Automatic Tracking of Measurability
Introduction
The automatic tracking of measurability has been part of finmath-lib since its earliest version (≈ 2004). It adds a method to random variables to inspect their measurability. Its concept and implementation is similar to a forward mode automatic differentiation, in the sense that each operator is augmented with a corresponding operation on measurability and certain random variables (constants, Brownian increments) are initialized with specific values.
Definition
For a random variable X we define a map T : X → [-∞ , ∞) such that for for t ≥ T(X) it is guaranteed that X is Ft-measurable. It is not guaranteed that T(X) is the smallest such number, although in most cases the implementation can provide the smallest time.
Applications
Originally the concept was introduces as a safe-guard in least-square-estimates of the conditional expectation (American Monte-Carlo), to ensure that the regression basis functions are Ft-measurable if used in an Ft-conditional expectation. However, the concept allows for important optimizations in the context of stochastic automatic differentiation: it may be used to detect cases where the computationally expensive conditional expectation operator can be avoided. It is:
Implementation
Then random variable interface net.finmath.stochastic.RandomVariable provides a method getFiltrationTime() implementing T(X) such that for a random variable X represented by the object X and t calculated by t = X.getFiltrationTime() it is guaranteed that X is Ft-measurable (note again: it is not guaranteed that t is the smallest such number, although in most cases the implementation can provide the smallest time).
The implementation is similar to a forward mode automatic differentiation, where the operator on random variables is augmented by additional operations on the filtration time:
Let T(X) denote the filtration time of X, i.e., T(X) = X.getFiltrationTime(). We apply the following definitions:
- For constant (deterministic) random variables C we set T(C) = -∞
- For Brownian increments W(t+Δt)-W(t) we set T(W(t+Δt)-W(t)) = t+Δ
- For operators f on random variables X1,…,Xn with Z = f(X1,…,Xn) we set T(Z) = max(T(X1),…,T(Xn))
This definition of T already fulfills the requirement that t = T(Z) is a time guaranteeing that Z is Ft-measurable, however it may not give the smallest such filtration time. There are a few optimizations or special cases which can be implemented, e.g. for Z = 0 * X we have T(Z) = T(0) (instead of T(Z) = max(T(0),T(X)) = T(X)).
Note: This definition already covers the generation of the correct filtration times for an Euler-scheme of an Ito process.
An implementation may achieve this by augmenting (extending) the type X by (X,t) with t = T(X) and overloading operators on X.
Example
The method is implemented in RandomVariableFromDoubleArray implementing the interface RandomVariable in finmath-lib. The interface is given as (extract)
public interface RandomVariable {
double getFiltrationTime();
public RandomVariable add(RandomVariable randomVariable);
// ... (declaration of other methods)
}
and the implementation is given as (extract)
public class RandomVariableFromDoubleArray implements RandomVariable {
private final double time; // Time (filtration)
@Override
public double getFiltrationTime() {
return time;
}
// ... (implementation of other methods)
@Override
public RandomVariable add(RandomVariable randomVariable) {
// Set time of this random variable to maximum of time with respect to which measurability is known.
double newTime = Math.max(time, randomVariable.getFiltrationTime());
// ... (calculate newRealizations as sum of this and randomVariable)
return new RandomVariable(newTime, newRealizations);
}
}
References
For an application to AAD for American Monte-Carlo simulation see
Fries, Christian P., Automatic Backward Differentiation for American Monte-Carlo Algorithms - ADD for Conditional Expectations and Indicator Functions (June 27, 2017). Available at SSRN: https://ssrn.com/abstract=3000822