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djalil.chafai.net | ||
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almostsuremath.com
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| | | | | According to Kolmogorov's axioms, to define a probability space we start with a set ? and an event space consisting of a sigma-algebra F? on ?. A probability measure ? on this gives the probability space (?,?F?,??), on which we can define random variables as measurable maps from ? to the reals or other measurable... | |
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almostsuremath.com
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| | | | | In the previous two posts of the stochastic calculus notes, I began by introducing the basic concepts of a stochastic process and filtrations. As we often observe stochastic processes at a random time, a further definition is required. A stopping time is a random time which is adapted to the underlying filtration. As discussed in... | |
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almostsuremath.com
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| | | | | The monotone class theorem is a very helpful and frequently used tool in measure theory. As measurable functions are a rather general construct, and can be difficult to describe explicitly, it is common to prove results by initially considering just a very simple class of functions. For example, we would start by looking at continuous... | |
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www.jeremykun.com
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| | | This post is intended for people with a little bit of programming experience and no prior mathematical background. So let's talk about numbers. Numbers are curious things. On one hand, they represent one of the most natural things known to humans, which is quantity. It's so natural to humans that even newborn babies are in tune with the difference between quantities of objects between 1 and 3, in that they notice when quantity changes much more vividly than other features like color or shape. | ||
