Infinitesimal Calculus of Random Walk and Poisson Processes
Infinitesimal Calculus of Random Walk and Poisson Processes
Infinitesimal Calculus of Random Walk and Poisson Processes
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Gauge Institute Journal,<br />
H. Vic Dannon<br />
3.<br />
Integral <strong>of</strong> Hyper-real Function<br />
In [Dan3], we defined the integral <strong>of</strong> a Hyper-real Function.<br />
Let f () x be a hyper-real function on the interval [ ab] , .<br />
The interval may not be bounded.<br />
f () x may take infinite hyper-real values, <strong>and</strong> need not be<br />
bounded.<br />
At each<br />
a<br />
≤<br />
x<br />
≤b,<br />
there is a rectangle with base<br />
dx<br />
[ x − , x + 2<br />
], height f () x ,<br />
dx<br />
2<br />
<strong>and</strong> area<br />
f ( xdx. )<br />
We form the Integration Sum <strong>of</strong> all the areas for the x ’s<br />
that start at x = a, <strong>and</strong> end at x = b,<br />
∑ f ( xdx ) .<br />
x∈[ a, b]<br />
If for any infinitesimal dx , the Integration Sum has the<br />
same hyper-real value, then f () x is integrable over the<br />
interval [ ab] , .<br />
12
Change language