ARCH 2013.1 Proceedings August 1- 4, 2012 James G. Bridgeman - PDF document

Article from: ARCH 2013.1 Proceedings August 1- 4, 2012 James G. Bridgeman

Combinatorics for Moments of a Randomly Stopped Quadratic Variation Process James G. Bridgeman, FSA, CERA University of Connecticut 196 Auditorium Rd.U-3009 Storrs CT 06269-3009 bridgeman@math.uconn.edu August 4, 2012 Abstract A random process that includes jumps will in general have a quadratic variation that itself forms a non-trivial random process. One might be interested in moments of the quadratic variation process, for example in order to characterize it or in order to approximate it with a known process. The paper proposes a combinatoric approach to express higher moments of the quadratic variation process in terms of higher order variations of the original process and higher order autocovariations of the variations of the original process. These have lent themselves to direct calculation by Laplace transforms in the examples that gave rise to this work. h ( x 1 + x 2 ) 2 i Suppose x 1 and x 2 are random variables and we want to calculate E . One way to proceed might be h ( x 1 + x 2 ) 2 i �� �� x 2 1 + x 2 E = E + 2 E [ x 1 x 2 ] 2 � � � � �� x 2 x 2 = E + E + 2 � ( E [ x 1 ] E [ x 2 ]) 1 2 where � , de…ned as satisfying E [ x 1 x 2 ] = � E [ x 1 ] E [ x 2 ] , can be called a covaria- tion coe¢cient and � E [ x 1 ] 2 + E [ x 2 ] 2 � 8 9 h ( x 1 + x 2 ) 2 i < = ( E [ x 1 ] + E [ x 2 ]) 2 � � � � �� x 2 x 2 E = E + E +2 � � 1 2 : ; 2 2 The purpose of this paper is to state and prove Theorem 1 below which gener- alizes this simple example to an arbitrary (possibly random) number of terms x 1 + x 2 + ::: + x J and beyond 2 to an arbitrary moment E [( x 1 + x 2 + ::: ) n ] . The problem arose in work where f x j g were squared increments of a randomly stopped jump process and each term on the right was summable. The theo- rem will apply, however, to increments of discrete random processes generally, so long as they satisfy the assumptions of the theorem to allow an application of Fubini’s theorem when needed and to require covariation coe¢cients of all orders among the f x j g to satisfy a global uniformity condition. 1

Theorem 1 If either 1 or 2 : 1. x j � 0 almost always for all j , or 2. �X II � � � � � � x j 1 ; 1 � � � x j 1 ;i 1 x 2 j 2 ; 1 � � � x 2 j 2 ;i 2 � � � x l j l; 1 � � � x l E j l;il � �� � < 1 , ( ) X for all sets of indexed non-negative integers i l : l � i l = n where, for l X II each such f i l g , is taken over all indexed sets of permutations of sets � � of non-negative integers f j l;i : 1 � i � i l g l in which no two integers j l;i , j l 0 ;i 0 are equal, and if all covariation coe¢cients of all orders among the f x j g are global, not depending upon the speci…c subscripts j and j 0 for any two distinct x j and x j 0 , as speci…ed in the statement of Lemma 7 below then 2 0 1 n 3 @X 4 A 5 = E x j j X ! 2 3 j m X Y ! i l;m � 1 ! 6 7 X I X IV Y i l;m � 1 X 6 � � i l;m 7 n ! 1 6 l 7 x l Y Y = l ! i l � f i l g ( � 1) E l 6 7 j j m ! 4 i l;m ! 5 m j l l l ( ) X I X where is taken over all sets of indexed non-negative integers i l : l � i l = n , l for each such f i l g the covariation coe¢cient � f i l g is as de…ned in Lemma 7 below X IV and for each such f i l g the is taken over all sets of indexed non-negative ( ) X integers j m ; i l;m : j m � i l;m = i l for all l . m Proof. The proof will be assembled as a series of Lemmata and Remarks. Remark 2 If f x j g constitute squared increments of a discrete random process, for example of a randomly stopped jump process, then they satisfy hypothesis 1 of Theorem 1. If f x j g constitute increments of a stopped discrete stochastic process and if f x j g have …nite absolute moments of all orders then they satisfy hypothesis 2 of Theorem 1 since in that case there will be only a …nite number of x j . 2

2 0 1 n 3 @X 4 A 5 in Theorem 1 is Remark 3 Everything in the expression for E x j j Y ! X � � i l;m x l combinatoric with the exception of all of the � f i l g and E , j j l which carry the probabilistic content. In the applications from which this work arose, these probabilistic expressions are, respectively, directly calculable and directly summable for each f i l g and f i l;m g in the combinatorics. Lemma 4 (Multinomial Theorem - slightly restated) 0 1 n @X X I X II n ! A x j 1 ; 1 � � � x j 1 ;i 1 x 2 j 2 ; 1 � � � x 2 j 2 ;i 2 � � � x l j l; 1 � � � x l x j = Y j l;il � �� i l ! l ! i l j l ( ) X I X where is taken over all sets of indexed non-negative integers i l : l � i l = n l X II and, for each such set f i l g , is taken over all indexed sets of permutations � � of sets of non-negative integers f j l;i : 1 � i � i l g l in which no two integers j l;i , j l 0 ;i 0 are equal. (Compared to the usual statement of the multinomial the- orem, here we treat each permutation of each f j l;i : 1 � i � i l g l as creating a X II distinct monomial in .) X I X II Proof. The monomials de…ned in and include all (and only) the 0 1 n @X A monomials that can occur in the expansion of x j . Given such a mono- j mial x j 1 ; 1 � � � x j 1 ;i 1 x 2 j 2 ; 1 � � � x 2 j 2 ;i 2 � � � x l j l; 1 � � � x l j l;il � �� , without regard to the ordering among the x l j l; 1 � � � x l j l;il for each l , how many times does it occur in 0 1 n @X A the expansion of x j ? Any x j l;i can be chosen from any one of the n j 0 1 0 1 n @X @X A of A factors x j x j , but no two x j l;i in the same monomial can be j j 0 1 0 1 n @X @X A of A chosen from the same factor x j x j . So each such occurence j j 0 1 n @X A of the monomial in the expansion of x j is an assignment for all l of a j 3

0 1 n @X A to each particular x l unique l -element subset of the n factors in x j j l;i j Those are the factors which contribute that particular x l in the monomial. j l;i Y n ! to the monomial. The expression l ! il for " n -choose :::; l; l; :::; l; ::: " where l l runs over all positive integers and each l occurs i l times is the correct count- ing of the number of ways to make such an assignment without regard to the ordering among the x l j l; 1 � � � x l j l;il for each l . However, for each l , there are i l ! distinct permutations of each f j l;i : 1 � i � i l g l so dividing by each i l ! gives the correct count when each permutation is treated as creating a distinct monomial X II in . n ! Y Remark 5 The coe¢cient i l ! l ! il in Lemma 4 is the same as appears in Faá l di Bruno’s formula for the chain rule for higher derivatives, and comes from the same combinatorics. Lemma 6 If either 1 or 2 1. x j � 0 almost always for all j , or 2. �X II � � � � � � x j 1 ; 1 � � � x j 1 ;i 1 x 2 j 2 ; 1 � � � x 2 j 2 ;i 2 � � � x l j l; 1 � � � x l E j l;il � �� � < 1 , ( ) X for all sets of indexed non-negative integers i l : l � i l = n where, for l X II each such f i l g , is taken over all indexed sets of permutations of sets � � of non-negative integers f j l;i : 1 � i � i l g l in which no two integers j l;i , j l 0 ;i 0 are equal, then 2 0 1 n 3 @X 4 A 5 = E x j j h i h i h i h i X I X II � � � � n ! x 2 x 2 x l x l Y = � f j l;i g E x j 1 ; 1 ��� E x j 1 ;i 1 E ��� E ��� E ��� E ��� j 2 ; 1 j 2 ;i 2 j l; 1 j l;il i l ! l ! i l l � � , in which no two integers j l;i , j l 0 ;i 0 are equal, where for each f j l;i : 1 � i � i l g l � f j l;i g is the covariation coe¢cient de…ned by h i x j 1 ; 1 � � � x j 1 ;i 1 x 2 j 2 ; 1 � � � x 2 j 2 ;i 2 � � � x l j l; 1 � � � x l E j l;il � �� h i h i h i h i � � � � x 2 x 2 x l x l = � f j l;i g E x j 1 ; 1 � � � E x j 1 ;i 1 E � � � E � � � E � � � E � �� j 2 ; 1 j 2 ;i 2 j l; 1 j l;il 4