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 I. Basic math.
 1 Conditional probability.
 2 Normal distribution.
 3 Brownian motion.
 4 Poisson process.
 5 Ito integral.
 6 Ito calculus.
 7 Change of measure.
 8 Girsanov's theorem.
 9 Forward Kolmogorov's equation.
 10 Backward Kolmogorov's equation.
 11 Optimal control, Bellman equation, Dynamic programming.
 A. Deterministic optimal control problem.
 B. Stochastic optimal control problem.
 C. Optimal stopping time problem. Free boundary problem.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

Optimal stopping time problem. Free boundary problem.

e are investigating a model with a state variable given by the SDE where is a standard Brownian motion in , , .

Let be an open subset of , and be the time of first exit of from :

Let be the filtration generated by and denote a stopping time with respect to .

We introduce the following cost function The four summation terms above correspond to the following combinatorial situations:

1. Stop or exit before maturity .

2. Stop before both exit and maturity.

3. Exit before both stop and maturity.

4. Maturity before both exit and stop.

We introduce the function

Let

We proceed to calculate the PDE for . For motivation, review the section ( Backward induction ). There are two cases. In the event of the stopping at we have If the stopping time does not occur at then where Therefore

Note that only one of equalities or is true at all times. If the stopping does occur then thus, by way of repeating the most recent calculation, we obtain Recall that the motivation comes from the section ( Backward induction ). We are doing induction backwards in time. There is defined after present time and is its diffusion state variable. Thus, we have smoothness and the Ito formula applies. We finish the calculation as before: thus If the stopping does not occur then and .

Summary

The function satisfies on the conditions where exactly one of the inequalities is strict at all times, thus The boundary and final conditions are

Given the optimal stopping strategy is defined by

Let then

 (Free boundary problem 1)

 Notation. Index. Contents.