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.

## Extensions of Sobolev spaces.

roposition

(Extension theorem) Let are bounded and . Assume that admits a locally continuously differentiable parametrization. Then there exists a bounded linear "extension" operator such that for any a.s. in and the support of lies within .

Proof

The may be -expanded across a flat boundary by the reflection of the form If the boundary is not flat then there exists a change of variables that makes it locally flat. Then such procedure extends globally using the partition of unity (see the proof of the proposition ( Global approximation by smooth functions ) for an example of the technique). The partition of unity insures that the support of the result is localized.

 Notation. Index. Contents.