### $\sigma$-localization and $\sigma$-martingales

Jan Kallsen, Jan Kallsen
2003 Teorija verojatnostej i ee primenenija
сг-LOCALIZATION AND сг-MARTINGALES В статье вводится понятие сг-локализации, обобщающее понятие ло кализации в общей теории случайных процессов, сг-локализационный класс, связанный с множеством мартингалов, есть класс сг-мартингалов, который играет важную роль в финансовой математике. Подробно рас сматриваются эти процессы и соответствующие а-мартингалъные меры. Обобщая понятие стохастического интеграла по компенсированным слу чайным мерам, мы выводим каноническое представление для сг-мартин
more » ... ов. Ключевые слова и фразы: а-локализация, сг-мартингал, стохасти ческий интеграл, каноническое представление, сг-мартингальная мера. Introduction, сг-Martingales have been introduced by Chou [3] and were inves tigated further by Emery [6] . They play a key role in the general statement of the fun damental theorems of asset pricing in [5], [12], and [2]. a-Martingales can be interpreted quite naturally as semimartingales with vanishing drift. Similar to local martingales, the set of cr-martingales may be obtained from the class of martingales by a localization procedure, but here localization has to be understood in a broader sense than usually (cf. [4, Lid]). This concept of cr-localization is introduced in Section 2. The subsequent section treats the set of сг-martingales and their properties. By extending the stochastic integral relative to compensated random measures, the canonical local martingale repre sentation X = X 0 + X е + x * (p -v) is generalized to cr-martingales in Section 4. Finally, <7-martingale measures are characterized in terms of semimartingale characteristics. Throughout the paper, we use the notation of [4] and [9], [10]. In particular, we work with a filtered probability space (П,^", (^)t6R+>^)-The transposed of a vector x or matrix is denoted by x T and its components by superscripts. Increasing processes are identified with their corresponding Lebesgue-Stieltjes measure. Kalis en J. 2. cr-Localization. For any semimartingale X and any predictable set D С П x R+, we write X D := X 0 1D(0) + ID • X, where 1d(0)(u>) := 1d((^,0)) for € In particular, we have X tt°'TI1 = X T for any stopping time T (cf. [4, 1.4.37]). Definition 2.1. For any class of semimartingales we define the o~-localized class as follows: A process X belongs to % if and only if there exists an increasing sequence (Z> n )n€N of predictable sets such that D n f П x R+ up to an evanescent set and X Dn G ^ for any n G N. Definition 2.2. A class of semimartingales is called stable under cr-stopping if X D G ^ for any X G ^ and any predictable set D. Lemma 2.1. // ^ гз stable under stopping, then (^ioc) n П [0,nj. For any n G N there exist a localizing sequence of stopping times (T(n,p)) pG N and Pn G N such that (I Dn ) r(n,p) G *«f for any p G N and P(T(n,p n ) < n) ^ 2~n. Let D n := £>n П (Пт^ДО»^7 71^™ )!)-Observe that (6 n ) n eN is an increasing sequence of predictable sets. Let к G N and (UJ, t) G ftx [0, /с] with (w, t) G (|J n € N D n )\( (J nGN Ai) = limsup n _ >00 (D n \ D n )-Obviously, this holds (up to evanescence) also for (w,k) instead of (cj,t). Since the Borel-Cantelli lemma yields P({UJ G ft: (u,k) G limsup n _ )>00 (£>n \ Dn)}) = 0 and hence ft x [0, к] С |J n € N .D n up to an evanescent set. Since ^ is stable under stopping and X 3n = ((X D -) T(n ' p -) ) A -^-+i T(m ' Pm) , it follows that X 5n G for any n G N. Hence X G ^o-. Lemma 2.1 is proved. The following result serves primarily as a preparation to Corollary 2.1 because most classes of interest (in particular Ж) are not stable under cr-stopping. Proposition 2.1. If^ is stable under a-stopping, then (4oo)n)n€N a localizing sequence of predictable sets such that X Dn G ^ for any nGN. Let the characteristics of X be given in the form (3.1) below. Without loss of generality we may modify A so that AQ-= 0, AAo = AQ -A 0 -> 0, and Aoo = 1, which implies that P A is a probability measure on (ft x R+, £P). For any nGN there exist a localizing sequence of predictable sets (D(n,p)) pe jsi and p n G N such that (X Dn ) D{n > p) G for any p G N and (P 0 Л)(£>(п,р п ) С ) ^ 2" n . Let D n := D n П (Г) m > n D(m,pm))-Observe that (D n ) n eisi is an increasing sequence of pre dictable sets and hence (Un6N D n ) \ (U n e N D n ) = limsup n _^0 0 ( J D n \ Dn)-Since (P 0 A)(D n \ 5 n ) ^ ZM>n( p ® A)(D(m,p r n) c ) ^ E m^n 2 " m = 2-( n ' 1 \ the Borel-Cantelli lemma yields (P® A)(limsu Prwoo (£> n \Аг)) =0. Therefore, Z> := (ft x R+)\( |J n € N D n ) is a (P ® vi)-null set. By [14, Lemma 2.5], this implies that -XoId(O) + Id • X -0 and hence X DnUD = X Dn up to indistinguishability. Since ^ is stable under cr-stopping and X 3nUD = X 5 " = ((X D -) D(n ' p -) ) n -^"+ lD(m ' Pm) , it follows that X 5nUD G for any nGN. Hence X G Proposition 2.1 is proved. Corollary 2.1. // ^ is stable under stopping and ^i oc is stable under a-stopping, then is stable under cr-stopping and (Ч> а )о -^a-Proof. It is easy to see that (^ioc)o-is stable under cr-stopping, which implies that 4>o is stable under cr-stopping (cf. Lemma 2.1). By Lemma 2 .1 and Proposition 2.1, we have {^a) a = ((%,<:)*)* = (%c)a -£ P(T(m,pm) < m) < 2 _ m = 2~(П_1) ' cr-localization and cr-martingales