 Journal of Mathematical Finance, 2013, 3, 383-391 http://dx.doi.org/10.4236/jmf.2013.33039 Published Online August 2013 (http://www.scirp.org/journal/jmf) Risk Measures and Nonlinear Expectations* Zengjing Chen1,2, Kun He3, Reg Kulperger4 1School of Mathematics, Shandong University, Jinan, China 2Department of Financial Engineering, Ajou University, Suwon, Korea 3Department of Mathematics, Donghua University, Shanghai, China 4Department of Statistical and Actuarial Science, The University of Western Ontario London, Ontario, Canada Email: zjchen@sdu.edu.cn Received November 20, 2012; revised May 22, 2013; accepted June 16, 2013 Copyright © 2013 Zengjing Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT Coherent and convex risk measures, Choquet expectation and Peng’s g-expectation are all generalizations of mathe- matical expectation. All have been widely used to assess financial riskiness under uncertainty. In this paper, we investi- gate differences amongst these risk measures and expectations. For this purpose, we constrain our attention of coherent and convex risk measures, and Choquet expectation to the domain of g-expectation. Some differences among coherent and convex risk measures and Choquet expectations are accounted for in the framework of g-expectations. We show that in the family of convex risk measures, only coherent risk measures satisfy Jensen’s inequality. In mathematical finance, risk measures and Choquet expectations are typically used in the pricing of contingent claims over families of measures. The different risk measures will typically yield different pricing. In this paper, we show that the coherent pricing is always less than the corresponding Choquet pricing. This property and inequality fails in general when one uses pricing by convex risk measures. We also discuss the relation between static risk measure and dynamic risk meas- ure in the framework of g-expectations. We show that if g-expectations yield coherent (convex) risk measures then the corresponding conditional g-expectations or equivalently the dynamic risk measure is also coherent (convex). To prove these results, we establish a new converse of the comparison theorem of g-expectations. Keywords: Risk Measure; Coherent Risk; Convex Risk; Choquet Expectation; g-Expectation; Backward Stochastic Differential Equation; Converse Comparison Theorem; BSDE; Jensen’s Inequality 1. Introduction The choice of financial risk measures is very important in the assessment of the riskiness of financial positions. For this reason, several classes of financial risk measures have been proposed in the literature. Among these are coherent and convex risk measures, Choquet expecta- tions and Peng’s g-expectations. Coherent risk measures were first introduced by Artzner, Delbaen, Eber and Heath [1] and Delbaen [2]. As an extension of coherent risk measures, convex risk measures in general probabil- ity spaces were introduced by Föllmer & Schied [3] and Frittelli & Rosazza Gianin [4]. g-expectations were in- troduced by Peng [5] via a class of nonlinear backward stochastic differential equations (BSDEs), this class of nonlinear BSDEs being introduced earlier by Pardoux and Peng [6]. Choquet [7] extended probability measures to nonadditive probability measures (capacity), and in- troduced the so called Choquet expectation. Our interest in this paper is to explore the relations among risk measures and expectations. To do so, we re- strict our attention of coherent and convex risk measures and Choquet expectations to the domain of g-expecta- tions. The distinctions between coherent risk measure and convex risk measure are accounted for intuitively in the framework of g-expectations. We show that 1) in the family of convex risk measures, only coherent risk measures satisfy Jensen’s inequality; 2) coherent risk measures are always bounded by the corresponding Choquet expectation, but such an inequality in general fails for convex risk measures. In finance, coherent and convex risk measures and Choquet expectations are often used in the pricing of a contingent claim. Result 2) im- plies coherent pricing is always less than Choquet pricing, but the pricing by a convex risk measure no longer has *This work is supported partly by the National Natural Science Founda- tion of China (11231005; 11171062) and WCU(World Class University ogram of the Korea Science and Engineering Foundation (R31-20007 and the NSERC grant of RK. This work has been supported by grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada. C opyright © 2013 SciRes. JMF  Z. J. CHEN ET AL. 384 this property. We also study the relation between static and dynamic risk measures. We establish that if g-ex- pectations are coherent (convex) risk measures, then the same is true for the corresponding conditional g-expecta- tions or dynamic risk. In order to prove these results, we establish in Section 3, Theorem 1, a new converse com- parison theorem of g-expectations. Jiang [8] studies g- expectation and shows that some cases give rise to risk measures. Here we are able to show, in the case of g- expectations, that coherent risk measures are bounded by Choquet expectation but this relation fails for convex risk measures; see Theorem 4. Also we show that convex risk measures obey Jensen’s inequality; see Theorem 3. The paper is organized as follows. Section 2 reviews and gives the various definitions needed here. Section 3 gives the main results and proofs. Section 4 gives a summary of the results, putting them into a Table form for convenience of the various relations. 2. Expectations and Risk Measures In this section, we briefly recall the definitions of g-ex- pectation, Choquet expectation, coherent and convex risk measures. 2.1. g-Expectation Peng [5] introduced g-expectation via a class of back- ward stochastic differential equations (BSDE). Some of the relevant definition and notation are given here. Fix 0,T and let 0 ttT be a -dimen- sional standard Brownian motion defined on a completed probability space Wd ,,P. Suppose tT0 t is the natural filtration generated by , that is 0tT t W ; ts Ws t. We also assume . Denote T 2 2,,: is measurable random variables with <, 0,; tt LPE t — T 2 2 0 0,,: is valued,adapted processes with d< T dd ts LT XXEXs —— Let : 0, dT ,, d yz satisfy (H1) For any 0t is a continuous progressively measurable process with ,,g yzt 2 0,,d < T Egyzss . (H2) There exists a constant such that for any 0K 112 2 ,, ,d yzyz 112 2 12 12 ,, ,, , 0,. gyzt gyzt yyzz tT (H3) ,0,0, ,0,. yt ytT In Section 3, Corollary 3 we will consider a special case of with . d 1d Under the assumptions of (H1) and (H2), Pardoux and Peng [6] showed that for any , the BSDE 2,,L P T ,,dd, 0 TT tssss tt ygyzsszWt (1) has a unique pair solution 22 0 ,0,,0,, d tt t yzLT LT . Using the solution t of BSDE (1), which depends on y , Peng [5] introduced the notion of g-expectations. Definition 1 Assume that (H1), (H2) and (H3) hold on g and . Let 2,,LP , s z be the solution of BSDE (1). g defined by 0 : g is called the g-ex- pectation of the random variable . t defined by : tt is called the conditional g-expectation of the random variable y . Peng [5] also showed that g-expectation g and conditional -expectation t preserve most of basic properties of mathematical expectation, except for linearity. The basic properties are summarized in the next Lemma. Lemma 1 (Peng) Suppose that 2 12 ,, ,,LP . 1) Preservation of constants: For any constant ,c gcc . 2) Monotonicity: If 12 , then 12gg . 3) Strict monotonicity: If 21 , and , then 12 >>P 0 12 > gg . 4) Consistency: For any 0,tT, gg tg . 5) If does not depend on , and y is - measurable, then t . gtgt In particular, 0 ggtt . 6) Continuity: If n as n in 2,,LP, then limngn g . The following lemma is from Briand et al. [9, Theo- rem 2.1]. We can rewrite it as follows. Copyright © 2013 SciRes. JMF  Z. J. CHEN ET AL. 385 Lemma 2 (Briand et al. ) Suppose that t is of the form 0d, t tss xW ,0tT where t is a continuous bounded proces s. Then ,,, 0, lim gst st st tt XEX gXtt st where the limit is in the sense of . 2,,LP 2.2. Choquet Expectation Choquet [7] extended the notion of a probability measure to nonadditive probability (called capacity) and defined a kind of nonlinear expectation, which is now called Cho- quet expectation. Definitio n 2 1) A real valued set function :0V,1 is called a capacity if a) 0, 1;VV b) , whenever and VA VB,AB B ,,L . 2) Let V be a capacity. For any , the Choquet expectation P 2 V is defined by 0 0 :1d VVttVt dt Remark 1 A property of Choquet expectation is posi- tive homogeneity, i.e. for any constant 0, a . VV aa 2.3. Risk Measures A risk measure is a map :, where is in- terpreted as the “habitat” of the financial positions whose riskiness has to be quantified. In this paper, we shall con- sider . 2,,LP The following modifications of coherent risk measures (Artzner et al.[1]) is from Roorda et. al. [10]. Definition 3 A risk measure is said to be coherent if it satisfies 1) Subadditivity: 121 2 XXX , , ; 12 2) Positive homogeneity: ,XX X for all real number 0; 3) Monotonicity: , Y whenever ; Y 4) Translation invariance: XX for all real number . As an extension of coherent risk measures, Föllmer and Schied [3] introduced the axiomatic setting for con- vex risk measures. The following modifications of con- vex risk measures of Föllmer and Schied [3] is from Frittelli and Rosazza Gianin [4]. Definition 4 A risk measure is said to be convex if it satisfies 1) Convexity: 121 11 2 XX X , 0,1 , 12 ,XX ; 2) Normality: 00 ; 3) Properties (3) and (4) in Definition 3. A functional in Definitions 3 and 4 is usually called a static risk measure. Obviously, a coherent risk measure is a convex risk measure. As an extension of such a functional Artzner et al. [11,12], Frittelli and Rosazza Gianin [13] introduced the notion of dynamic risk measure which is random and depends on a time parameter . , , t t Definition 5 A dynamic risk measure 22 :,,,, tt LPL P is a random functional which depends on , such that for each it is a risk measure. If satisfies for each t t 0,tT the conditions in Definition 3, we say t is a dynamic coherent risk measure. Similarly if t satisfies for each 0,tT the conditions in De- finition 4, we sa y t is a dynamic convex risk meas- ure. 3. Main Results In order to prove our main results, we establish a general converse comparison theorem of g-expectation. This theorem plays an important role in this paper. Theorem 1 Suppose that 1 , g and 2 satisfy (H1), (H2) and (H3). Then the following conclusions are equivalent. 1) For any 2 ,,,LP , 12 . ggg 2) For any 1122 ,,,,,0, , d yztyztT 1212 1112 22 ,, ,,,,. gyyzzt yztg yzt (2) Proof: We first show that inequality (2) implies ine- quality 3). Let 11 ,, tt yz 22 , tt yz and be the solutions of the following BSDE corresponding to the terminal value , tt YZ , X and , and the generator 12 , gg and , respectively ,,dd. T tss tt yXgyzss zW T ss (3) Then 1 1 0, gy 2 2 0, gy 0. gY For fixed 11 ,yz tt , consider the BSDE 11 11 21 ,, ,, d. t T ssss ss t T ss t y d yyzzsgyzss zW (4) Copyright © 2013 SciRes. JMF  Z. J. CHEN ET AL. 386 It is easy to check that is the solu- tion of the BSDE (4). 1212 , tttt yyzz Comparing BSDEs (4) and (3) with X and g, assumption (2), (2) then yields 1212 11 1 22 2 ,, ,, ,,, 0. tttt tt tt yyzztgyzt gyztt Applying the comparison theorem of BSDE in Peng [5], we have Taking , thus by the definition of 12 , 0. ttt Yyyt 0t -expectation, the proof of this part is complete. We now prove that inequality (1) implies (2). We dis- tinguish two cases: the former where does not de- pend on , the latter where y may depend on . y Case 1, does not depend on . The proof of this case 1 is done in two steps. y Case 1, Step 1: We now show that for any 0,tT, we have 12 2 , ,,,. gtgtgt LP Indeed, for 0,tT , set 12 :> tg tgtgt A . If for all 0,tT, we have then we ob- tain our result. 0, t PA If not, then there exists 0,tT such that >0 t PA . We will now obtain a contradiction. For this , t 12 > tt AgtAgtgt II . That is 12 |> Agtgtgt t I 0. Taking g-expectation on both sides of the above inequality, and apply the strict monotonicity of -ex- pectation in Lemma 1 (3), it follows 12 >0. t gAgt gt gt I But by Lemma 1 (4) and (5), 12 12. t ttt gAgt gtgt gAgAtgA t I III Note that by Lemma 1(v) 0, 1,2. it it gAgA t II i Thus 12 12 11 22 0< 0 ttt ttt t tt tt gAg AtgAt gA gAtA gAt gAgAt gAgA t III IIII II II This induces a contradiction, thus concluding the proof of this Step 1. Case 1, Step 2: For any ,0,tT with t 1,i and , let us choose d i z , it W i XzW 2 . Obviously, . 2,, i LP By Step 1, 1 2 12 1 2, 0,. tg t gt XX X XtT Thus 12 12 11 1 22 2. tt gt t gt t XXEXX t XEX t XEX t Let ,t ,y applying Lemma 2, since g does not de- pend on we rewrite ,, yzt simply as ,, zt thus 1 2 tg z 2 ,,,, tg ztt 12 gz z10. The proof of Case 1 is complete. Case 2, g depends on y. The proof is similar to the proof of Theorem 2.1 in Coquet et al. [14]. For each >0 and 112 2 ,, ,d yzyz , define the stopping time 11 2 2 1112 22 1212 ,;, inf0; ,,,, ,, . yzyz tgyztgyzt gyyzztT , Obviously, if for each 112 2 ,, ,d yzyz 1122 ,; ,<0PyzyzT , for all , then the proof is done. If it is not the case, then there exist >0 and 112 2 ,, ,d yzyz, such that 1122 ,; ,<>0PyzyzT . Fix ,,, 1,2, ii yzi and consider the following (forward) SDEs defined on the interval ,T d,,d , , 1,2 ii iii ii Yt gYtzttzW Yyti d, t Copyright © 2013 SciRes. JMF  Z. J. CHEN ET AL. 387 and 33 12 12 3 12 d,,d , . t YtgYtzzttz zW Yyyt d, Obviously, the above equations admit a unique solu- tion which is progressively measurable with i Y 2 0<. sup i tT EYt Define the following stopping time 12 112 2 3 12 :inf; ,,,, ,, . 2 tgYtztgYtzt gYtzz tT It is clear that T and note that T whenever ,T <>0 thus, Hence . <T <. P Moreover, we can prove 12 3 >, on <YY Y . In fact, setting 312 ˆ YtYt Yt Yt, then 3 12 12 112 2 ˆ d,, ,,, ,d. YtgYt zzt Ytzt gYtztt Thus ˆ dˆ , ,, 0. d2 Yt tY t It follows that on , , ˆ<0. 2 Y This implies 312 <<PYY YP >0 a are . (5) By the definition of and , the pair pro- cesses nd 3 12 Y the solutions of the following BSDEs with terminal values 12 ,YY 12 ,tz 3 Y 12 ,,Ytz Y ,tz z 1,YT 2 YT and 3 YT, ,,dd, 1,2 i ti siis tt yYTgyzss zWi TT and 3 12 12 ,,d d TT ts tt yYT gyzzsszzW . s Hence, 11 11 1, gg YYT y 22 22 2 gg YYT y and 33 12 . gg YYT yy Applying the strict comparison theorem of BSDE and inequality (5), by the assumptions of this Theorem, we have 12 312 12 12 12 < . gg gg yy YYY YYy y This induces a contradiction. The proof is complete. Lemma 3 Suppose t hat g satisfies (H1) , (H2) and (H3). For any constant , let =0c 11 ,,,, yztcgy zt cc . Then for any 2,,LP , . gg cc Proof: Letting tg t yc , then t y is the solu- tion of BSDE ,,dd. TT tss tt ycgyzss zW ss Since 11 ,,,, yztcgy zt cc , the above BSDE can be rewritten as 11 ,,dd TT tss tt yccg y zsszW cc . ss (6) Let tg t y , then satisfies t cy ,,dd . TT tss tt cyccgyzssczW ss (7) Comparing with BSDE (6) and BSDE (7), by the uniqueness of the solution of BSDE, we have ,, tt tt cy czy z. Let 0,t then 00 cy y . The conclusion of the Lemma now follows by the definition of g-expectation. This concludes the proof. Applying Theorem 1 and Lemma 3, immediately, we obtain several relations between g-expectation g and . These are given in the following Corollaries. Corollary 1 The g-expectation g is the classical mathematical expectation if and only if g does not de- pend on and is linear in . y z Proof: Applying Theorem 1, g is linear if and only if ,, yzt is linear in , z. By assumption (H3), that is , 0t,0gy for all , t. Thus does not depend on . The proof is complete. y Corollary 2 The -expectation g is a convex risk measure if and only if does not depend on and is convex in . y z Proof: Obviously, -expectation g 0,1 is convex risk measure if and only if for any Copyright © 2013 SciRes. JMF  Z. J. CHEN ET AL. 388 2 11 ,,,. gg LP , g (8) For a fixed , let 0,1 1 2 11 ,,, ,, 11 ,,1,,. 11 g yztgyzt yztgy zt Applying Lemma 3, 12 , 11. gggg Inequality (8) becomes 12 2 1 ,,,. ggg LP 1, Applying Theorem 1, for any ,,0,, 1,2, d ii yztTi 1212 111 222 112 2 1,1, ,,11 , ,,1, , gyyzzt yztgyzt gyzt gyzt which then implies that g is convex. By the explanation of Remark for Lemma 4.5 in Briand et al. [9], the convexity of and the assumption (H3) imply that does not depend on . The proof is complete. y The function is positively homogeneous in if for any , z 0a ,, ,, azag z . Corollary 3 The -expectation g is a coherent risk measure if and only if does not depend on and it is convex and positively homogenous in . In particular, if , y z 1d is of the form ,tt zta zbz with . 0a Proof: By Corollary 2, the -expectation g is a convex risk measure if and only if does not depend on and is convex in . Applying Theorem 1 and Lemma 3 again, it is easy to check that y z -expectation g is positively homogeneous if and only if is positively homogeneous (that is for all and >0a, gg aa , ,, if and only if for any , 0a , azag z d. In particular, if , notice the fact that 1 is con- vex and positively homogeneous on , and that does not depend on . We write it as y , zt then 00 0 ,, , 1,1, . zz z g ztg ztIgztI gtzIgt zI 0 z (9) Note that , , but 0z zI z 0z zIz , . 22 zz zz zz Thus from (9) 1, 1,1, 1, ,. 22 tg tgtg t ztz z Defining 1, 1, :2 t tg t a , 1, 1, :2 t tg t b . Obviously since the convexity of 0,a yields 1,1, 0, 0. 2 gt gtgt The proof is complete. Remark 2 Corollaries 2 and 3 give us an intuitive explanation for the distinction between coherent and convex risk measures. In the framework of g-expectations, convex risk measures are generated by convex functions, while coherent measures are generated only by convex and positively homogenous functions. In particu lar , if d = 1, it is generated only by the family ,tt zta zbz with . Thus the family of coherent risk measures is much smaller than the family of convex risk measures. 0a Jensen’s inequality for mathematical inequality is im- portant in probability theory. Chen et al. [15] studied Jensen’s inequality for g-expectation. We say that -expectation satisfies Jensen’s inequality if for any convex function :, then 2 , whenever ,,,. gg LP (10) Lemma 4 [Chen et al. [15] Theorem 3.1] Let be a convex function and satisfy 1 , 2H and 3 . Then 1) Jensen’s inequality (10) holds for -expectations if and only if does not depend on and is posi- tively homogeneous in y ; 2) If 1,d the necessary and sufficient condition for Jensen’s inequality (10) to hold is that there exist two adapted processes and such that 0ab ,tt zta zbz. Now we can easily obtain our main results. Theorem 2 below shows the relation between static risk measures and dynamic risk measures. Theorem 2 If -expectation g static convex (coherent) risk measure, then the corresponding condi- tional g-expectation is a t .T namic convex (co- herent) risk measure for each t is dy 0, Proof: This follows directly direct from Theorem 1. Theorem 3 below shows that in the family of convex risk measure, only coherent risk measure satisfies Jen- Copyright © 2013 SciRes. JMF  Z. J. CHEN ET AL. 389 sen’s inequality. Theorem 3 Suppose that g is a convex risk measure. Then g is a coherent risk measure if and only if g satisfies Jensen’s inequality. Proof: If g is a convex risk measure, then by Corollary 2, is convex. Applying Lemma 4, g satisfies Jensen’s inequality if and only if is posi- tively homogenous. By Corollary 2, the corresponding g is coherent risk measure. The proof is complete. Theorem 4 and Counterexample 1 below give the rela- tion between risk measures and Choquet expectation. Theorem 4 If g is a coherent risk measure, then g is bounded by the corresponding Choquet expec- tation, that is 2 , ,, gV L P where A IVA . If g is a convex risk measure then inequality above fails in general. By construction there exists a convex risk measure and random variables 1 and 2 such that 11 2 and > gVg V2 The prove this theorem uses the following lemma. Lemma 5 Suppose that does not depend on . Suppose that the y -expectation g satisfies (1) , B I, gA Bg II a A I <1 g (2) For any positive constant , AB 2 , ,,. gg aaL P Then for any the 2,,L P -expectation g is bounded by the corresponding Choquet expec- tation, that is 0 0 1d d. gg g xx xI x (11) Proof: The proof is done in three steps. Step 1. We show that if 0 is bounded by , then inequality (11) holds. >0N In fact, for the fixed , denote N n by 21 1 0 22 :. 2 n nn n iN n iiN iN I Then in , nn n 2,, .LP Moreover, can be rewritten as 2 1 2 . 2 n n n niN i NI But by the assumptions (1) and (2) in this lemma, we have 2 12 2 12 2 . 2 n n n n n gg niN i g niN i NI NI Note that 2 10 2 d, . 2 n n gg x niN i NIIx n and , . n gg n Thus, taking limits on both sides of inequality (12), it follows that 0d. gg x x The proof of Step 1 is complete. Step 2. We show that if is bounded by , that is >0N N , then inequality (11) holds. Let then Applying Step 1, , NN 0 2 NN. x 0d. gg Nx NI (13) But by Lemma 1(v), . gg NN On the other hand, 2 00 0 0 dd d d d. N gg Nx xN N gx N gx N N gx xIx x x x .x Thus by (13) 0 0 dd N ggg xx N NIxI Therefore 0 0 1d d. N gg g xx N xI x Step 3. For any 2,, .LP let , N N I then NN . By Step 2, 0 0 1d d. N N gg g xx N xI x Letting , it follows that N 0 0 1d d. gg g xx xI x The proof is complete. Proof of Theorem 4: If the -expectation g is a coherent risk measure, then it is easy to check that the -expectation g satisfies the conditions of Lemma 5. Let A VA I A . By Lemma 5 and the de- finition of Choquet expectation, we have . gV The first part of this theorem is complete. Counterexample 1 shows that this property of coherent risk measures fails in general for more general convex risk measures. This completes the proof of Theorem 4. (12) Counterexample 1 Suppose that is 1-dimen- sional Brownian motion (i.e. d = 1). Let t W 1gzz where max, 0 x . Then g is a convex risk Copyright © 2013 SciRes. JMF  Z. J. CHEN ET AL. 390 measure. Let 11 1 2T W I . and 21 2 T W I Then 1V1g However 2 > gV2 . V Here the capacity in the Choquet expectation is given by V .I g VAA Proof of the Inequality in Counterexample 1: The convex function satisfies (H1), (H2) and(H3). Thus, by Corollary 2, 1gzz -expectation g is a convex risk measure. This together with the property of Choquet expectation in Remark 1 implies 111 1 11 11 22 11 . 22 TT TT gg g WW VV V WW II II Moreover, since by Corollary 3, 1d g is a convex risk measure rather than a coherent risk measure. We now prove that 2 >V2g In fact, since 11 222 TT VVg WW W III 1 , T . we only need to show 11 2>2 TT gg WW II Let , z be the solution of the BSDE 1 21d T TT ts Wtt yIz szW d. ss 0, (14) First we prove that ,0,:>1> t LP tTz (15) where is Lebesgue measure on L 0, .T If it is not true, then a.e. 1 t z 0,tT and BSDE (14) becomes 1 2d T T ts Wt yI zW . s Thus 11 22 TTt t tt WWW W yEI EI . t By the Markov property, 21 tTttt yPWWWW . Recall that and are independent and . Thus Tt WW 0,Ttt W Tt WWN 1 2d t txW x yyy , where is the density function of the normal distribution . Thus 0,NTt Secondly we prove that 11 2>2 TT gg WW II . Let , tt YZ be the solution of the BSDE 1 221dd 2 T TT s tss Wtt Z YI sZW . (16) Obviously, 11dd , 22 T TT ts 2 s Wtt YZZ sW which means , 22 ts YZ is the solution of BSDE 11dd . T TT ts Wtt yIzs zW ss But 1. T tgt W yI Thus by the uniqueness of the solution of BSDE, 1. 2T tg W YI t On the other hand, let , tt z be the solution of the BSDE 1 21d T TT ts Wtt yIz szW d. ss (17) Comparing BSDE(17) with BSDE (16), notice (15) and the fact 12 1 2 z z and gV g whenever z >1. By the strict comparison theorem of BSDE, we have >, 0,. tt Yt T Setting 0t , thus 11 2>22 TT ggV WW III 1 T W . The proof is complete. Remark 3 In mathematical finance, coherent and convex risk measures and Choquet expectation are used in the pricing of contingent claim. Theorem 4 shows that coherent pricing is always less than Choquet pricing, while Counterexample 1 demonstrates that pricing by a convex risk measure no longer has this property. In fact the convex risk price may be greater than or less than the Choquet expectation. 4. Summary Coherent risk measures are a generalization of mathe- matical expectations, while convex risk measures are a generalization of coherent risk measures. In the frame- work of -expectation, the summary of our results is given in Table 1. In that Table, the Choquet expectation is : 21 , tttt zDy W t z t W where is the Malliavin derivative. Thus can be greater than 1 whenever is near 0 and is near 0. Thus (15) holds, which contradicts the assumption a.e. t Dt 1 t z 0,tT. A Counterexample 1 shows that convex risk may be or VA I. Choquet expectation. Only in the case of coherent Copyright © 2013 SciRes. JMF  Z. J. CHEN ET AL. Copyright © 2013 SciRes. JMF 391 Table 1. Relations among coherent and convex risk mea- sures g , choquet expectation and Jensen’s inequality. 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