Asymptotic Mathematics

is not a common name. IMO, it suggests that striving for exact solutions, one could decompose a domain so that exact solutions are possible in the sub-domains, when impossible in the domain.
Above you see the famous Black-Scholes-Equation for the pricing of European Call options (with an end condition you can solve it exactly). But its assumptions: constant volatility, no early exercise, discrete dividend payments, ... are unrealistic.
In UnRisk we use integration techniques for the numerical treatment of deal types within a generalized Black-Scholes world. We decompose (time, underlying,..) in such a way that we can calculate so called Green-Functions which we integrate. The total solution comes from an asymptotic recomposition.
It is a proprietary method, which we call Adaptive Integration.
Mathematica offers comprehensive numerical and integrated symbolic computation and it often uses symbolic techniques behind the scenes to optimize numerical computations for time and accuracy. Again, it is the AND effect that matters.

1 comment:

  1. That is not to suggest that many of these models were intended to be purely linear, as Wilson et al. (ibid.) suggest with reference to Polya's four element model. Hence the deliberate choice of the term "element" rather than "stage" in this text. how to master math