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Quantitative Analysis
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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 MATH given by the SDE MATH where $W_{t}$ is a standard Brownian motion in $\QTR{cal}{R}^{n}$ , MATH , MATH .

Let $U$ be an open subset of $\QTR{cal}{R}^{n}$ , $X_{0}=x\in U$ and $\tau$ be the time of first exit of $X_{t}$ from $U$ : MATH

Let $\QTR{cal}{F}_{t}$ be the filtration generated by $X_{t}$ and $\theta$ denote a stopping time with respect to $\QTR{cal}{F}_{t}$ .

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

1. Stop $\theta$ or exit $\tau$ before maturity $T$ .

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 MATH

Let MATH

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

Note that only one of equalities $u=\psi$ or $0=f+u_{t}+Lu$ is true at all times. If the stopping does occur then MATH thus, by way of repeating the most recent calculation, we obtain MATH Recall that the motivation comes from the section ( Backward induction ). We are doing induction backwards in time. There is MATH defined after present time $t$ and $X_{t}$ is its diffusion state variable. Thus, we have smoothness and the Ito formula applies. We finish the calculation as before: MATH thus MATH If the stopping does not occur then MATH and $0=f+u_{t}+Lu$ .

Summary

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

Given MATH the optimal stopping strategy is defined by MATH

Let MATH then

MATH (Free boundary problem 1)





Notation. Index. Contents.


















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