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The argument of the delta function only vanishes at the upper limit of this integral What is the value of an integral like this If we rewrite this integral, changing its upper limit to some instant after t and multiplying the integrand by the step function 8(t - t'), we see that the problem here is equivalent to that discussed in Appendix F: how to determine the value ofthe product 8(u)8( u) To prevent such ambiguities, the idea of a stochastic differential equation had to be put into a more rigorous basis than that provided by Langevin's pioneering paper Here we discuss briefly this more rigorous formulation We will see that there are an infinite number of ways in which one can construct a rigorous stochastic calculus where equations such as (841) are no longer ambiguous.

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But the two most popular alternative ways are those proposed by Ito [312] and Stratonovich [580] In Ito calculus, (841) vanishes as Langevin had assumed originally On the other hand, the ordinary calculus chain rule changes In Stratonovich calculus, the ordinary rules of calculus are maintained, but (841) is taken to be D /2 A completely rigorous mathematical presentation of stochastic calculus in the usual axioms-theorem-proof style would be completely out of touch with the spirit of this book and its intended readership Instead, we have a more concrete discussion of stochastic calculus based on an approach developed by Gillespie [232] This discussion differs from Gillespie's only because we deal explicitly with the inconsistency problem that leads to Stratonovich's and Ito's calculus, whereas [232] does not.

The reader interested in a more mathematical approach can find some useful references in the recommended reading section at the end of the chapter, where even some books about applications of stochastic calculus in quantitative finance are mentioned Any differential equation can be rewritten as a system of first-order differential equations Let us look at the simplest case, where there is just one equation in this system and see how we can generalize it to make it stochastic The general form of a first-order differential equation is.

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To support research and development in the life sci ences, Singapore established the Biomedical Research Council of the Agency for Science, Technology and Research (A*STAR), which funds ba sic and applied research, scholarships for advanced studies in biotechnol.

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