4.4 Dual Representation and Inf-Convolution
4.4.3 Examples of inf-convolution
In this Section, we use the previous result on the inf-convolution of g-expectations to treat several particular examples.
4.4.3.1 Quadratic and Quadratic
We first study the inf-convolution of two dynamic entropic risk-measure. This example is treated by Barrieu and El Karoui [6] by a direct method, they find that the optimal risk transfer isproportional in the sense that there exists a∈(0,1)such that
(Eg1Eg2)(ξT) =Eg1(aξT) +Eg2((1−a)ξT).
We retrieve here this result using Theorem 4.4.2. For this, we first need to study the inf-convolution of the two corresponding generators gi,i= 1,2
gti(z, u) := 1
Lemma 4.4.1 Let g1 and g2 be the two convex generators defined in equation (4.4.9). For any bounded FT-measurable random variableξT, we have,
(Eg1Eg2)(ξT) =Eg1 The first infimum above is easy to calculate and is attained for
v∗ := γ1 γ1+γ2z.
For the second one, we postulate similarly that it should be attained for w∗:= γ1
γ1+γ2
u.
In order to verify this result, it is sufficient to prove that for all(x, y)∈R2 (γ1+γ2)
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Setλ:=γ1/(γ1+γ2),a:=x/γ1,b:=y/γ2 and h(x) :=ex−1−x, this is equivalent to h(λa+ (1−λ)b)≤λh(a) + (1−λ)h(b),
which is clear by convexity of the functionh.
Therefore, we finally obtain
Using the notations of Theorem 4.4.2, we can compute the quantity FT(2), giving the optimal risk transfer We calculate similarly FT(1) and obtain
FT(1) = γ1
Here, we assume that d= 1. We study the inf-convolution of a dynamic entropic risk-measure with a linear one corresponding to a linear BSDE. In this case, we want to calculate the inf-convolution of the two corresponding generatorsg1 and g2 given by
g1t(z, u) := 1
Lemma 4.4.2 Let g1 and g2 be defined in the two previous equations. We have, for any bounded FT-measurable random variableξT,
(Eg1Eg2)(ξT) =Eg1(FT(1)) +Eg2(FT(2)),
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where
Remark 4.4.3 Notice that FT(2) has no longer the linear form with respect to ξT obtained in the previous example. Now, the agent 2 receives the value ξT perturbed by a random value only depending on the data contained in the filtration, i.e the Brownian motion B and the random measures µ and (νt).
Proof. We start by computing the inf-convolution in(z, u) of the generators:
(g1g2)(t, z, u) = inf The first infimum above is easy to calculate and is attained for
v∗ :=αγ.
Similarly, it is easy to show that the functionw→γ
ewγ −1−wγ
Notice that all the quantities appearing in g1g2 are finite. Indeed, we first have for any u ∈ L2(ν)∩L∞(ν)
|β(1∧ |x|)u(x)| ≤β2(1∧ |x|2) + (u(x))2, (4.4.11) and this quantity is thereforeνt-integrable for allt. Then, since β≥ −1 +δ, it is also clear that for some constant Cδ >0
0≤(1 +β(1∧ |x|)) ln(1 +β(1∧ |x|))−β(1∧ |x|)≤Cδ(1∧ |x|2), (4.4.12) and thus the second integral is also finite.
We can now compute for instance the quantityFT(2), which is given after some easy calculations by FT(2)=−
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Recall that(Z, U) is part of the solution of the BSDE with generatorg1g2 and terminal condition
And finally, we can conclude that the optimal risk transfer takes the form FT(2)=ξT +1
4.4.3.3 Absolute value and Quadratic
Here, we study again the example of the entropic risk measure, i.e. we keep the generatorg1 given by
whereγ >0. The generatorg2 correspond to the second example of risk measure given in Examples 4.3.1, i.e.
These infima are easy to calculate and are attained at(v∗, w∗) defined by v∗=µγ1{z>µγ}−µγ1{z<−µγ}+z1{−µγ≤z≤µγ}
w∗(x) =ψ(η, x)1{u(x)>ψ(η,x)}+ψ(C1, x)1{u(x)≤ψ(C1,x)}+u(x)1{ψ(C1,x)<u(x)≤ψ(η,x)}. whereψ(r, x) :=γln (1 +r(1∧ |x|)). We finally obtain
(g1g2)(t, z, u) =A(z) +B(u),
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where
A(z) :=
µz−1
2µ2γ
1{z>µγ}−
µz+1 2µ2γ
1{z<−µγ}+ 1
2γz21{−µγ≤z≤µγ}
B(u) :=
Z
E
C1(1∧ |x|)(γ+u(x))−ψ(C1, x)(1 +C1(1∧ |x|))1{u(x)≤ψ(C1,x)}
νt(dx) +
Z
E
η(1∧ |x|)(γ+u(x))−ψ(η, x)(1 +η(1∧ |x|))1{u(x)>ψ(η,x)}
νt(dx) +
Z
E
γ
e
u(x)
γ −1−u(x) γ
1{ψ(C1,x)<u(x)≤ψ(η,x)}
νt(dx),
that is to say this inf-convolution is equal to a linear function outside a given set, and is quadratic inside the set (inz andu). Notice that all the quantities appearing in B(u)are finite, thanks to the inequalities (4.4.11) and (4.4.12) given in the previous example.
Now to have the optimal risk transfer, we calculateFT(1) andFT(2) using their definition. We obtain for example for FT(2)
FT(2) = Z T
0
(µ2γ−µZt)1{|Zt|>µγ}dt
− Z T
0
Z
E
h
η(1∧ |x|) ψ(η, x)1{Ut(x)>ψ(η,x)}+Ut(x)1{0<Ut(x)<ψ(η,x)}
+C1(1∧ |x|) ψ(C1, x)1{Ut(x)<ψ(C1,x)}+Ut(x)1{ψ(C1,x)<Ut(x)<0}i
νt(dx)dt +
Z T 0
Z
E
Ut(x)−ψ(η, x)1{Ut(x)>ψ(η,x)}+ψ(C1, x)1{Ut(x)≤ψ(C1,x)}
+Ut(x)1{ψ(C1,x)<Ut(x)≤ψ(η,x)}
µ(dt, dx),e
where(Z, U) is part of the solution of the BSDE with generatorg1g2 and terminal condition ξT. And we have of courseFT(1)=ξT −FT(2).
Remark 4.4.4 In this example, the optimal transfer has again no longer the proportional form found in example 1. Moreover, this time we are only able to calculate FT(1) and FT(2) in function of (Z, U), where, as defined before, (Et1,2(ξT), Zt, Ut) denotes the solution of the BSDE with generator g1g2 and terminal condition ξT.
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Model Uncertainty and 2BSDEs with Jumps
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Second Order Backward SDEs with Jumps
Contents
5.1 Introduction . . . 127 5.2 Issues related to aggregation. . . 129 5.2.1 The stochastic basis . . . 129 5.2.2 The main problem . . . 130 5.2.3 Characterization by martingale problems . . . 131 5.2.4 Notations and definitions . . . 133 5.3 Preliminaries on 2BSDEs. . . 142 5.3.1 The non-linear Generator . . . 142 5.3.2 The Spaces and Norms. . . 145 5.3.3 Formulation. . . 146 5.3.4 Connection with standard backward SDEs with jumps . . . 147 5.3.5 Connection withG-expectations andG-Lévy processes . . . 148 5.4 Uniqueness result . . . 150 5.4.1 Representation of the solution. . . 150 5.4.2 A priori estimates and uniqueness of the solution . . . 154
5.1 Introduction
Recall from the previous chapter that(Y, Z, U)is a solution of the BSDE with jumps, with generator F and terminal conditionξ if for allt∈[0, T], we have P−a.s.
Yt=ξ+ Z T
t
F(s, Ys, Zs, Us)ds− Z T
t
ZsdBs− Z T
t
Z
Rd\{0}
Us(x)(µ−ν)(ds, dx), (5.1.1) whereµis an integer valued random measure with compensator ν.
Our aim in this chapter is to generalize (5.1.1) to the second order, as introduced recently by Soner, Touzi and Zhang [112]. Their key idea in the definition of the second order BSDE is that the equation defining the solution has to holdP-almost surely, for everyPin a class of non dominated probability measures. Furthermore, they prove a uniqueness result using a representation result of the 2BSDEs
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as essential supremum of standard BSDEs. We will extend these definitions and properties to the case with jumps. For the sake of clarity, we will now briefly outline the main differences and difficulties due to the presence of jumps.
(i)First, we will introduce an equation similar to (5.1.1), for which the driverF depends also on the quadratic variation ofBcon the one hand, and on the compensator of the jump measure ofBdon the other hand, where BcandBd denote respectively the continuous and the purely discontinuous parts of B. Recall that the generator of standard BSDEs does not depend on the quadratic variation of the canonical process, and that in the continuous second order case it only depends on the quadratic variation of the Brownian martingale part. In our case, since we have an additional jump martingale term in the equation, we also have an additional dependence in the driver.
(ii) A second major difference with (5.1.1) in the second order case, is, as we recalled earlier, that the BSDE has to hold P-almost surely for every probability measureP lying in a wide family of probability measures. Under each P,Bc and Bd have a prescribed quadratic variation and jump measure compensator. This is why we can intuitively understand the second order BSDE with jumps (2BSDEJ in the sequel) (5.3.3) as a backward SDE with model uncertainty, where the uncertainty affects both the quadratic variation and the jump measure of the process driving the equation.
(iii)The last major difference with (5.1.1) in the second order case is the presence of an additional increasing process K in the equation. To have an intuition for K, one has to have in mind the representation (5.4.1) that we prove in Theorem 5.4.1, stating that the Y part of a solution of a 2BSDEJ is an essential supremum of solutions of standard backward SDEs with jumps. The processK maintains Y above any standard solutionyP of a BSDE with jumps, with given quadratic variation and jump measure under P. The process K is then formally analogous to the increasing process appearing in reflected BSDEs (as defined in [46] for instance).
There are many other possible approaches in the literature to handle volatility and / or jump measure uncertainty in stochastic models ([3], [39], [37], [14]). Among them, Peng [101] introduced a notion of Brownian motion with uncertain variance structure, called G-Brownian motion. This process is defined without making reference to a given probability measure. It refers instead to the G-Gaussian law, defined by a partial differential equation (see [102] for a detailed exposition and references).
Finally, recall that Pardoux and Peng [97] proved that if the randomness in g andξ is induced by the current value of a state process defined by a forward stochastic differential equation, the solution to the BSDE (4.2.1) could be linked to the solution of a semilinear PDE by means of a generalized Feynman-Kac formula. Similarly, Soner, Touzi and Zhang [112] showed that 2BSDEs generalized the point of view of Pardoux and Peng, in the sense that they are connected to the larger class of fully non linear PDEs. In this context, the 2BSDEJs are the natural candidates for a probabilistic solution of fully non linear integro-differential equations. Note that in a recent preprint, Bion-Nadal [13], by taking a dual point of view, connects the probabilistic construction of time consistent convex dynamic procedures in a non dominated setting, to a class of non local and fully non linear second order partial differential equations.
The rest of this chapter is organised as follows. In Section5.2, we introduce the set of probability
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measures on the Skorohod spaceDthat we will work with. Using the notion of martingale problems onD, we construct probability measures under which the canonical process has given characteristics.
Then we prove an aggregation result under this family. In Section 5.3, we define the notion of 2BSDEJs and show how it is linked to standard BSDEs with jumps. Section 5.4 is devoted to our uniqueness result and some a priori estimates.