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.
 5 Convex Analysis.
 A. Basic concepts of convex analysis.
 B. Caratheodory's theorem.
 C. Relative interior.
 D. Recession cone.
 E. Intersection of nested convex sets.
 F. Preservation of closeness under linear transformation.
 G. Weierstrass Theorem.
 H. Local minima of convex function.
 I. Projection on convex set.
 J. Existence of solution of convex optimization problem.
 K. Partial minimization of convex functions.
 L. Hyperplanes and separation.
 M. Nonvertical separation.
 N. Minimal common and maximal crossing points.
 O. Minimax theory.
 Q. Polar cones.
 R. Polyhedral cones.
 S. Extreme points.
 T. Directional derivative and subdifferential.
 U. Feasible direction cone, tangent cone and normal cone.
 V. Optimality conditions.
 W. Lagrange multipliers for equality constraints.
 X. Fritz John optimality conditions.
 Y. Pseudonormality.
 Z. Lagrangian duality.
 [. Conjugate duality.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

et and be nonempty convex subsets of and respectively and let be a function . We introduce the following notations.

The following statements are consequences of the propositions ( Minimax theorem ) and ( Partial minimization result ).

Proposition

(Saddle point result 1) Assume that

1. the function is convex and closed,

2. the function is convex and closed,

3. .

Then the minimax equality holds and is nonempty under any of the following conditions.

0. The level sets of the function are compact.

1. The recession cone and the constancy space of the function are equal.

2. The function has the form with being a closed proper convex function and set being given by the linear constraints

and .

3. where are symmetric matrices, is positive semidefinite, is positive definite, where the are positive semidefinite matrixes.

In addition, if (0) holds then is compact.

Proposition

(Saddle point result 2). Assume that

1. the function is convex and closed,

2. the function is convex and closed,

3. Either or .

Then

(a) If the level sets of functions and are compact then the set of saddle points of is nonempty and compact.

(b) If and then the set of saddle points of is nonempty.

Proposition

1. the function is convex and closed,

2. the function is convex and closed,

Then the set of saddle points of is nonempty and compact if any of the following conditions are satisfied

1. and are compact.

2. is compact and is nonempty and compact for some and .

3. is compact and is nonempty and compact for some and .

4. and are nonempty and compact for some , and .

 Notation. Index. Contents.