in Table 1, Table 4 indicates the performance of the IS estimate of J 1 = x under the parameter configuration treated in § §6.2 and 6.3.…”

“…For comparison, we also consider the IS scheme of Giesecke and Shkolnik (2010), which rescales the constituent intensities proportionally so that J is a stopped Poisson process with rate x. While SISR˜ uses 1 000 particles, IS uses 10 000 simulation trials.…”

“…We have also considered an alternative ad-hoc IS scheme IS˜ that uses˜ as the importance measure rather than the measure identified by Giesecke and Shkolnik (2010). To roughly match the CPU time of SISR˜ , IS˜ is based on 4 000 simulation runs.…”

“…Moreover, the SISR scheme yields meaningful estimates of very small probabilities. If the portfolio is relatively heterogeneous, the SISR scheme can also outperform the logarithmically efficient IS scheme of Giesecke and Shkolnik (2010).…”