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Some results of g-evaluation with comonotonic additivity1

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Some results of g-evaluation with comonotonic additivity1

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    1Some results of g-evaluation with comonotonic additivity 11112ZHU Dong-yun, TIAN De-jian, ZHAO Man, DENG Feng, FENG Shi-qiang

    1 Department of Mathematics, China University of Mining and Technology, Xuzhou (221008)

     2 College of Mathematic ; Information, China West Normal University, Nanchong (637002)

    E-mail:zhudongyun1212@sina.com

    Abstract

    This paper studies some properties of g-evaluation with comonotonic additivity and proves that if an g- evaluation satisfies the property of comonotonic additivity, then it also satisfies the property of positively homogeneous and the corresponding generator g must be positively homogeneous with respect to y and z. Furthermore this paper gets a sufficient condition for g is additive with respect to y and z and gets the form of g on condition that the dimension of Brownian motion is one. Key words: Backward stochastic differential equation; g-Evaluation; Comonotonic additivity

1. Introduction

    It is by now well-known that there exists a unique adapted and square integrable solution to a backward stochastic differential equation (BSDE in short) of type

     T T Y = ξ + g (Y s, , Z)ds ? Z dBs, t ? [0, T ] , (1)? ? t s s s t t

     ξ providing that the function g is Lipschitz with respect to y and z , and that and ( g (t, 0, 0)) are square integrable. g is said to be generator of BSDE(1). We denote thet? [0, ] T

    unique adapted and square integrable solution of BSDE(1) by (Y( g , T , ξ ), Z( g , T , ξ )). t t t?[0,T ]

    When g also satisfies for any (t, y) , then, Y g (t , y, 0) ? 0( g ,T ,ξ ) , denoted byEξ ] , is[ g 0

    , denoted by called g -expectation of ξ ; Y( g,T ,ξ )E[ξ | F] , is called conditional g t t

    [1] g -expectation of ξ . If g doesnt satisfyg (t , y, 0) ? 0 for any (t, y), then

     2 g , we denote. The systemL ?t ? [0, T ], ξ ?(, F , P), s ? [0, t ]Y( g , t,ξ )by E [ξ ]s s,t t g [2]? ? E [?]is called g -evaluation .? ?, s t ? ? 0?s?t?T

    The notion of g -expectation can be considered as a nonlinear extension of the well-known Girsanov transformations, the original motivation for studying g -expectation comes from the

    theory of expected utility, which is the foundation of modern mathematical economics. Since the notion of g -expectation was introduced, many properties of g -expectation have been studied in

    [1, 3, 4, 5,6, 7, 8, 9]. In [7], the authors investigated a very interesting problem: what kind of g -expectation may be a Choquet expectation? The essential character of Choquet expectation is

     comonotonic additivity. In [10], the author studies some properties of g - expectation with

    comonotonic additivity and proves that if an g -expectation E[?] satisfies the property of g

    comonotonic additivity, then the corresponding generator g must be independent of y and be

    positively homogeneous with respect to z .

    For g - evaluation, what results we will get if it satisfies the property of comonotonic

    additivity? This paper studies some properties of g-evaluation with comonotonic additivity and

     1 Foundation item: Supported by the NSFC (No. 10671205) and Jiangsu Province Universities “blue project”.

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    proves that if an g- evaluation satisfies the property of comonotonic additivity, then it also satisfies the property of positively homogeneous and the corresponding generator g must be positively homogeneous with respect to y and z. Furthermore this paper gets a sufficient condition for g is additive with respect to y and z and gets the form of g on condition that the dimension of Brownian motion is one.

2. Preliminaries

    In this section, we introduce some Notations, Assumptions, Definitions and Lemmas which will be useful in this paper. be a d-dimensional standard Brownian Let (, F , P)be a probability space and ( B ) t t ? 0 B = 0 , let ( F )be the filtration generated by this Brownianmotion on this space such thatt ?00 t

     motion and augmented by the set of all P -null subsets N :

     F = σ{ B; 0 ? s ? t} ? N , t ? [0, T ] , t s n T > 0z ? R, | z |whereis a fixed real number. For any positive integer n anddenotes its

    Euclidean norm.

    We define the following usual spaces of processes:

     2 2 ?? S (0, T ; R) := {ψ continuous and progressively measurable;E supψ <?} || 0?t?Tt??F ??T 2 n 2 <?} .H (0, T ; R) := {ψ progressively measurable;||ψ||=E2|ψ |dt

    ? ? ? t

    ? ?F 2 0

    ? ?

    The generator g of a BSDE is a function

    d g ω , t , y, z : × [0, T ] × R × R ? R,()

    such that the process ( g (t , y, z)) is progressively measurable for each pair ( y, z ) in t ? [0, T ]

    d R × R. We list below possible assumptions on g : K ? 0 such thatdP × dt ? a.s. ( A1) (Lipschitz condition). There exists a constant

     d ?y , y? R, z , z? R, | g (t , y , z ) ? g (, t y , z) |? K (| y ? y| + | z ? z|) . 1 2 1 2 1 1 2 2 1 2 1 2 2 ( A2) .The process ( g (t , 0, 0)) .? H (0, T ; R) t ? [0, T ]F ( A3) . dP × dt ? a.s, . ?y ? R , g (t , y, 0) ? 0 . ( A4) . dP × dt ?a .s. , g (t , 0, 0) ? 0 .

     2 Let g satisfies the assumptions ( A1)L ξ ? (, Fand ( A2) . Then for each, P) , byT [11]of adapted processes Pardoux-Peng , there exists a unique pair (Y( g , T , ξ ), Z( g , T , ξ )) t t t?[0,T ]

     2 2 d solving the BSDE(1). in S (0, T ; R) × H (0, T ; R )F F

     In the following Definitions 2.1-2.2, g is assumed to satisfy ( A1)and ( A3) .

     Definition 2.1. The g -expectation g

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2 is E [?] : L(, F, P) ? RT defined by

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     E[ξ ]= Y ( g ,T ,ξ ). g 0

    Definition 2.2. The conditional g -expectation of ξ with respect to is defined byFt Eξ | F] = Y( g , T , ξ ) .[g t t

     2 Given t ?T andξ ? L (, F , P) . We let (Y( g , t , ξ ), Z ( g , t , ξ ))denote the solutions s s?[0,t ]t

    of the following BSDE:

     t t Y = ξ + g (Y r, , Z)ds ? Z dBr , s ? [0, t ] .? ? s r r r s s

     Definition 2.3. Let g satisfies the assumptions ( A1)and ( A2). For

     2 g L ?t ? [0, T ], ξ ?(, F , P), s ? [0, t ] . We denote byE [ξ ] := Y( g, t,ξ ) :s s,t t

     g 2 2 ?] : L(F ) ? L(F ) , 0 ? s ? t ?T .E [s t s,t g ? ? The systemE [?]is called g -evaluation.? ?, s t ? ? 0?s?t?T

     Remark 2.1. If g also satisfies the assumption ( A3). Then for

     2 gg L , we have, i.e. the?t ? [0, T ], ξ (, F , P), s ? [0, t ]E [ξ ] = E [ξ ] = E[ξ | F]?g t s,t s,T t

     g -evaluation is the conditional g -expectation.

     d and let Let t , y, z ? [0,T [ × R × Rε ? 0, T ?t ; let() ]]

    (Y( g , t + ε , y + z ? ( B? B)), Y( g , t + ε , y + z ? ( B? B))s t +ε t s t +ε t s?[0,t +ε ]

    denote the solution of the following BSDE:

    + ε+ εt t ε ε ε ε Y = y + z ? ( B? B) +, Z )dr ?dBr, s ? [0, t + ε ] .g (r , Y Z ??t +ε t s r r r s s

    [8]Lemma 2.1 (Representation Theorem) . Let ( A1) ( A2) and hold for g ; let 1 ? p < 2 .

     d Then for each pair ( y, z ) ? R × R, the following equality

     1 p ??g (t, y, z) = L?Y ( g , t + ε , y + z ? ( B? B )) ? y lim ? t t +ε t ?+ ε ε ?0 holds for every t ? [0,T [ . [12]Lemma 2.2 . Let ( A1) and ( A2) hold for g . Then the following two statements are

     equivalent:

    (1) dP × dt ? a.s. .g (t , 0, 0) ? 0

    g (2) ?0 ? t ? T ,E [0] ? 0 . , t T [8] Lemma 2.3(Uniqueness Theorem for g -Expectation) . Let ( A1) and ( A4)hold for two

    g generators andg. Then the following statements (1) and (2) are equivalent: 1 2 d . (1) dP × dt ? a.s. (t , y, z, ) ?( y, z ) ? R × R, g (t , y, z ) = g1 2 2

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(2) Y( g, t , ξ ) = Y( gξ, t ) , ?t ? [0, T ], ξ ? L(, F , P) . 0 10 2 t

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3. Properties of g- evaluation with comonotonic additivity

    Choquet(1953/1954) proposed the notion of Choquet capacity, which is a kind of

    comonotonic set function. Through Choquet capacity, Choquet gave the definition of Choquet

    expectation. Choquet expectation is a kind of well-known nonlinear expectation, the essential

    [10] character of Choquet expectation is comonotonic additivity. Jiangstudies some properties of g - expectation with comonotonic additivity. In this section, we study some properties of g -

     evaluation with comonotonic additivity.

     In this section, we always suppose g satisfies ( A1)and ( A2) .

     Definition 3.1. Two real-valued random variables ξ and η are called comonotonic if there exists a subset S ? such thatP( \ S ) = 0 and [ξ () ()][ () ()] 0,, Sω ? ξ ω η ω?η ω? ?ω ω ? . 12 12 12

     g ? ? Definition 3.2. An g - evaluationE [?]satisfies the property of comonotonic? ?, s t ? ? 0?s?t?T

     2 [0, T ] , comonotonic random variablesξ ,η L (, F , P) ,s ? [0, t ] ,?t ? ?additivity if fort

    g g g E [ξ + η ] = E [ξ ] + E [η ] . s,t s,t s,t

     g ? ? Theorem 3.1. Suppose an g - evaluationE [?]satisfies the property of? ? s t , ? ? 0?s?t?T

    comonotonic additivity.Then

    g (1) E [?] is positively homogeneous.s,t

    (2) g is positively homogeneous with respect to y and z .

     (3) dP × dt ? a.s. g (t , 0, 0) ? 0 , i.e. g satisfies ( A4) .

     2 ? L (, F , P) , since ξ and ξ are comonotonic, we haveξ Proof. (1) For any t g g g g E [2ξ ] = E [ξ ] + E [ξ ] = 2E [ξ ] . (2)s,t s,t s,t s,t

     Thus for any positive integerm, n , we can conclude that n n g g g gEξ ] = nEξ ] , E[ ξ ] = Eξ ] .[n[[s,s,s,s,t t t t m m Then in view of the continuity of BSDEs with respect to terminal data, we deduce that g g 2 (, F , P) .Eξ ] = aEξ ] , ?a ? 0 , ξ ? L[a[ s,t s,t t Thus (1) holds.

     d (2) For anya ? 0 , ( y, z) ? R × R, we set 1 z 1 lim S := t ? 0,T ; g (t, y, z) = L?Y ( g , t + ε , y + z ? (B? B )) ? y ,[[ ?? ? ? ? ?

    y ? t t +ε t ? + ε ε ?0? ?

    1 az 1 ? ?B )) ?a y . lim S := t ? 0,T ; g (t, ay, az) = L? ? [[ ??Y ( g , t + ε , ay + az ? (B

    ? ?

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    { }ay ? t t k +ε t ? k =1+ ε ε ?0??

    ??z az ? a.s. , Then for each such that P nof {} t ? S ? S , there exists a subsequence n

    =1 n

    y ay

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