I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 1 Black-Scholes formula.
 2 Change of variables for Kolmogorov equation.
 3 Mean reverting equation.
 4 Affine SDE.
 5 Heston equations.
 6 Displaced Heston equations.
 7 Stochastic volatility.
 8 Markovian projection.
 A. Markovian projection on displaced diffusion.
 B. Markovian projection on Heston model.
 9 Hamilton-Jacobi Equations.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Markovian projection.

e saw in the section ( Gyongy lemma ) the possibility to replicate one dimensional distributions of a complex process of the generic form with the local volatility process of much simpler form provided that the function is given by the relationship

 (Gyongy 2)
The section ( Time series forecasting ) contains a generic proof that the conditional expectation minimizes unconditional expectation of the square difference. Speaking in terms of this section, the function should be guessed to be a solution of the following minimization problem:
 (Markovian projection)
where the expectation is unconditional and is taken at the observation time. We note that the expectation is taken with respect to the original process and not with respect to : we do not evaluate . This point is not obvious from considerations of the section ( Time series forecasting ). Hence, we proceed with derivation of the ( Markovian projection ).

We define the functional for some stochastic process and consider the problem The solution has to satisfy for any smooth deterministic function . We calculate the derivative: Next, we use the operation consistent with the foundation formulas ( Chain rule ) and ( Total probability rule ). We slice the event space by the values of : Hence, it suffices to have Comparing the last relationship with the ( Gyongy 2 ) we see that indeed, the recipes ( Gyongy 2 ) and ( Markovian projection ) are equivalent.

It is important to note that the class of functions in the minimization problem ( Markovian projection ) has to be restricted only by existence requirements for the process . It is under such freedom that the conditional expectation and the minimization problem are equivalent. For general the process is not always analytically tractable. We will be restricting our attention to some subclasses of functions that provide analytical tractability. Thus, we will be sacrificing equivalence to gain analytical tractability.

We present several examples of this technique in the following sections.

 A. Markovian projection on displaced diffusion.
 B. Markovian projection on Heston model.
 Notation. Index. Contents.
 Copyright 2007