Portfolio Size in Stochastic Portfolio Networks Using Digital Portfolio Theory ()
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 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.
2. Expectations and Risk Measures
In this section, we briefly recall the definitions of g-expectation, Choquet expectation, coherent and convex risk measures.
2.1. 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 
and let 
be a 
-dimensional standard Brownian motion defined on a completed probability space
. Suppose 
is the natural filtration generated by
, that is 
We also assume
. Denote
Let 
satisfy
(H1) For any 
is a continuous progressively measurable process with
.
(H2) There exists a constant 
such that for any
(H3) 
In Section 3, Corollary 3 we will consider a special case of 
with
.
Under the assumptions of (H1) and (H2), Pardoux and Peng [6] showed that for any
, the BSDE
(1)
has a unique pair solution
.
Using the solution 
of BSDE (1), which depends on
, Peng [5] introduced the notion of g-expectations.
Definition 1 Assume that (H1), (H2) and (H3) hold on g and
. Let 
be the solution of BSDE (1).
defined by 
is called the g-expectation of the random variable
.
defined by 
is called the conditional g-expectation of the random variable
.
Peng [5] also showed that g-expectation 
and conditional 
-expectation 
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
.
1) Preservation of constants: For any constant 
.
2) Monotonicity: If
, then
.
3) Strict monotonicity: If
, and
, then
.
4) Consistency: For any
,
.
5) If 
does not depend on
, and 
is 
- measurable, then
In particular,
.
6) Continuity: If 
as 
in
, then
.
The following lemma is from Briand et al. [9, Theorem 2.1]. We can rewrite it as follows.
Lemma 2 (Briand et al. ) Suppose that 
is of the form 
where 
is a continuous bounded process. Then
where the limit is in the sense of
.
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 Choquet expectation.
Definition 2
1) A real valued set function 
is called a capacity if a) 
b)
, whenever 
and
.
2) Let 
be a capacity. For any
, the Choquet expectation 
is defined by
Remark 1 A property of Choquet expectation is positive homogeneity, i.e. for any constant 
2.3. 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
.
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:
,
;
2) Positive homogeneity: 
for all real number 
3) Monotonicity: 
whenever 
4) Translation invariance: 
for 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:
,
;
2) Normality:
;
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
.
Definition 5 A dynamic risk measure
is a random functional which depends on
, such that for each 
it is a risk measure. If 
satisfies for each 
the conditions in Definition 3, we say 
is a dynamic coherent risk measure. Similarly if 
satisfies for each 
the conditions in Definition 4, we say 
is a dynamic convex risk measure.
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 
and 
satisfy (H1), (H2) and (H3). Then the following conclusions are equivalent.
1) For any 
2) For any 
(2)
Proof: We first show that inequality (2) implies inequality 3).
Let 
and 
be the solutions of the following BSDE corresponding to the terminal value 
and 
and the generator 
and
, respectively
(3)
Then 
For fixed
, consider the BSDE
(4)
It is easy to check that 
is the solution of the BSDE (4).
Comparing BSDEs (4) and (3) with 
and
, assumption (2), (2) then yields
Applying the comparison theorem of BSDE in Peng [5], we have 
Taking
, thus by the definition of 
-expectation, the proof of this part is complete.
We now prove that inequality (1) implies (2). We distinguish two cases: the former where 
does not depend on
, the latter where 
may depend on
.
Case 1, 
does not depend on
. The proof of this case 1 is done in two steps.
Case 1, Step 1: We now show that for any
, we have
Indeed, for
, set
If for all
, we have 
then we obtain our result.
If not, then there exists 
such that
. We will now obtain a contradiction.
For this
,
That is
Taking g-expectation on both sides of the above inequality, and apply the strict monotonicity of 
-expectation in Lemma 1 (3), it follows
But by Lemma 1 (4) and (5),
Note that by Lemma 1(v)
Thus
This induces a contradiction, thus concluding the proof of this Step 1.
Case 1, Step 2: For any 
with 
and
, let us choose 
. Obviously, 
By Step 1,
Thus
Let 
applying Lemma 2, since g does not depend on 
we rewrite 
simply as 
thus 
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 
and 
define the stopping time
Obviously, if for each 
, for all 
then the proof is done. If it is not the case, then there exist 
and
such that
.
Fix 
and consider the following (forward) SDEs defined on the interval 
and
Obviously, the above equations admit a unique solution 
which is progressively measurable with
Define the following stopping time
It is clear that 
and note that 
whenever 
thus, 
Hence
.
Moreover, we can prove
In fact, setting
then
Thus 
It follows that on
, 
This implies
(5)
By the definition of 
and
, the pair processes 
and 
are the solutions of the following BSDEs with terminal values 
and
,
and
Hence,
and
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
, let
.
Then for any
, 
Proof: Letting
, then 
is the solution of BSDE
Since
the above BSDE can be rewritten as
(6)
Let
, then 
satisfies
(7)
Comparing with BSDE (6) and BSDE (7), by the uniqueness of the solution of BSDE, we have
Let 
then
. 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 
and
. These are given in the following Corollaries.
Corollary 1 The g-expectation 
is the classical mathematical expectation if and only if g does not depend on 
and is linear in
.
Proof: Applying Theorem 1, 
is linear if and only if 
is linear in
. By assumption (H3), that is 
for all
. Thus 
does not depend on
. The proof is complete.
Corollary 2 The 
-expectation 
is a convex risk measure if and only if 
does not depend on 
and is convex in
.
Proof: Obviously, 
-expectation 
is convex risk measure if and only if for any 
(8)
For a fixed
, let
Applying Lemma 3,
Inequality (8) becomes
Applying Theorem 1, for any
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.
The function 
is positively homogeneous in 
if for any
,
.
Corollary 3 The 
-expectation 
is a coherent risk measure if and only if 
does not depend on 
and it is convex and positively homogenous in
. In particular, if
, 
is of the form
with
.
Proof: By Corollary 2, the 
-expectation 
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 
-expectation 
is positively homogeneous if and only if 
is positively homogeneous (that is for all 
and 
if and only if for any
,
.
In particular, if
, notice the fact that 
is convex and positively homogeneous on
, and that 
does not depend on
. We write it as 
then
(9)
Note that
, 
, but
Thus from (9)
Defining
,
.
Obviously 
since the convexity of 
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 
with
. Thus the family of coherent risk measures is much smaller than the family of convex risk measures.
Jensen’s inequality for mathematical inequality is important 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
(10)
Lemma 4 [Chen et al. [15] Theorem 3.1] Let 
be a convex function and satisfy
, 
and
. Then 1) Jensen’s inequality (10) holds for 
-expectations if and only if 
does not depend on 
and is positively homogeneous in
;
2) If 
the necessary and sufficient condition for Jensen’s inequality (10) to hold is that there exist two adapted processes 
and 
such that
.
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 
is a static convex (coherent) risk measure, then the corresponding conditional g-expectation 
is dynamic convex (coherent) risk measure for each 
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 Jensen’s inequality.
Theorem 3 Suppose that 
is a convex risk measure. Then 
is a coherent risk measure if and only if 
satisfies Jensen’s inequality.
Proof: If 
is a convex risk measure, then by Corollary 2, 
is convex. Applying Lemma 4, 
satisfies Jensen’s inequality if and only if 
is positively homogenous. By Corollary 2, the corresponding 
is coherent risk measure. The proof is complete.
Theorem 4 and Counterexample 1 below give the relation between risk measures and Choquet expectation.
Theorem 4 If 
is a coherent risk measure, then 
is bounded by the corresponding Choquet expectation, that is 
where
. If 
is a convex risk measure then inequality above fails in general. By construction there exists a convex risk measure and random variables 
and 
such that
The prove this theorem uses the following lemma.
Lemma 5 Suppose that 
does not depend on
. Suppose that the 
-expectation 
satisfies (1) 
(2) For any positive constant
,
Then for any 
the 
-expectation 
is bounded by the corresponding Choquet expectation, that is
(11)
Proof: The proof is done in three steps.
Step 1. We show that if 
is bounded by
, then inequality (11) holds.
In fact, for the fixed
, denote 
by
Then 
in 
Moreover, 
can be rewritten as
But by the assumptions (1) and (2) in this lemma, we have
(12)
Note that
and 
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
, then inequality (11) holds.
Let 
then 
Applying Step 1,
(13)
But by Lemma 1(v), 
On the other hand,
Thus by (13)
Therefore
Step 3. For any 
let 
then
. By Step 2,
Letting
, it follows that
The proof is complete.
Proof of Theorem 4: If the 
-expectation 
is a coherent risk measure, then it is easy to check that the 
-expectation 
satisfies the conditions of Lemma 5.
Let 
. 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.
Counterexample 1 Suppose that 
is 1-dimensional Brownian motion (i.e. d = 1). Let 
where
. Then 
is a convex risk measure. Let 
and 
Then
However 
Here the capacity 
in the Choquet expectation 
is given by 
Proof of the Inequality in Counterexample 1: The convex function 
satisfies (H1), (H2) and(H3). Thus, by Corollary 2, 
-expectation 
is a convex risk measure. This together with the property of Choquet expectation in Remark 1 implies
Moreover, since 
by Corollary 3, 
is a convex risk measure rather than a coherent risk measure. We now prove that 
In fact, since
we only need to show
Let 
be the solution of the BSDE
(14)
First we prove that
(15)
where 
is Lebesgue measure on 
If it is not true, then 
a.e. 
and BSDE (14) becomes
Thus
By the Markov property,
Recall that 
and 
are independent and
. Thus
where 
is the density function of the normal distribution
. Thus 
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.
.
Secondly we prove that
Let 
be the solution of the BSDE
(16)
Obviously,
which means 
is the solution of BSDE
But 
Thus by the uniqueness of the solution of BSDE, 
On the other hand, let 
be the solution of the BSDE
(17)
Comparing BSDE(17) with BSDE (16), notice (15) and the fact
and 
whenever z >1. By the strict comparison theorem of BSDE, we have 
Setting
, 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.
4. 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 
-expectation, the summary of our results is given in Table 1. In that Table, the Choquet expectation is
.
Counterexample 1 shows that convex risk may be 
or 
Choquet expectation. Only in the case of coherent
Table 1. Relations among coherent and convex risk measures
, choquet expectation and Jensen’s inequality.
risk there is an inequality relation with Choquet expectation.
NOTES
This work has been supported by grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada.