Risk Measures and Nonlinear Expectations *

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 investigate 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.


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 expectations 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 probability spaces were introduced by Föllmer & Schied [3] and Frittelli & Rosazza Gianin [4].g-expectations were introduced 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 introduced the so called Choquet expectation.
Our interest in this paper is to explore the relations among risk measures and expectations.To do so, we restrict our attention of coherent and convex risk measures and Choquet expectations to the domain of g-expectations.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) implies coherent pricing is always less than Choquet pricing, but the pricing by a convex risk measure no longer has this property.We also study the relation between static and dynamic risk measures.We establish that if g-expectations are coherent (convex) risk measures, then the same is true for the corresponding conditional g-expectations or dynamic risk.In order to prove these results, we establish in Section 3, Theorem 1, a new converse comparison theorem of g-expectations.Jiang [8] studies gexpectation and shows that some cases give rise to risk measures.Here we are able to show, in the case of gexpectations, 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.

Expectations and Risk Measures
In this section, we briefly recall the definitions of g-expectation, Choquet expectation, coherent and convex risk measures.

g-Expectation
Peng [5] introduced g-expectation via a class of backward stochastic differential equations (BSDE).Some of the relevant definition and notation are given here.Fix be a -dimensional standard Brownian motion defined on a completed probability space is the natural filtration generated by   , that is , , : is measurable random variables with < , 0, ; (H2) There exists a constant such that for any 0 , , , g y z t g y z t    In Section 3, Corollary 3 we will consider a special case of with .
d  1 d  Under the assumptions of (H1) and (H2), Pardoux and Peng [6] showed that for any , the BSDE has a unique pair solution Using the solution t of BSDE (1), which depends on y  , Peng [5] introduced the notion of g-expectations.
 defined by : [5] also showed that g-expectation   g   and conditional g -expectation g t preserve most of basic properties of mathematical expectation, except for linearity.The basic properties are summarized in the next Lemma.

 
 , and , then   5) If g does not depend on , and In particular, 0 where the limit is in the sense of .

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 Choquet expectation.
Definition 2 1) A real valued set function , whenever and Let V be a capacity.For any , the Choquet expectation

Risk Measures
A risk measure is a map : , where  is interpreted as the "habitat" of the financial positions whose riskiness has to be quantified.In this paper, we shall consider .
Definition 3 A risk measure  is said to be coherent if it satisfies 1) Subadditivity: all real number  .
As an extension of coherent risk measures, Föllmer and Schied [3] introduced the axiomatic setting for convex risk measures.The following modifications of convex 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: 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 .

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.
and  be the solutions of the following BSDE corresponding to the terminal value It is easy to check that is the solution of the BSDE (4).
   We now prove that inequality (1) implies ( 2).We distinguish two cases: the former where g does not depend on , the latter where y g may depend on .y Case 1, g does not depend on .The proof of this case 1 is done in two steps.
, we have then we obtain our result.
  0, If not, then there exists   0, t T such that   > 0 t P A .We will now obtain a contradiction.
For this , t Taking g-expectation on both sides of the above inequality, and apply the strict monotonicity of g -expectation in Lemma 1 (3), it follows But by Lemma 1 (4) and ( 5), This induces a contradiction, thus concluding the proof of this Step 1.
Case 1, Step 2: For any applying Lemma 2, since g does not depend on we rewrite  , ,  g y z t simply as   , , 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     , , , y z y z t g y z t g y z t g y y z z t T , , ,  , for all ,  then the proof is done.If it is not the case, then there exist > 0  and , , , Obviously, the above equations admit a unique solution which is progressively measurable with Define the following stopping time  

Y t g Y t z z t g Y t z t g Y t z t t
By the definition of and , the pair processes Y the solutions of the following BSDEs with terminal values Applying the strict comparison theorem of BSDE and inequality (5), by the assumptions of this Theorem, we have This induces a contradiction.The proof is complete.Lemma 3 Suppose that g satisfies (H1), (H2) and (H3).For any constant Then for any the above BSDE can be rewritten as For a fixed , let 0,1 g y z t g y z t g y z t g y z t , , , .
which then implies that g is convex.By the explanation of Remark for Lemma 4.5 in Briand et al. [9], the convexity of g and the assumption (H3) imply that g does not depend on .The proof is complete. .
Obviously since the convexity of 0,  a g yields 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 particular, if d = 1, it is generated only by the family   We say that g -expectation satisfies Jensen's inequality if for any convex function :  , whenever , , , .
is coherent risk measure.The proof is complete.Theorem 4 and Counterexample 1 below give the relation between risk measures and Choquet expectation.
The prove this theorem uses the following lemma.Lemma 5 Suppose that g does not depend on .Suppose that the y g -expectation For any positive constant , Then for any the is bounded by the corresponding Choquet expectation, that is x (11) Proof: The proof is done in three steps.
Step 1.We show that if 0   is bounded by , then inequality (11) holds.

> 0 N
In fact, for the fixed , denote But by the assumptions (1) and (2) in this lemma, we have  Thus, taking limits on both sides of inequality (12), it follows that The proof of Step 1 is complete.
Step 2. We show that if  is bounded by , that is But by Lemma 1(v), On the other hand, Thus by ( 13) Step 3.For any The proof is complete.

Proof of Theorem 4:
By Lemma 5 and the definition of Choquet expectation, we have 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.
Here the capacity in the Choquet expectation is given by V    .
is a convex risk measure rather than a coherent risk measure.We now prove that In fact, since First we prove that where is Lebesgue measure on


Recall that and are independent and .Thus 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.

Summary
Coherent risk measures are a generalization of mathematical expectations, while convex risk measures are a generalization of coherent risk measures.In the framework of g -expectation, the summary of our results is given in

y 1 , Step 1 :
Case We now show that for any following (forward) SDEs defined on the interval   ,T  family of coherent risk measures is much smaller than the family of convex risk measures.0 a  Jensen's inequality for mathematical inequality is important in probability theory.Chen et al.[15] studied Jensen's inequality for g-expectation.
measure.This together with the property of Choquet expectation in Remark 1 implies

Lemma 2 (Briand et al. ) Suppose that  
following lemma is from Briand et al. [9, Theorem 2.1].We can rewrite it as follows.
risk measure then inequality above fails in general.By construction there exists a convex risk measure and random variables 1

.
In thatTable, the Choquet expectation Choquet expectation.Only in the case of coherent