 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  and Delbaen . As an extension of coherent risk measures, convex risk measures in general probabil- ity spaces were introduced by Föllmer & Schied  and Frittelli & Rosazza Gianin . g-expectations were in- troduced by Peng  via a class of nonlinear backward stochastic differential equations (BSDEs), this class of nonlinear BSDEs being introduced earlier by Pardoux and Peng . Choquet  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 Universityprogram of the Korea Science and Engineering Foundation (R31-20007and the NSERC grant of RK. This work has been supported by grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Copyright © 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  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  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 0ttT be a -dimen- sional standard Brownian motion defined on a completed probability space Wd,,P. Suppose tT0t is the natural filtration generated by , that is 0tTtW;tsWs t. We also assume . Denote T 22,,: is measurable random variables with <, 0,;ttLPE t  — T 2200,,: is valued,adapted processes with d>P012>gg. 4) Consistency: For any 0,tT, gg tg . 5) If g does not depend on , and y is - measurable, then t.gtgt In particular, 0ggtt. 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. 385Lemma 2 (Briand et al. ) Suppose that tX is of the form 0d,ttssXxW ,0tT where t is a continuous bounded proces s. Then ,,, 0,lim gst stst ttXEX gXttst  where the limit is in the sense of . 2,,LP2.2. Choquet Expectation Choquet  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 AB,,L. 2) Let V be a capacity. For any , the Choquet expectation P2V is defined by   00:1dVVttVt dt Remark 1 A property of Choquet expectation is posi- tive homogeneity, i.e. for any constant 0,a .VVaa 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.) is from Roorda et. al. . Definition 3 A risk measure  is said to be coherent if it satisfies 1) Subadditivity: 121 2XXXX  ,, ; 122) Positive homogeneity: ,XXXX  for all real number 0; 3) Monotonicity: ,XY whenever ;XY 4) Translation invariance: XX  for all real number . As an extension of coherent risk measures, Föllmer and Schied  introduced the axiomatic setting for con- vex risk measures. The following modifications of con- vex risk measures of Föllmer and Schied  is from Frittelli and Rosazza Gianin . Definition 4 A risk measure is said to be convex if it satisfies 1) Convexity: 121112XXX 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  introduced the notion of dynamic risk measure which is random and depends on a time parameter . ,,ttDefinition 5 A dynamic risk measure 22:,,,,ttLPL P is a random functional which depends on t, 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,gg and 2g satisfy (H1), (H2) and (H3). Then the following conclusions are equivalent. 1) For any 2,,,LP, 12.ggg  2) For any 1122,,,,,0, ,dyztyztT 12121112 22,,,,,,.gyyzztgyztg yzt (2) Proof: We first show that inequality (2) implies ine- quality 3). Let 11,,ttyz 22,ttyz and  be the solutions of the following BSDE corresponding to the terminal value ,ttYZ, X and , and the generator 12, ggg and g, respectively ,,dd.TtssttyXgyzss zW Tss (3) Then 110,gy 220,gy 0.gY For fixed 11,yztt, consider the BSDE 11 1121,, ,,d.tTssss sstTsstydgyyzzsgyzsszW (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,ttttyyzzComparing BSDEs (4) and (3) with X and gg, assumption (2), (2) then yields 1212 111222,, ,,,,, 0.tttt ttttgyyzztgyztgyztt Applying the comparison theorem of BSDE in Peng , we have Taking , thus by the definition of 12, 0.tttYyyt 0tg-expectation, the proof of this part is complete. We now prove that inequality (1) implies (2). We dis- tinguish two cases: the former where g does not de- pend on , the latter where yg may depend on . yCase 1, g does not depend on . The proof of this case 1 is done in two steps. yCase 1, Step 1: We now show that for any 0,tT, we have 122, ,,,.gtgtgtLP     Indeed, for 0,tT , set 12:>tg tgtgtA . If for all 0,tT, we have then we ob- tain our result. 0,tPA If not, then there exists 0,tT such that >0tPA . We will now obtain a contradiction. For this , t12>ttAgtAgtgtII    . That is 12|>AgtgtgttI   0. Taking g-expectation on both sides of the above inequality, and apply the strict monotonicity of g-ex- pectation in Lemma 1 (3), it follows 12>0.tgAgt gt gtI     But by Lemma 1 (4) and (5),  1212.ttttgAgt gtgtgAgAtgA tIIII       Note that by Lemma 1(v) 0, 1,2.it itgAgA tII i  Thus 121211220<0tttttt tttttgAg AtgAtgA gAtA gAtgAgAtgAgA tIIIIIIIIIII       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 diz,itWiXzW2. Obviously, .2,,iXLP By Step 1, 1212 12, 0,.gtg tgtXX XXtT   Thus 12 12111222.gttgt tgt tXXEXXtXEXtXEXt    Let ,t,y applying Lemma 2, since g does not de- pend on we rewrite ,,gyzt simply as ,,gzt thus 1 2tg z2,,,, tg ztt12gz 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. . For each >0 and 112 2,, ,dyzyz , define the stopping time 11 2 21112 221212,;,inf0; ,,,,,, .yzyztgyztgyztgyyzztT , Obviously, if for each  112 2,, ,dyzyz1122,; ,<0PyzyzT, for all , then the proof is done. If it is not the case, then there exist >0 and 112 2,, ,dyzyz, such that 1122,; ,<>0PyzyzT. Fix ,,, 1,2,iiyzi and consider the following (forward) SDEs defined on the interval ,T  d,,d, , 1,2iiiiiiiYt gYtzttzWYyti d,t Copyright © 2013 SciRes. JMF Z. J. CHEN ET AL. 387and  3312 12312d,,d, .tYtgYtzzttz zWYyyt  d, Obviously, the above equations admit a unique solu- tion which is progressively measurable with iY20<.sup itTEYt Define the following stopping time 12112 2312:inf; ,,,,,, .2tgYtztgYtztgYtzz tT  It is clear that T and note that T whenever ,T<>0 thus, Hence . , on 0a are. (5) By the definition of and , the pair pro- cesses nd 312Y the solutions of the following BSDEs with terminal values 12,YY12 ,tz 3Y12,,Ytz Y,tz z1,YT2YT and 3YT, ,,dd, 1,2itisiisttyYTgyzss zWi TT and 312 12,,d dTTtsttyYT gyzzsszzW . sHence, 11111,ggYYT y 22222ggYYT y and 3312.ggYYT yy Applying the strict comparison theorem of BSDE and inequality (5), by the assumptions of this Theorem, we have  12312121212<.ggggyy YYYYYy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,,,,gyztcgy ztcc. Then for any 2,,LP, .ggcc Proof: Letting tg tyc, then ty is the solu- tion of BSDE ,,dd.TTtssttycgyzss zW ss Since 11,,,,gyztcgy ztcc, the above BSDE can be rewritten as 11,,ddTTtssttyccg y zsszWcc .ss (6) Let tg ty, then satisfies tcy,,dd .TTtssttcyccgyzssczW ss (7) Comparing with BSDE (6) and BSDE (7), by the uniqueness of the solution of BSDE, we have ,,tt ttcy czy z. Let 0,t then 00cy 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 g. 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 zProof: Applying Theorem 1, g is linear if and only if ,,gyzt is linear in ,yz. By assumption (H3), that is , 0t,0gy for all ,yt. Thus g does not depend on . The proof is complete. yCorollary 2 The g-expectation g is a convex risk measure if and only if g does not depend on and is convex in . yzProof: Obviously, g-expectation g0,1 is convex risk measure if and only if for any Copyright © 2013 SciRes. JMF Z. J. CHEN ET AL. 388 211 ,,,.ggLP  ,g (8) For a fixed , let 0,11211,,, ,, 11,,1,,.11g yztgyztgyztgy zt  Applying Lemma 3,  12, 11.gggg Inequality (8) becomes 1221 ,,,.gggLP     1,  Applying Theorem 1, for any ,,0,, 1,2,diiyztTi    1212111 222112 21,1,,,11 ,,,1, ,gyyzztgyztgyztgyzt gyzt  which then implies that g is convex. By the explanation of Remark for Lemma 4.5 in Briand et al. , the convexity of g and the assumption (H3) imply that g does not depend on . The proof is complete. yThe function g is positively homogeneous in if for any , z0a,, ,,gazag z. Corollary 3 The g-expectation g is a coherent risk measure if and only if g does not depend on and it is convex and positively homogenous in . In particular, if , yz1dg is of the form ,ttgzta zbz with . 0aProof: By Corollary 2, the g-expectation g is a convex risk measure if and only if g does not depend on and is convex in . Applying Theorem 1 and Lemma 3 again, it is easy to check that y zg-expectation g is positively homogeneous if and only if g is positively homogeneous (that is for all and >0a, ggaa,,, if and only if for any , 0a,gazag z d. In particular, if , notice the fact that 1g is con- vex and positively homogeneous on , and that g does not depend on . We write it as y,gzt then  000,, ,1,1, .zzzg ztg ztIgztIgtzIgt zI0z (9) Note that , , but 0zzI z0zzIz, .22zz zzzz Thus from (9)  1, 1,1, 1,,.22gtg tgtg tgztz z  Defining 1, 1,:2tgtg ta,  1, 1,:2tgtg tb. Obviously since the convexity of 0,ag yields 1,1, 0, 0.2gt 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 ,ttgzta 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.  studied Jensen’s inequality for g-expectation. We say that g-expectation satisfies Jensen’s inequality if for any convex function :, then  2,whenever ,,,.ggLP   (10) Lemma 4 [Chen et al.  Theorem 3.1] Let g be a convex function and satisfy 1H, 2H and 3H. Then 1) Jensen’s inequality (10) holds for g-expectations if and only if g does not depend on and is posi- tively homogeneous in yz; 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 0ab,ttgzta 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 g-expectation g static convex (coherent) risk measure, then the corresponding condi- tional g-expectation  is agt.T  namic convex (co- herent) risk measure for each t is dy0,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. 389sen’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, g is convex. Applying Lemma 4, g satisfies Jensen’s inequality if and only if g 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 gAIVA . 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 g does not depend on . Suppose that the yg-expectation g satisfies (1) , BI,gA BgII aAI<1g (2) For any positive constant , AB2, ,,.ggaaL P  Then for any the 2,,LPg-expectation g is bounded by the corresponding Choquet expec- tation, that is  001d d.gg gxxIxI    x (11) Proof: The proof is done in three steps. Step 1. We show that if 0 is bounded by , then inequality (11) holds. >0NIn fact, for the fixed , denote Nn by  211022:.2nnnniNniiNiN I Then in ,nnn2,, .LPMoreover,  can be rewritten as 212.2nnnniNiNI But by the assumptions (1) and (2) in this lemma, we have 2122122.2nnnnnggniNigniNiNINI  Note that 2102d, .2nnggxniNiNIIx n and , .nggnThus, taking limits on both sides of inequality (12), it follows that 0d.ggxIx The proof of Step 1 is complete. Step 2. We show that if  is bounded by , that is >0NN, then inequality (11) holds. Let then Applying Step 1, ,NN 02NN.x0d.ggNxNI (13) But by Lemma 1(v), .ggNN  On the other hand,  20000ddddd.NggNx xNNgxNgxNNgxIxIxIxIxIx .x Thus by (13)  00ddNgggxxNNIxI  Therefore  001d d.Ngg gxxNIxIx  Step 3. For any 2,, .LP let ,NNI then NN. By Step 2,  001d d.NNgg gxxNIxI x   Letting , it follows that N001d d.gg gxxIxI x  The proof is complete. Proof of Theorem 4: If the g-expectation g is a coherent risk measure, then it is easy to check that the g-expectation g satisfies the conditions of Lemma 5. Let gAVA 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 tW 1gzz where max, 0xx. Then g is a convex risk Copyright © 2013 SciRes. JMF Z. J. CHEN ET AL. 390 measure. Let 1112TWI. and 212TWI Then 1V1g However 2>gV2.V Here the capacity in the Choquet expectation is given by V.IgVAAProof of the Inequality in Counterexample 1: The convex function satisfies (H1), (H2) and(H3). Thus, by Corollary 2,  1gzz g-expectation g is a convex risk measure. This together with the property of Choquet expectation in Remark 1 implies 111111112211.22TTTTgg gWWVV VWWIIII    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 11222TTVVgWW WIII1,T. we only need to show  112>2TTggWWII Let ,yz be the solution of the BSDE 121dTTTtsWttyIz szWd.ss0, (14) First we prove that ,0,:>1>tLP tTz (15) where is Lebesgue measure on L0, .T If it is not true, then a.e. 1tz0,tT and BSDE (14) becomes 12dTTtsWtyI zW.s Thus 1122TTt tttWWW WyEI EI.t By the Markov property, 21tTtttyPWWWW. Recall that and are independent and . Thus TtWW0,TttWTtWWN12dttxWxyyy, where x is the density function of the normal distribution . Thus 0,NTtSecondly we prove that  112>2TTggWWII. Let ,ttYZ be the solution of the BSDE 1221dd2TTTstssWttZYI sZW . (16) Obviously, 11dd ,22TTTts2ssWttYZZIsW  which means ,22tsYZ is the solution of BSDE 11dd .TTTtsWttyIzs zWss But 1.TtgtWyI Thus by the uniqueness of the solution of BSDE, 1.2TtgWYIt On the other hand, let ,ttyz be the solution of the BSDE 121dTTTtsWttyIz szWd.ss (17) Comparing BSDE(17) with BSDE (16), notice (15) and the fact 12 12zz  and gVg whenever z >1. By the strict comparison theorem of BSDE, we have >, 0,.ttyYt T Setting 0t, thus  112>22TTggVWWIII1TW. 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 g-expectation, the summary of our results is given in Table 1. In that Table, the Choquet expectation is :21 ,ttttzDy WtztW 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. tDt1tz0,tT. gACounterexample 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 391Table 1. Relations among coherent and convex risk mea- sures g, choquet expectation and Jensen’s inequality. Risk Measures Relation to Choquet Expectation Jensen inequality g is linear math. expectation gV true g is convex and positively homogeneous coherent gV true (*) g is convex convex Neither ≤ nor ≥ not true except (*) risk there is an inequality relation with Choquet expec- tation. REFERENCES  P. Artzner, F. Delbaen, J. M. Eber and D. Heath, “Co- herent Measures of Risk,” Mathematical Finance, Vol. 9, No. 3, 1999, pp. 203-228. doi:10.1111/1467-9965.00068  F. Delbaen, “Coherent Risk Measures on General Prob- ability,” In: K. Sandmann and P. Schonbucher, Eds., Ad- vances in Finance and Stochastics, Springer-Verlag, Ber- lin, 2002, pp. 1-37. doi:10.1007/978-3-662-04790-3_1  H. Föllmer and A. Schied, “Convex Measures of Risk and Trading Constraints,” Finance and Stochastics, Vol. 6, No. 4, 2002, pp. 429-447. doi:10.1007/s007800200072  M. Frittelli and E. R. Gianin, “Putting Order in Risk Mea- sures,” Journal of Banking Finance, Vol. 26, No. 7, 2002, pp. 1474-1486. doi:10.1016/S0378-4266(02)00270-4  S. G. Peng, “BSDE and Related g-Expectation,” Pitman Research Notes in Mathematics Series, Vol. 364, 1997, pp. 141-159.  E. Pardoux and S. G. Peng, “Adapted Solution of a Back- ward Stochastic Differential Equation,” Systems and Con- trol Letters, Vol. 14, No. 1, 1990, pp. 55-61. doi:10.1016/0167-6911(90)90082-6  G. Choquet, “Theory of Capacities,” Annales de l'institut Fourier (Grenoble), Vol. 5, 1953, pp. 131-195. doi:10.5802/aif.53  L. Jiang, “Convexity, Translation Invariance and Subad- ditivity for g-Expectations and Related Risk Measures,” Annals of Applied Probability, Vol. 18, No. 1, 2008, pp. 245-258. doi:10.1214/105051607000000294  P. Briand, F. Coquet, Y. Hu, J. Mémin and S. G. Peng, “A Converse Comparison Theorem for BSDEs and Re- lated Problems of g-Expectation,” Electronic Communi- cations in Probability, Vol. 5, 2000, pp. 101-117. doi:10.1214/ECP.v5-1025  B. Roorda, J. M. Schumacher and J. Engwerda, “Coherent Acceptability Measures in Multi-Period Models,” Mathe- matical Finance, Vol. 15, No. 4, 2005, pp. 589-612. doi:10.1111/j.1467-9965.2005.00252.x  P. Artzner, F. Delbaen, J. M. Eber, D. Heath and H. Ku, “Multiperiod Risk and Coherent Multiperiod Risk Meas- urement,” E.T.H.Zürich, Preprint, 2002.  P. Artzner, F. Delbaen, J. M. Eber, D. Heath and H. Ku, “Coherent Multiperiod Risk Adjusted Values and Bell- man’s Principle,” Annals of Operations Research, Vol. 152, No. 1, 2007, pp. 5-22. doi:10.1007/s10479-006-0132-6  M. Frittelli and E. R. Gianin, “Dynamic Convex Risk Measures,” In: G. Szegö, Ed., Risk Measures for the 21st Century, John Wiley, Hoboken, 2004, pp. 227-248.  F. Coquet, Y. Hu, J. Mémin and S. G. Peng, “A General Converse Comparison Theorem for Backward Stochastic Differential Equations,” Comptes Rendus de l’Académie des Sciences Paris, Series I, Vol. 333, 2001, pp. 577-581.  Z. J. Chen, R. Kulperger and L. Jiang, “Jensen’s Inequal- ity for g-Expectation: Part 1,” Comptes Rendus de l’Aca- démie des Sciences Paris, Series I, Vol. 337, 2003, pp. 725-730.