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By Fwu-Ranq Chang

Lots of the present books on stochastic keep an eye on conception are written for college students in arithmetic or finance. This advent is designed, although, for these drawn to the relevance and functions of the theory's mathematical ideas to economics. as a result, mathematical equipment are mentioned intuitively and illustrated with monetary examples. extra importantly, mathematical recommendations are brought in language and terminology generic to graduate scholars in economics.

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Example text

First, we recognize that, in the end, we want to make sense of expressions like E [E [X | E1 ] | E2 ], where E1 and E2 are two σ -algebras contained in F, and X is a random variable. If we denote Y = E [X | E1 ] , then E [E [X | E1 ] | E2 ] = E [Y | E2 ] . Thus, if E [E [X | E1 ] | E2 ] is defined, Y = E [X | E1 ] has to be a random variable. That is, the conditional expectation is a random variable. Second, we recognize that expectations and probabilities are connected through the indicator function I A (ω) = 1 if ω ∈ A, 0 if ω ∈ / A.

1. ti Now look at the following two sets: F1 = ω : sup Wti = + ∞ ti and F2 = ω : inf Wti = −∞. = ω : sup (−Wti ) = + ∞ . ti ti Since {−Wt : t ∈ I } is a version of {Wt : t ∈ I }, F1 and F2 differ by a set of probability zero. From the fact that Fi ⊂ and the fact that P ( P (Fi ) ≤ P ( − A ⊂ F1 ∪ F2 , i = 1, 2, − A) = 1, we have − A) = 1 ≤ P (F1 ∪ F2 ) = P (Fi ) . Hence, P (F1 ) = P (F2 ) = 1. 2 Why can we assume, without√loss of generality, 1 ∈ D? Hint. Otherwise √ let b ∈ D. Choose c = b/t in Brownian scaling.

2 Verify the following equality: if µ = E [X ], then E [(X − µ)2 ] = E {(X − E [X | Y ])2 } + E {(E [X | Y ] − µ)2 }. Hint. It suffices to show that E {(X − E [X | Y ]) (E [X | Y ] − µ)} = 0. For this, you need to show E {X E [X | Y ]} = E {(E [X | Y ])2 }. 20 Probability Theory To complete the proof, you should recognize that E {X E [X | Y ]} = E {E [X E [X | Y ] | Y ]} and that E [X | Y ] is σ (Y )-measurable, implying E [X E [X | Y ] |Y ] = E [X | Y ] · E [X | Y ] . 3 Change of Variables Next, we turn our attention to the case of change of variables.

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