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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
1. Calculational Linear Algebra.
2. Wavelet Analysis.
3. Finite element method.
4. Construction of approximation spaces.
5. Time discretization.
6. Variational inequalities.
A. Stationary variational inequalities.
B. Evolutionary variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Variational inequalities.


n this chapter we consider a technique that extends the finite elements (discussed in the chapter ( Finite elements )) to financial instruments involving stochastic optimal control (American feature). The reference is [Bensoussan] .

We previously introduced a backward induction and Bellman equation in the sections ( Backward induction ) and ( Bellman equation ). We now study variational inequalities because direct application of Bellman equation is not possible in combination with Finite Element technique. Indeed, when conducting Finite Element calculations, we evolve a vector of coordinates with respect to some finite element basis. Bellman equation requires us to continuously take maximum among some function values. We would like to avoid assembling and disassembling basis decompositions on every time step. In multidimensional situation such operation is not even feasible.

To see how a stochastic control problem may lead to a variational inequality consider first the following problem of evaluation of $u\left( x\right) $ : MATH where the $W_{t}$ is the standard Brownian motion, the MATH is the first exit time of the process $x+W_{t}$ from $U$ for $x\in U$ , $U$ is a bounded set with smooth boundary and $f$ is a function MATH and $f|_{\partial U}=0$ , MATH . We saw in the section ( Representation of solution for elliptic PDE using stochastic process ) that the $u\left( x\right) $ solves the following problem MATH We apply such result in context of the section ( Optimal stopping time problem ). We modify the task to find the function $u\left( x\right) $ defined by the relationships MATH Combining the calculations of the sections ( Representation of solution for elliptic PDE using stochastic process ) and ( Optimal stopping time problem ) we conclude that such $u\left( x\right) $ solves the following free boundary problem MATH almost everywhere in $U$ and MATH

Observe that the above problem may be rewritten as

MATH (Variational inequality example)
where the bilinear form $B$ (compare with the section ( Elliptic PDE section )) is given by MATH and the class of functions $K$ is defined as MATH

To see the equivalence of the two formulations consider the area where $u<\psi$ . We can find two functions $v_{1}$ and $v_{2}$ from $K$ such that $v_{1}<u$ and $v_{2}>u$ . Then MATH for $i=1,2$ implies MATH . One the other hand, in the area where $u=\psi$ we always have $v-u$ is nonpositive for $v\in K$ and thus MATH implies MATH .




A. Stationary variational inequalities.
B. Evolutionary variational inequalities.

Notation. Index. Contents.


















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