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Continuous Stochastic Calculus with Applications to Finance

Обложка книги Continuous Stochastic Calculus with Applications to Finance

Continuous Stochastic Calculus with Applications to Finance

Time spent to read the book in detail: Four weeks
The book, 295 pages, is ordered as follows:

Chapter 1 (First 50 pages):
These cover discreet time martingale theory.

Expectation/Conditional expectation: The coverage here is unusual and I found it irritating. The author defines conditional expectation of variables in e(P) - the space of extended random variables for which the expectation is defined - i.e. either E(X+) or E(X-) is defined - rather than the more traditional space L^1(R) - the space of integrable random variables. The source of irritation is that the former is not a vector space. Thus given a variable X in e(P) and another variable Y, in general X+Y will not be defined, for example if EX+ = infinity, EY= - infinity. As a result, one is constantly having to worry about whether one can add variables or not, a real pain. Perhaps an example might help:

Suppose I have two variables X1 AND X2. If I am in the space L^1 then I know both are finite almost everywhere (a.e) and so I can create a third variable Y through addition by setting say Y = X1+X2. In the treatment here however, I have to be careful since it is not a priori clear that X1+X2 is defined a.e. What I need is - one of the proofs in the book - that E(X1)+E(X2) be defined (i.e. it is not the case that one is + infinity the other -infinity). If both E(X1)and E(X2) are finite this reduces to the L^1 case. However, because the Author chooses to work in e(P), we still have, in order to show even this basic result, quite a bit of boring work to do. Specifically: if E(X1) = +infinity then we must have, recall the definition of e(P), that E(X1^+)= +infinity AND E(X1-) < -infinity and also, because E(X1)+E(X2) is defined E(X2)> -infinity and so , since X2 is in e(P), that E(X2^-)< -infinity. Now since,
(X1+X2)^-
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