April 20th, 2015, 10:45 am
QuoteOriginally posted by: emacQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: emacQuoteOriginally posted by: CuchulainnAnyhoo, it would see the issues are 1) Computability of a numerical approximation 2) How to define and measure accuracy, 3) the difficulty of partitioning functions spaces over random processes as is done with deterministic FEM. 1) This is not a problem for most MC schemes -- they are usually easy to implement.2) Usual measures are, for example, in L^2 norm in expectation (MSE)3) I don't see why this is an issue for MC -- just because it is for FEM does not mean it is for MC too.QuoteOriginally posted by: CuchulainnLooking at Euler for SDE, the ensemble of all realisations is uncountable it would seem, so heuristically this by definition a shaky process. If we are dealing with normal SDE with Brownian noise, your U is the set of continuous paths on [0,T], starting from 0. We want to integrate a functional over this space, i.e.[$]E [f(X_T)] = \int_U f(X_T(\omega)) P (d \omega)[$]where P is the Wiener measure.Typically, we do not have access to [$]X_T(\omega)[$] unless the SDE is exactly solvable. Instead, we replace by an approximation (e.g. the Euler-Maruyama scheme), call it [$]X^h_T(\omega)[$], with [$]h[$] being some discretisation parameter e.g. step size.We also cannot perform integration over U with respect to the Wiener measure, instead, we replace [$]P[$] with a measure which is equally weighted on all the MC paths. Call the paths [$]\omega^1, ... \omega^N[$] and the new MC measure [$]P^N:= \frac{1}{N} \sum_{i=1}^N \delta_{\omega^i}[$]Then, we can write the MC approximation as [$] \int_U f(X^h_T(\omega)) P^N (d \omega)[$].We have replaced integration over U w.r.t the measure P by integration with respect to a sum of Dirac masses (i.e. Summation!). You can then split the error into the part due to approximating [$]X_T(\omega)[$] by [$]X^h_T(\omega)[$] (the bias) and the part due to approximating P by [$]P^N[$] (the variance).There is nothing shaky going on. There is no concept of 'function' missing.The part in bold is the part I precisely don't want to use; it doesn't work; instead I want to express the process as a truncated Fourier-Hermite expansion. It is very much like FEM but I have not seen it done for SDE, but it is used for SPDE (it's called the stochastic Galerkin method). This article is probably what I am looking for..But what do you mean by "it doesn't work"?It does work!!!! In theory and in practice I take it back: Euler is not even wrong. Some claim that there are better methods, so I am interested in knowing what they might be.QuoteP.S. -- the bit in bold is discretisation of an SDE. It is separate from generic MC methodology.Indeed. We use Euler FDM to compute S at expiration and then we plug the value into the payoff.My question is: instead of Euler to compute S can we use Stochastic Galerkin?
Last edited by
Cuchulainn on April 19th, 2015, 10:00 pm, edited 1 time in total.