scholarly journals Final Report: Adaptive Numerical Algorithms for Partial Differential Equations, May 1, 1994 - April 30, 1998

2000 ◽  
Author(s):  
Phillip Colella
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Algorithms ◽  
Final Report ◽  
Partial Differential

Pricing futures by deterministic methods

Acta Numerica ◽  
2012 ◽  
Vol 21 ◽  
pp. 577-671
Author(s):  
Olivier Pironneau
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Narrow Range ◽  
Numerical Algorithms ◽  
The Other ◽  
Financial Mathematics ◽  
Pragmatic Approach ◽  
Partial Differential ◽  
Deterministic Methods ◽  
Better Than

In this article we will focus on only a small part of financial mathematics, namely the use of partial differential equations for pricing futures. Even within this narrow range it is hard to be systematic and complete, or even to do better than existing books such as Wilmott, Howison and Dewynne (1995), Achdou and Pironneau (2005), or software manuals such as Lapeyre, Martini and Sulem (2010). So this article may be valuable only to the extent that it reflects ten years of teaching, conferences and interaction with the protagonists of financial mathematics.Also, because the theory of partial differential equations is not always well known, we have chosen a pragmatic approach and left out the details of the theory or the proofs of some results, and refer the reader to other books. The numerical algorithms, on the other hand, are given in detail.

scholarly journals Multilevel Picard iterations for solving smooth semilinear parabolic heat equations

2021 ◽  
Vol 2 (6) ◽  
Author(s):  
Weinan E ◽  
Martin Hutzenthaler ◽  
Arnulf Jentzen ◽  
Thomas Kruse
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Algorithms ◽  
High Dimensional ◽  
Heat Equations ◽  
Partial Differential ◽  
New Family ◽  
Dimensional Approximation ◽  
Analytical Tools ◽  
Semilinear Parabolic

AbstractWe introduce a new family of numerical algorithms for approximating solutions of general high-dimensional semilinear parabolic partial differential equations at single space-time points. The algorithm is obtained through a delicate combination of the Feynman–Kac and the Bismut–Elworthy–Li formulas, and an approximate decomposition of the Picard fixed-point iteration with multilevel accuracy. The algorithm has been tested on a variety of semilinear partial differential equations that arise in physics and finance, with satisfactory results. Analytical tools needed for the analysis of such algorithms, including a semilinear Feynman–Kac formula, a new class of seminorms and their recursive inequalities, are also introduced. They allow us to prove for semilinear heat equations with gradient-independent nonlinearities that the computational complexity of the proposed algorithm is bounded by $$O(d\,{\varepsilon }^{-(4+\delta )})$$ O ( d ε - ( 4 + δ ) ) for any $$\delta \in (0,\infty )$$ δ ∈ ( 0 , ∞ ) under suitable assumptions, where $$d\in {{\mathbb {N}}}$$ d ∈ N is the dimensionality of the problem and $${\varepsilon }\in (0,\infty )$$ ε ∈ ( 0 , ∞ ) is the prescribed accuracy. Moreover, the introduced class of numerical algorithms is also powerful for proving high-dimensional approximation capacities for deep neural networks.

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