Volume 9, Issue 4
Tricks of the Trade
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Stochastic Integrals and Their Expectations
ItoIntegralRules implements properties for ItoIntegral[dX] using an approach based on a family of simplification rules, exported by the package ItoIntegralRules.m. Rule names are prefixed by ItoIntegral or ItoExpect to avoid name clashes. The rule-based approach is preferred to canonical simplification techniques because, as a consequence of Itô's lemma, there will typically be inequivalent simplification strategies. For example, if B is Brownian motion, then Itô's lemma can be applied together with (or, in differential form, ) to show
and the preferred choice between the two equivalent forms will depend on context, in particular whether it is more convenient for expressions to contain stochastic or classical integrals. We now survey the major rules and briefly illustrate their use in simplification. More detailed information and unit tests can be found in ItoIntegralTests.nb.
Additivity and Linearity
ItoIntegralRules implements additivity: is applied automatically once the package is loaded. We exemplify this by considering ItoIntegral[a dB-b dB], representing the stochastic integral .
It would normally be convenient to extract constant coefficients a and b: ItoIntegrationRules defines ItoIntegralExpandRule to perform this. In this particular case, Itovsn3 can apply the original rules for ItoIntegral after linear expansion to deliver a complete solution.
Relationship to Classical Integration
If the Itô integral involves no semimartingale terms other than the time term t (and its differential dt), then it can be rewritten as a classical time integral. ItoIntegralRules supplies ItoIntegralClassicRule to make this transformation and then the integral may possibly evaluate. (An implementation issue should be noted here. Itovsn3 is based on the total differentiation Dt operation, which assumes dependence unless explicitly stated otherwise. Integrate assumes symbols are constant by default. As long as the only quantities to vary in time are semimartingales, which would be the case in normal use of Itovsn3, this presents no problems.)
A Simplification Strategy
These rules can be applied in a variety of ways. In important special cases their application can be systematized. Here is a simple example. Consider expressions formed from just one Brownian motion B, as defined earlier, and time t using only addition, multiplication, and (possibly iterated) integration with respect to B and t, which we shall call "polynomial semimartingales." These can be reduced to expressions that involve classical integrals alone (no Itô integrals) by repeated application of specific formulas derived from stochastic integration by parts, itself derived from Itô's lemma:
Taking into account the various structural variations ( as in the lists l1, l2 following) for monomials p and q, there are 80 different rules to be considered! It is therefore convenient (and more reliable) to construct the various resulting rules automatically as follows (the rule set is also tested automatically in the accompanying notebook ItoIntegralTests.nb).
The rule set must be applied iteratively to suitable expressions until they stop changing, so we use FixedPoint to construct an appropriate function. (Note the argument iter, controlling maximum number of iterations, is set by default to Infinity since there is no a priori upper bound on the number of iterations required to simplify a general polynomial semimartingale.)
We can test this simplification procedure on a famous result from stochastic calculus: the family of Hermite polynomials forms a structure that is preserved by Itô integration.
With this definition, we have
and here we test this for the first 10 values of n.
Variations on this approach can be devised for polynomial semimartingales based on time t and n independent Brownian motions: see Gaines [9, 10] who applies the notion of Lyndon bases for shuffle products on free algebras. Rather than pursuing this, we now turn to consider expectations of stochastic integrals.
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