Riemann Integrals
In this chapter we deal with stochastic Riemann integrals, i.e. with ordinary Riemann integrals with a stochastic process as the integrand. Mathematically, these constructs are relatively unsophisticated, they can be defined pathwise for continuous functi
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Riemann Integrals
8.1
Summary
In this chapter we deal with stochastic Riemann integrals, i.e. with ordinary Riemann integrals with a stochastic process as the integrand.1 Mathematically, these constructs are relatively unsophisticated, they can be defined pathwise for continuous functions as in conventional (deterministic) calculus. However, this pathwise definition will not be possible any longer for e.g. Ito integrals in the chapter after next. Hence, at this point we propose a way of defining integrals as a limit (in mean square) which will be useful later on. If the stochastic integrand is in particular a Wiener process, then the Riemann integral follows a Gaussian distribution with zero expectation and the familiar formula for the variance. A number of examples will facilitate the understanding of this chapter.
8.2
Definition and Fubini’s Theorem
As one has done in deterministic calculus, we will define the Riemann integral by an adequate partition as the limit of a sum.
Partition In order to define an integral of a function from 0 to t, we decompose the interval into n adjacent, non-overlapping subintervals which are allowed to intersect at the
1 Bernhard Riemann (1826–1866) studied with Gauss in Göttingen where he himself became a professor. Already before his day, integration had been used as a technique which reverses differentiation by forming an antiderivative. However, Riemann explained for the first time under which conditions a function possesses an antiderivative at all.
© Springer International Publishing Switzerland 2016 U. Hassler, Stochastic Processes and Calculus, Springer Texts in Business and Economics, DOI 10.1007/978-3-319-23428-1_8
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8 Riemann Integrals
endpoints: Pn .Œ0; t/ W
0 D s0 < s1 < : : : < sn D t :
(8.1)
In the following, we always assume that the partition Pn .Œ0; t/ becomes increasingly fine with n growing (“adequate partition”): max .si si1 / ! 0 for n ! 1 :
1in
(8.2)
By si we denote an arbitrary point in the i-th interval, si 2 Œsi1 ; si ;
i D 1; : : : ; n :
Occasionally, we will sum up the lengths of the subintervals. Obviously, it holds that n X
.si si1 / D sn s0 D t :
iD1
In general, for a function ' one obtains: n X
.'.si / '.si1 // D '.t/ '.0/ :
(8.3)
iD1
Sometimes, we will operate with the example of the equidistant partition. It is given by si D it=n: 0 D s0 < s1 D
t n1 < : : : < sn1 D t < sn D t : n n
Due to si si1 D 1=n the required refinement from (8.2) for n ! 1 is guaranteed.
Definition and Existence Now, the product of a deterministic function f and a stochastic process X is to be integrated. To this end, the Riemann sum is defined by means of the notation introduced: Rn D
n X
f .si / X.si / .si si1 /:
(8.4)
iD1
Here, we have a sum of rectangular areas, each with a width of .si si1 / and the height f .si / X.si /. With n growing, the area beneath f .s/ X.s/ on Œ0; t is to be
8.2 Definition and Fubini’s Theorem
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approximated all the better. If the limit of this sum for n ! 1 exis
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