I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 1 Real Variable.
 2 Laws of large numbers.
 3 Characteristic function.
 4 Central limit theorem (CLT) II.
 5 Random walk.
 6 Conditional probability II.
 7 Martingales and stopping times.
 8 Markov process.
 9 Levy process.
 10 Weak derivative. Fundamental solution. Calculus of distributions.
 11 Functional Analysis.
 12 Fourier analysis.
 13 Sobolev spaces.
 A. Convolution and smoothing.
 B. Approximation by smooth functions.
 C. Extensions of Sobolev spaces.
 D. Traces of Sobolev spaces.
 E. Sobolev inequalities.
 F. Compact embedding of Sobolev spaces.
 G. Dual Sobolev spaces.
 H. Sobolev spaces involving time.
 I. Poincare inequality and Friedrich lemma.
 14 Elliptic PDE.
 15 Parabolic PDE.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Convolution and smoothing.

his section contains a recipe for building a smooth approximation to a measurable function.

The is a subset of . The is the boundary of . For an we denote , where the is the distance from to the boundary .

Definition

(Standard mollifier definition).

1.

2. For

is the "standard mollifier".

3. If is locally integrable then we define the "mollification" for .

Proposition

(Properties of mollifiers).

1. .

2. almost surely as .

3. uniformly on every compact subset of .

4. in .

Definition

(Partition of unity definition). Let be a subset of and be a finite or countable collection of subsets of such that The collection of functions is called a smooth partition of unity subordinated to .

 Notation. Index. Contents.