Time spent to read the book in detail: Four weeksThe 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)^- <= (X1)^- +(X2)^-, we have E(X1+X2)- less than infinity which shows that a)X1+X2 is defined a.e. and b) it is in e(P).A little more work shows that, E(X1)+E(X2) =E(X1)+E(X2).When one introduces conditioning the above irritation continues. We have that if X is in e(P) that the conditional expectation E(X|L) exist and is in , not as is standard in the literatureL^1, but rather, in e(P). Consequently we can no longer carry out simple operations, normally done without thinking, such as E(X1|L)+ E(X2|L)= E(X1+X2|L), but rather have to pause to check if as in the example above that E(X1|L)+ E(X2|L) is defined etc, etc.Submartingale , Supermartingales ,Martingales: The definitions here again are a little unusual. The variables for both Sub and Super martingales are taken to be, yet again, in e(P). This in turn forces the definition:A submartingale is an adapted process X = (Xn,Fn) such that: 1) E(Xn^+)<ยฅ ( The Standard in the literature is to have E(Xn)<ยฅ 2) E( Xn+1|Fn)>=XnLikewise for a supermartingale we get:A supermartingale is an adapted process X = (Xn,Fn) such that:1) E(Xn^-)<ยฅ ( The Standard in the literature is to have E(Xn)<ยฅ 2) E( Xn+1|Fn)<=XnThese definitions, along with the fact that a martingale is both a supermartingale and submartingale, lead then to the standard - as appears in the literature - definition of a martingale.Stopping Times, Upcrossing Lemmas, Modes of Convergence: The treatment here is quite nice - modulo the e(P)- inconvenience. The proofs are all given in detail. And the level is at that of Chung's "A Course in Probability Theory", Chapter 9.Optional Sampling Theorem, Maximal Inequalities: A very rigorous treatment of the Optional Sampling Theorem (OST) is given. The need for closure is emphasized in order for OST to be applied in its full generality. In the absence of closure - the author emphasizes why - it is shown how the OST still applies if the optional times are taken to be bounded. The author then uses these results to show how stopped smartingales START TRANSACTION WITH CONSISTENT SNAPSHOT; /* 1576 = 220b4fc5db4c8600114e11151c0da98e