Risk Measures and Nonlinear Expectations*

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)

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: [email protected]

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 mathematical 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 measure 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 nonlinear BSDEs being introduced earlier by Pardoux and Peng [6]. Choquet [7] extended probability measures The choice of financial risk measures is very important in to nonadditive probability measures (capacity), and in- the assessment of the riskiness of financial positions. For troduced the so called Choquet expectation. this reason, several classes of financial risk measures Our interest in this paper is to explore the relations have been proposed in the literature. Among these are among risk measures and expectations. To do so, we re- coherent and convex risk measures, Choquet expecta- strict our attention of coherent and convex risk measures tions and Peng’s g-expectations. Coherent risk measures and Choquet expectations to the domain of g-expecta- were first introduced by Artzner, Delbaen, Eber and tions. The distinctions between coherent risk measure Heath [1] and Delbaen [2]. As an extension of coherent and convex risk measure are accounted for intuitively in risk measures, convex risk measures in general probabil- the framework of g-expectations. We show that 1) in the ity spaces were introduced by Föllmer & Schied [3] and family of convex risk measures, only coherent risk Frittelli & Rosazza Gianin [4]. g-expectations were in- measures satisfy Jensen’s inequality; 2) coherent risk troduced by Peng [5] via a class of nonlinear backward measures are always bounded by the corresponding stochastic differential equations (BSDEs), this class of Choquet expectation, but such an inequality in general *This work is supported partly by the National Natural Science Founda- fails for convex risk measures. In finance, coherent and tion of China (11231005; 11171062) and WCU(World Class University convex risk measures and Choquet expectations are often program of the Korea Science and Engineering Foundation (R31-20007 used in the pricing of a contingent claim. Result 2) im- and the NSERC grant of RK. This work has been supported by grants from the Natural Sciences and plies coherent pricing is always less than Choquet pricing, Engineering Research Council (NSERC) of Canada. but the pricing by a convex risk measure no longer has

Copyright © 2013 SciRes. JMF 384 Z. J. CHEN ET AL. this property. We also study the relation between static for convenience of the various relations. and dynamic risk measures. We establish that if g-expectations are coherent (convex) risk measures, then the 2. Expectations and Risk Measures same is true for the corresponding conditional g-expecta- In this section, we briefly recall the definitions of g-ex- tions or dynamic risk. In order to prove these results, we pectation, Choquet expectation, coherent and convex risk establish in Section 3, Theorem 1, a new converse com- measures. parison theorem of g-expectations. Jiang [8] studies g- expectation and shows that some cases give rise to risk 2.1. g-Expectation measures. Here we are able to show, in the case of g- expectations, that coherent risk measures are bounded by Peng [5] introduced g-expectation via a class of back- Choquet expectation but this relation fails for convex risk ward stochastic differential equations (BSDE). Some of measures; see Theorem 4. Also we show that convex risk the relevant definition and notation are given here. measures obey Jensen’s inequality; see Theorem 3. Fix T  0, and let W be a d -dimen-   t 0tT The paper is organized as follows. Section 2 reviews sional standard Brownian motion defined on a completed and gives the various definitions needed here. Section 3 probability space ,, P . Suppose  is the   t 0tT gives the main results and proofs. Section 4 gives a natural filtration generated by W , that is t 0tT summary of the results, putting them into a Table form ts  Ws; t. We also assume T  . Denote

2 — 2 LP,tt , : is measurable random variables with E < , t 0,T ;

T LT2 0, ,dd XX : is —— valued, adapted processes with EXs2 d <  ts0

Let g :  d 0,T    satisfy pectation of the random variable  . (H1) For any yz,,d g yzt,, is a   defined by  :y is called the   t0 g  t  g tt continuous progressively measurable process with conditional g-expectation of the random variable  .

T 2 Peng [5] also showed that g-expectation g  and Egyzss,, d < . 0 conditional g -expectation g t   preserve most of basic properties of mathematical expectation, except for (H2) There exists a constant K  0 such that for any linearity. The basic properties are summarized in the next d Lemma. yz11,, yz 2, 2 Lemma 1 (Peng) Suppose that

gyzt11,, gy 2 ,, zt 2 2 ,,12  LP ,, . K yy12 zz 12, t0, T . 1) Preservation of constants: For any constant c,

(H3) g yt,0, 0,  yt,   0, T . g cc  .

In Section 3, Corollary 3 we will consider a special 2) Monotonicity: If 1 2, then gg12    . d case of  with d  1. 3) Strict monotonicity: If    , and P>>0, Under the assumptions of (H1) and (H2), Pardoux and 12 12 2 then gg12 >   . Peng [6] showed that for any  L,, P, the 4) Consistency: For any t0,T , BSDE  TT gg  t  g  . ygyzsszWt ,,d d , 0 T (1)  tsssstt 5) If g does not depend on y , and  is t - has a unique pair solution measurable, then 22d yztt,0,,0,, LT LT. t0 gtgt . Using the solution y of BSDE (1), which depends t In particular,  0 . on  , Peng [5] introduced the notion of g-expectations. ggtt 2 Definition 1 Assume that (H1), (H2) and (H3) hold on 6) Continuity: If n   as n  in LP,,  , 2 g and  LP,, . Let  ys , zs  be the solution of then lim ngn  g  . BSDE (1). The following lemma is from Briand et al. [9, Theo-

g   defined by g   : y0 is called the g-ex- rem 2.1]. We can rewrite it as follows.

Lemma 2 (Briand et al. ) Suppose that X is of  t   X12111  XX      X2 , t the form X xW d, 0 tT , where  is a ts0 s  t   0,1 , XX12,  ; continuous bounded process. Then 2) Normality:  00  ; 3) Properties (3) and (4) in Definition 3. XEX  gst st A functional   in Definitions 3 and 4 is usually lim st gXtt,,, t t0,   st called a static risk measure. Obviously, a coherent risk where the limit is in the sense of LP2 ,, . measure is a convex risk measure. As an extension of such a functional   , Artzner et 2.2. Choquet Expectation al. [11,12], Frittelli and Rosazza Gianin [13] introduced the notion of dynamic risk measure t  , which is Choquet [7] extended the notion of a probability measure random and depends on a time parameter t . to nonadditive probability (called capacity) and defined a Definition 5 A dynamic risk measure kind of nonlinear expectation, which is now called Cho- 22 quet expectation. tt :,,LPL   ,,P Definition 2 is a random functional which depends on t , such that 1) A real valued set function V :0  ,1 is called   for each t it is a risk measure. If   satisfies for a capacity if t  each t0,T the conditions in Definition 3, we say a) VV0,   1; t  is a dynamic coherent risk measure. Similarly if

b) VA VB , whenever AB,   and A  B . t  satisfies for each tT0,  the conditions in De- 2 2) Let V be a capacity. For any  L,, P, finition 4, we say t  is a dynamic convex risk meas- the Choquet expectation V  is defined by ure. 0   :1Vt  d tVt   dt V  0 3. Main Results Remark 1 A property of Choquet expectation is posi- In order to prove our main results, we establish a general tive homogeneity, i.e. for any constant a  0, converse comparison theorem of g-expectation. This theorem plays an important role in this paper. VVaa   . Theorem 1 Suppose that g, g1 and g2 satisfy (H1), (H2) and (H3). Then the following conclusions are 2.3. Risk Measures equivalent. 2 A risk measure is a map  :  , where  is in- 1) For any ,,,LP  , terpreted as the “habitat” of the financial positions whose ggg   . riskiness has to be quantified. In this paper, we shall con- 12 2 d sider LP,, . 2) For any  yzt11,,,  yzt 2 ,, 2  0,,T The following modifications of coherent risk measures gy 1212 y,, z zt (Artzner et al.[1]) is from Roorda et. al. [10]. (2) g yzt,, g yzt ,,. Definition 3 A risk measure  is said to be coherent 111 2 22 if it satisfies Proof: We first show that inequality (2) implies ine- 1) Subadditivity: X12XXX 1  2 , quality 3). XX,  ; 11 22 12 Let  yztt,,  yztt,  and YZtt,  be the solutions 2) Positive homogeneity: X   X , for all of the following BSDE corresponding to the terminal real number   0; value X  ,  and , and the generator g  gg12, 3) Monotonicity: X  Y , whenever X  Y; and g , respectively TT 4) Translation invariance:   XX    for yX gyzsszW,,d  d . (3) all real number  . tssttss As an extension of coherent risk measures, Föllmer Then    y1 ,    y2 ,  Y . g1   0 g2   0 g   0 and Schied [3] introduced the axiomatic setting for con- For fixed yz11, , consider the BSDE vex risk measures. The following modifications of con-  tt vex risk measures of Föllmer and Schied [3] is from yt    Frittelli and Rosazza Gianin [4]. T g yyzzsgyzss11,, 11 ,, d (4) Definition 4 A risk measure is said to be convex if it t  21ssss ss satisfies T  zWd. 1) Convexity: t ss

1212 It is easy to check that yyzztttt, is the solu- 0<III      gA tttg12 A t g A t  tion of the BSDE (4). II   II  Comparing BSDEs (4) and (3) with X   and gA tt g12 A t A t g A t t g  g , assumption (2), (2) then yields II gA11tt gA t g yyzztgyzt1212,, 11 ,, tttt1 tt II gA22tt gA t 22 gyztt2 tt,,, 0.  0 Applying the comparison theorem of BSDE in Peng This induces a contradiction, thus concluding the proof 12 of this Step 1. [5], we have Yyytttt, 0. Taking t  0 , thus by the definition of g -expectation, the proof of this part is Case 1, Step 2: For any  ,0,tT  with   t and d i complete. zi  , let us choose XzWitW, i  1,2 . i 2  We now prove that inequality (1) implies (2). We dis- Obviously, X LP ,,. tinguish two cases: the former where g does not de- By Step 1, pend on y , the latter where g may depend on y . XX12 X1 g  tg1  t Case 1, g does not depend on y . The proof of this case 1 is done in two steps. XtT2 , 0, . gt2   Case 1, Step 1: We now show that for any tT0,  , we have Thus        , XX12 EXX 12 gtgtgt 12   g  tt

  t  ,,,.LP2  XEX11  gt1  t Indeed, for tT0, , set      t A :>  . XEX22  tg tgtgt 12   gt t  2 .   t If for all tT0,  , we have PAt  0, then we ob- Let   t, applying Lemma 2, since g does not de- tain our result. pend on y, we rewrite g yzt,, simply as g zt,, If not, then there exists t 0,T such that PA  >0.      t thus gz z,,,,t g zt g z t t0. The proof We will now obtain a contradiction.  12 1 1 22 of Case 1 is complete. For this t , Case 2, g depends on y. The proof is similar to the

IIAg t > A g  t  g   t . proof of Theorem 2.1 in Coquet et al. [14]. For each  >0 tt 12   d and  yz11,,  yz 2, 2, define the stopping time That is   yzyz11,; 2 , 2 I Ag t  g |> t  g   t  0. t 12 inftgyztgyzt 0; 111, , 2 22 , ,  Taking g-expectation on both sides of the above gy1212 y,, z zt T . inequality, and apply the strict monotonicity of g -ex- d pectation in Lemma 1 (3), it follows Obviously, if for each yz11,, yz 2, 2, PyzyzT  11,; 2 , 2 <  0, for all , then the proof I           >0. gAt  g t  g12  t g  tis done. If it is not the case, then there exist  >0 and d But by Lemma 1 (4) and (5),  yz11,,  yz 2, 2 , I            such that gAt  g t  g12  t g  t PyzyzT  11,; 2 , 2 < >0. III   . gAttt g1  A t g2 A t Fix ,,,yzii i 1,2, and consider the following Note that by Lemma 1(v) (forward) SDEs defined on the interval  ,T   ii II0, i1, 2. d,Yt gYtzttzWiii  ,d  dt, gAit gA it t  Yytii , , 1,2 Thus  i

Copyright © 2013 SciRes. JMF Z. J. CHEN ET AL. 387 and YY33 Tyy. gg 12 d,,Yt33 gYtz  zttd  z  zd W,    12 12 t Applying the strict comparison theorem of BSDE and  3 inequality (5), by the assumptions of this Theorem, we Yyyt12,  . have Obviously, the above equations admit a unique solu- 312 i    tion Y which is progressively measurable with yy12gg Y< Y  Y 

12 i 2 YYy  y. EYtsup  <. gg1212 0tT This induces a contradiction. The proof is complete. Define the following stopping time Lemma 3 Suppose that g satisfies (H1), (H2) and (H3).

 12 For any constant c =0 , let :inftgYtztgYtzt ; 1122,,  , ,  11 g yzt,, cg y , zt , . 3   cc gY t,, z12 z t T . 2  2 Then for any  LP ,,  , ggcc    . Proof: Letting yc  , then y is the solu- It is clear that  T and note that   T tg t t tion of BSDE whenever   T, thus, <  <.T Hence P<>0. TT  yc gyzss,,d zW d . Moreover, we can prove tssttss 12 3 Since YY >, Y  on <. 11 In fact, setting g yzt,, cg y , zt , , cc Ytˆ  Y312  t Y  t Y  t, the above BSDE can be rewritten as then 3 TT11 d,,Ytˆ  gY  t z  z t yc cgy,,d zss  zW d. (6)  12 tssttss cc 12 g1122Ytzt,, gYtzt ,, d. t  Let ytg  t, then cyt satisfies TT dYtˆ   cy c cg y,,d z s s cz d W . (7) Thus , tY  , , ˆ   0. tssttss d2t   Comparing with BSDE (6) and BSDE (7), by the ˆ  It follows that on  ,  , Y   <0. uniqueness of the solution of BSDE, we have 2

This implies cytt,, cz   y tt z . PY312<< Y Y P >0. (5)     Let t  0, then cy0 y0. The conclusion of the Lemma now follows by the definition of g-expectation. By the definition of YY12, and Y 3 , the pair pro- This concludes the proof. cesses Ytz12,, Ytz, and Y3 tz,  z are  12  12Applying Theorem 1 and Lemma 3, immediately, we the solutions of the following BSDEs with terminal 1 2 3 obtain several relations between g-expectation   values YT, YT and YT, g      and g . These are given in the following Corollaries. i TT Corollary 1 The g-expectation   is the classical yYT gyzss,, d  zWi d ,  1,2 g   tittsiis mathematical expectation if and only if g does not de- and pend on y and is linear in z .

TT Proof: Applying Theorem 1, g  is linear if and yYT3 gyzzss,,d   zz d W. ts tt12 12 sonly if g  yzt,, is linear in y, z . By assumption (H3), that is gy ,0,t  0 for all y,t. Thus g does Hence, not depend on y . The proof is complete. 11 Corollary 2 The g -expectation   is a convex ggYYT y1, g   11 risk measure if and only if g does not depend on y YY22 Ty and is convex in z . gg22 2 Proof: Obviously, g -expectation g  is convex and risk measure if and only if for any  0,1

zz zz gg11       g ,  (8) zz, .  ,,,.LP2  22 Thus from (9) For a fixed  0,1, let g 1,tg  1, t gtg 1,   1, t  11 g zt,.z z g1  yzt,,  g y , zt , , 22  Defining 11 g yzt,, 1 g y , zt , . 2  g 1,tg  1, t g 1,tg  1, t  11 a : , b : . t 2 t 2 Applying Lemma 3, Obviously a  0, since the convexity of g yields     , 1   1   . ggg12 g gt1,  g 1, t gt0, 0. Inequality (8) becomes 2 1      1   , The proof is complete. gg 1   g  2 2 Remark 2 Corollaries 2 and 3 give us an intuitive  ,,,.LP explanation for the distinction between coherent and Applying Theorem 1, for any convex risk measures. In the framework of g-expectations, convex risk measures are generated by convex functions, d yztii,,  0,T , i 1, 2, while coherent measures are generated only by convex and positively homogenous functions. In particular, if d = gy1,1, y z  zt 1212   1, it is generated only by the family g zt,  att z bz with a  0 . Thus the family of coherent risk measures is g111yztg,,  2  1  y 2 1  zt 2 , much smaller than the family of convex risk measures. gyzt11,, 1 gy 2 ,, zt 2 Jensen’s inequality for mathematical inequality is important in probability theory. Chen et al. [15] studied which then implies that g is convex. By the explanation Jensen’s inequality for g-expectation. of Remark for Lemma 4.5 in Briand et al. [9], the We say that g -expectation satisfies Jensen’s convexity of g and the assumption (H3) imply that g inequality if for any convex function  : , then does not depend on y . The proof is complete. The function g is positively homogeneous in z if gg     , (10) for any a  0 , g ,,az ag ,, z  . 2 whenever  ,LP  , ,  . Corollary 3 The g -expectation g  is a coherent risk measure if and only if g does not depend on y Lemma 4 [Chen et al. [15] Theorem 3.1] Let g be a and it is convex and positively homogenous in z . In convex function and satisfy H1 , H 2 and H3 . particular, if d  1, g is of the form Then g zt,  a z bz with a  0 . 1) Jensen’s inequality (10) holds for g -expectations tt if and only if g does not depend on y and is posi-

Proof: By Corollary 2, the g -expectation g  is a tively homogeneous in z ; convex risk measure if and only if g does not depend 2) If d  1, the necessary and sufficient condition for on y and is convex in z . Applying Theorem 1 and Jensen’s inequality (10) to hold is that there exist two Lemma 3 again, it is easy to check that g -expectation adapted processes a  0 and b such that   is positively homogeneous if and only if g is g   g zt,  a z bz. positively homogeneous (that is for all a >0 and ,   tt

ggaa     if and only if for any a  0 , Now we can easily obtain our main results. Theorem 2 g ,,az ag, z,. below shows the relation between static risk measures In particular, if d  1, notice the fact that g is con- and dynamic risk measures.  vex and positively homogeneous on , and that g Theorem 2 If g -expectation g  is a static convex does not depend on y . We write it as g zt,  then (coherent) risk measure, then the corresponding conditional g-expectation g t  is dynamic convex (co- g zt,, g  zt Izz00 g zt , I (9) herent) risk measure for each t0,T. Proof: This follows directly direct from Theorem 1. gtzIg1,z0 1, t zIz0 . Theorem 3 below shows that in the family of convex   Note that zIz0  z , zIz0 z, but risk measure, only coherent risk measure satisfies Jen-

Copyright © 2013 SciRes. JMF Z. J. CHEN ET AL. 389 sen’s inequality. Note that Theorem 3 Suppose that   is a convex risk g    2n N  measure. Then g  is a coherent risk measure if and   i1 niNggIIx x d, n.   0  only if   satisfies Jensen’s inequality. 2 n g 2 Proof: If g  is a convex risk measure, then by and n , n  . Corollary 2, g is convex. Applying Lemma 4, g  gg  Thus, taking limits on both sides of inequality (12), it satisfies Jensen’s inequality if and only if g is posi-  follows that   I  d.x The proof of Step tively homogenous. By Corollary 2, the corresponding gg 0   x  1 is complete. g  is coherent risk measure. The proof is complete. Theorem 4 and Counterexample 1 below give the rela- Step 2. We show that if  is bounded by N >0, tion between risk measures and Choquet expectation. that is   N , then inequality (11) holds. Let N N, then 0 N 2N. Applying Step Theorem 4 If g  is a coherent risk measure, then 1, g  is bounded by the corresponding Choquet expec- 2 tation, that is , L,  ,P where  gV      NI  d.x (13) gg0  Nx VA g IA  . If g  is a convex risk measure then   inequality above fails in general. By construction there But by Lemma 1(v), ggNN   . On the exists a convex risk measure and random variables 1 other hand, and  such that 2  2N I ddxI x 00ggNx  xN gV11   and  g 2 >  V2   N   I dx The prove this theorem uses the following lemma.  N g  x

Lemma 5 Suppose that g does not depend on y . 0   g I x dx Suppose that the g -expectation g  satisfies (1)  N  III I, AB,  (2) For any N gA B g A g B   I d.x positive constant a <1, 0 g  x 2 Thus by (13) ggaa  , L,  , P . 0 N 2  NIxI dd  x. Then for any  L,, P the g -expectation gg N  xx0 g g  is bounded by the corresponding Choquet expectation, that is Therefore 0 N 0   I 1dxI d.x  I 1dxI  d.x (11) gg N  xx0 g gg xx0 g  2 N Step 3. For any  LP ,,  . let  I  N , Proof: The proof is done in three steps. N   Step 1. We show that if   0 is bounded by N >0, then   N . By Step 2, 0 N then inequality (11) holds.  N I 1dxI d.x In fact, for the fixed N , denote  n by gg N  xx0 g

21n  iN Letting N , it follows that  n :. I i0 n iN iN1  0  2     nn  I 1dxI d.x 22 gg  xx0 g n 2 Then , n  in LP,, . The proof is complete. Moreover,  n can be rewritten as Proof of Theorem 4: If the g -expectation g  is

2n N a coherent risk measure, then it is easy to check that the  n  I . i1 n iN g -expectation   satisfies the conditions of Lemma 2   g 2n 5.

But by the assumptions (1) and (2) in this lemma, we Let VA   g  IA  A  . By Lemma 5 and the de- have finition of Choquet expectation, we have gV    . The first part of this theorem is complete. 2n N Counterexample 1 shows that this property of coherent  n  I gg  n iN risk measures fails in general for more general convex i1 2   2n  (12) risk measures. This completes the proof of Theorem 4. 2n  N Counterexample 1 Suppose that Wt  is 1-dimen-   I .   n g iN sional Brownian motion (i.e. d = 1). Let gz  z 1  i1 2   2n  where x  maxx ,0 . Then g  is a convex risk

1 Secondly we prove that measure. Let   I and   2I . Then 1 WT 1 2 WT 1 2 2>2II. gg WWTT11  g1  V1 However gV2 >2. Here the  capacity V in the Choquet expectation V  is given Let YZtt,  be the solution of the BSDE by VA gIA.  TTZ Proof of the Inequality in Counterexample 1: The YI221ds  sZWd. (16)  tssWT 1  convex function gz  z 1  satisfies (H1), (H2) tt2 and(H3). Thus, by Corollary 2, g -expectation   is g   Obviously, a convex risk measure. This together with the property of  Choquet expectation in Remark 1 implies YZZtsTT s IW 1  1dsW ds , 22T tt2 11  gg1 II g 22WWTT11 YZts which means ,  is the solution of BSDE 11 22 VVIIWW11   V1 . 22TT  TT yI z1d s zW d . tsWT 1 tt ss Moreover, since d  1 by Corollary 3, g  is a convex risk measure rather than a coherent risk measure. But yI  . Thus by the uniqueness of We now prove that >  In fact, since tg WT 1 t g2 V2 Y the solution of BSDE, t  I . On the 22III 2, 2 gWT 1 t VVgWWWTT11 T1 other hand, let  ytt, z  be the solution of the BSDE we only need to show TT yI21 z d szWd. (17) 2>2II. tsWT 1 ttss ggWWTT11  Comparing BSDE(17) with BSDE (16), notice (15) Let y, z be the solution of the BSDE and the fact TT   yI21 zd szWd. (14)   tsWT 1  ss  z g   tt z 12  1 and gV   First we prove that 2 whenever z >1. By the strict comparison theorem of LP t,0,: T zt >1>0, (15) BSDE, we have ytt>,Yt  0,. T where L is Lebesgue measure on 0,T  . Setting t  0 , thus If it is not true, then z  1 a.e. t 0,T and BSDE t   2>2III 2. (14) becomes ggVWWTT11  WT1 T The proof is complete. yI2d zW. tsWT 1 t s Remark 3 In mathematical finance, coherent and Thus convex risk measures and Choquet expectation are used in the pricing of contingent claim. Theorem 4 shows that yEI22 EI. coherent pricing is always less than Choquet pricing, ttWT 1 WWTt1 W t t while Counterexample 1 demonstrates that pricing by a By the Markov property, convex risk measure no longer has this property. In fact the convex risk price may be greater than or less than the yPWWWWtTttt21  . Choquet expectation.

Recall that WWTt and Wt are independent and

WWNTt 0,Tt. Thus 4. Summary  yyy 2d , Coherent risk measures are a generalization of mathe- t   xW 1x t matical expectations, while convex risk measures are a where  x is the density function of the normal generalization of coherent risk measures. In the frame- distribution NTt0,  . Thus zDyttt21  W t , work of g -expectation, the summary of our results is where Dt is the Malliavin derivative. Thus zt can be given in Table 1. In that Table, the Choquet expectation greater than 1 whenever t is near 0 and Wt is near 0. is VA  : g  IA  . Thus (15) holds, which contradicts the assumption Counterexample 1 shows that convex risk may be  zt  1 a.e. tT0,  . or  Choquet expectation. Only in the case of coherent

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