Implementing Arrow Debreu Equilibria By Continuous Trading of Few Long Lived Sec

Cover Implementing Arrow Debreu Equilibria By Continuous Trading of Few Long Lived Sec
Implementing Arrow Debreu Equilibria By Continuous Trading of Few Long Lived Sec
Darrell Duffie
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T)« That is, 6. E R represents the random variable which takes the real number 6. -, if branch 1 is the realized event at this node. Let p = (p^, . . . , p ) eR denote the vector of conditional branching probabilities at this node.
The processes m, , . . • > hIm are then mutually orthogonal martingales if they satisfy the following two conditions at each node: (i) p^6. "=0, j = l, 2, . . . , N (zero mean jumps, the martingale property), and (ii) 5. [p](5, =0 Vj ^ k, where [p] denotes the diagon
...al matrix whole 1-th diagonal element is p^ (mutually uncorrelated jumps, implying mutually orthogonal martingales).
We construct the processes m, . . . , m^ by designing their jumps at each node of the event tree, in any order, taking m. (0) = ¥.. At a given node (with L branches), it is simple to choose non-zero -27- vectors &, ... , 6r_, in R satisfying Aj[p]6j = j = 1, . . . L - 1, (a. 4) where A. Is a i x L matrix whose first row is a vector of ones and J whose k-th row is iS _-■ . This cannot be done for j ^ L if 1/2 A, [p] is a full rank L x L matrix (its rows are non-zero and mutually orthogonal).


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