I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 1 Finite differences.
 2 Gauss-Hermite Integration.
 3 Asymptotic expansions.
 4 Monte-Carlo.
 A. Generation of random samples.
 B. Acceleration of convergence.
 a. Antithetic variables.
 b. Control variate.
 c. Importance sampling.
 d. Stratified sampling.
 C. Longstaff-Schwartz technique.
 D. Calculation of sensitivities.
 5 Convex Analysis.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Stratified sampling.

e introduce sets and separate our sampling of into : Let be allocation rates, and be probabilities of : . We proceed with identification of parameters. We want to keep estimation unbiased and reduce variance.

First, we seek weights such that Hence, we need to take so that to have an unbiased estimate.

We calculate the variance as follows

It remains to calculate two claims:

1. In case we have (follows from formula ( Jensen inequality )).

2. Optimal is up to normalization constant (direct verification).

 Notation. Index. Contents.