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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.
14. Elliptic PDE.
15. Parabolic PDE.
A. Galerkin approximation for parabolic Dirichlet problem.
B. Energy estimates for Galerkin approximate solution.
C. Existence of weak solution for parabolic Dirichlet problem.
D. Parabolic regularity.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Parabolic regularity.


e study the smoothness of the weak solution for the parabolic problem, see the definition ( Weak solution of parabolic Dirichlet problem ). The main tools are the elliptic regularity results, see the section ( Elliptic regularity section ) and the formulas ( Cauchy inequality with epsilon ),( Differential inequality 1 ),( Differential inequality 2 ). We assume that the coefficients of the operator $L$ are independent of $t$ to simplify the calculations.

Proposition

(Parabolic regularity 1). Assume that the functions MATH of the operator $L$ (see the definition ( Parabolic differential operator )) are smooth on $\bar{U}$ and independent of $t$ . Let $U$ be a bounded subset of $\QTR{cal}{R}^{n}$ . Let MATH and $u$ is a weak solution (see the definition ( Weak solution of parabolic Dirichlet problem )) of the problem ( Parabolic Dirichlet problem ).

Then MATH with the following bounds MATH where the constant $C$ depends only on $U,T~$ and the coefficients of $L$ .

Proof

We multiply the equations ( Galerkin problem ) with MATH and sum for $k$ . We obtain MATH where MATH We note that MATH According to the formula ( Cauchy inequality ), smoothness of $b^{i},c$ and boundedness of $U$ MATH Therefore, MATH for any $\varepsilon>0$ . We integrate the above for $\left[ 0,t\right] $ then and obtain (for some $\varepsilon>0$ ): MATH According to the proposition ( Energy estimates for the Galerkin approximate solution ), MATH and by uniform positive definiteness of the matrix MATH there is a constant $\theta>0$ such that MATH Therefore, MATH

We pass to the limit $m\rightarrow\infty$ and find the first and third desired inequalities.

To prove the remaining inequality we note that the identity MATH may be rewritten as MATH where the MATH is already shown to be in $L^{2}$ . Hence, the proposition ( Boundary elliptic regularity ) applies and we derive MATH we integrate for MATH and the obtained in this proof estimates to arrive to the second desired inequality.

Proposition

(Parabolic regularity 2). Let $U$ be a bounded subset of $\QTR{cal}{R}^{n}$ . Assume that the functions MATH of the operator $L$ (see the definition ( Parabolic differential operator )) are smooth on $\bar{U}$ and independent of $t$ . Let MATH Suppose that the following compatibility conditions hold: MATH Then the weak solution $u$ of the problem ( Parabolic Dirichlet problem ) satisfies the estimates MATH MATH where the constant $C$ depends only on $U,T~$ and the coefficients of $L$ .

The compatibility conditions may be understood as follows. Since MATH then we should have for smoothness $g|_{\partial U}=0$ . Since MATH then we should have MATH and so fourth. The assumption of time independency of the coefficients of $L$ keeps it from getting very verbose.





Notation. Index. Contents.


















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