Sampling the Multivariate Standard Normal Distribution under a Weighted Sum Constraint
Abstract
:1. Introduction
2. Derivation of the Conditional Density
- (i)
- is positive definite;
- (ii)
- its determinant is given by
- (iii)
- the -element of the inverse is given by
3. Discussion
4. Sampling Algorithms
Algorithm 1. Sampling of given . |
1. From the vector of weights and the constraint c, compute the -dimensional mean vector and symmetric matrix from (2) and (3); |
2. Compute the eigen decomposition of the covariance matrix ; |
3. Sample i.i.d. standard Normal variates ; |
4. Transform these variates using the mean vector and covariance matrix |
5. Enlarge the -dimensional vector with the n-th component to get , |
6. Return where for . |
Algorithm 2. Sampling of given . |
1. Draw a sample u from a Uniform- distribution; |
2. Draw a sample from the conditional law of given : |
3. Apply Algorithm 1 using as constraint (i.e., ); |
4. Return . |
5. Applications
5.1. Conditional Portfolio Distribution
5.2. Expected Shortfall of a Defaultable Portfolio
6. Conclusions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. General Expression of the Conditional Density
Appendix B. Proof of Lemma 1.
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Vrins, F. Sampling the Multivariate Standard Normal Distribution under a Weighted Sum Constraint. Risks 2018, 6, 64. https://doi.org/10.3390/risks6030064
Vrins F. Sampling the Multivariate Standard Normal Distribution under a Weighted Sum Constraint. Risks. 2018; 6(3):64. https://doi.org/10.3390/risks6030064
Chicago/Turabian StyleVrins, Frédéric. 2018. "Sampling the Multivariate Standard Normal Distribution under a Weighted Sum Constraint" Risks 6, no. 3: 64. https://doi.org/10.3390/risks6030064