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Acta Mathematicae Applicatae Sinica, English Series
Vol. 25, No. 1 (2009) 11–20
DOI: 10.1007/s10255-007-7035-4www.Applmath.com.cn
Acta Mathema�ca ApplicataeSinica, English Series© The Editorial Office of AMAS & Springer-Verlag 2009
Moment Inequality and Holder Inequality for BSDEsSheng-jun Fan
College of School, China University of Mining and Technology, Xuzhou 221008, China (E-mail: f s [email protected])
Abstract Under the Lipschitz and square integrable assumptions on the generator g of BSDEs, this paper
proves that if g is positively homogeneous in (y, z) and is decreasing in y, then the Moment inequality for BSDEs
with generator g holds in general, and if g is positively homogeneous and sub-additive in (y, z), then the Holder
inequality and Minkowski inequality for BSDEs with generator g hold in general.
Keywords Backward stochastic differential equation; moment inequality for bsdes; holder inequality for
bsdes; minkowski inequality for BSDEs
2000 MR Subject Classification 60H10
1 Introduction
It is well known that there exists a unique square-integrable and adapted solution to a backwardstochastic differential equation (BSDE for short in the remaining of the paper) of the type
ys = ξ +∫ T
s
g(u, yu, zu)du −∫ T
s
zu · dBu, s ∈ [0, T ], (1.1)
provided that g is Lipschitz in both variables y and z, and that ξ and the process (g(t, 0, 0))t∈[0,T ]
are square integrable. The g is called the generator of the BSDE (1.1). We denote the uniquesolution by (yT,g,ξ
s , zT,g,ξs )s∈[0,T ], and often denote yT,g,ξ
t by Egt,T [ξ] for each t ∈ [0, T ] which is
a kind of nonlinear evaluation according to [11]. In particular, if for any y ∈ R, g(t, y, 0) = 0,
a.s., a.e., then yT,g,ξ0 is called g-expectation of ξ, often denoted by Eg[ξ], and yT,g,ξ
t is calledconditional g-expectation of ξ with respect to Ft, often denoted by Eg[ξ|Ft] (see [9,10] fordetails).
The notion of g-expectation can be considered as a nonlinear extension of the well-knownGirsanov transformation. The original motivation for studying g-expectation comes from thetheory of expected utility, which is the foundation of modern mathematical economics. Chenand Epstein [1] gave an application of g-expectation to recursive utility, Peng[11,12], Resazza[13]
and Jiang[4] investigated some applications of g-expectations to static and dynamic pricingmechanisms and risk measures.
Since the notion of g-expectation was introduced, many properties of g-expectation havebeen studied in [4,5,8,10,11]. In [2,7], the Jensen’s inequality of g-expectation is studied, [15]studied the Moment inequality of g-expectation and [14] investigated Holder inequality of g-expectation. Furthermore, [6] investigated the Jensen inequality for BSDEs and [3] studied thecontinuous property on solutions of BSDEs.
Manuscript received March 29, 2007. Revised January 28, 2008.Supported by the National Natural Science Foundation of China (No. 10671205, ) and Youth Foundation ofChina University of Mining and Technology (No. 2006A041, 2007A029).
12 S.J. Fan
Unfortunately, in [15], the authors made a mistake in the proof of their Theorem 3 whichis the main result of [15], when using the Comparison Theorem of BSDEs to compare thesolutions of BSDE (4) and BSDE (5). According to the result of this paper, we know thatTheorem 3 in [15] is wrong. And, in [14], the author did not consider the case of conditionalg-expectation, which is greatly different from that of g-expectation. Motivated by the aboveresults with respect to g-expectation, this paper will further investigate the following questionon BSDEs not necessarily confined to g-expectation:
What conditions should be given to the generator g such that the Moment inequality, theHolder inequality and the Minkowski inequality for BSDEs with generator g hold in generalrespectively?
Under the Lipschitz and the square integrable assumptions on the generator g of BSDEs,in the present work, using a method different from [14,15], we shall prove that if g is positivelyhomogeneous in (y, z) and is decreasing in y, then the Moment inequality for BSDEs withgenerator g holds in general, and if g is positively homogeneous and sub-additive in (y, z), thenthe Holder inequality and the Minkowski inequality for BSDEs with generator g hold in general.
2 Preliminaries
Let (Ω,F ,P) be a probability space carrying a standard d-dimensional Brownian motion(Bt)t≥0, and let (Ft)t≥0 be the σ-algebra generated by (Bt)t≥0. We always assume that (Ft)t≥0
is right continuous and complete.Let T > 0 be a given real number. In this paper, we always work in the space (Ω,FT ,P),
and only consider processes indexed by t ∈ [0, T ]. Let 1A denote the indicator of event A, andR+ and R− denote respectively the set of non negative and non positive real number. Fornotational simplicity, we use L2(FT ) = L2(Ω,FT ,P), L2(Ft) = L2(Ω,Ft,P) when t ∈ [0, T ].For each positive integer n, let |z| denote Euclidean norm of z ∈ Rn, and define the followingstandard spaces of processes:
S2 = {φ ∈ R : φ progressivelymeasurable; E[ sup0≤t≤T
|φ(t)|2] < +∞},
Hn2 =
{φ ∈ Rn : φ progressivelymeasurable; ‖φ(t)‖2
2 = E[∫ T
0
|φ(t)|2dt]
< +∞}.
The generator g of a BSDE is a function
g(ω, t, y, z) : Ω × [0, T ]× R × Rd → R
such that the process (g(t, y, z))t∈[0,T ] is progressively measurable for each (y, z) in R×Rd andg satisfies the following assumptions:
(A1). There exists a constant μ ≥ 0, such that, a.s., a.e., we have,
∀ (yi, zi) ∈ R1+d(i = 1, 2), |g(t, y1, z1) − g(t, y2, z2)| ≤ μ(|y1 − y2| + |z1 − z2|).
(A2). The process (g(t, 0, 0))t∈[0,T ] belongs to H12.
Let g satisfy (A1) and (A2), by a result of Pardoux E. and Peng S.[9], for each ξ ∈ L2(FT ),there exists a unique pair of adapted processes in H1
2 ×Hd2, say (yT,g,ξ
s , zT,g,ξs )s∈[0,T ] that solves
the BSDE (1.1). In this paper, we denote yT,g,ξt by Eg
t,T [ξ] for each t ∈ [0, T ]. Actually, by someclassical results in this field, the process {Eg
t,T [ξ]}t∈[0,T ] belongs to S2.
Moment Inequality and Holder Inequality for BSDEs 13
Since nonlinear BSDEs were introduced by Peng S., many properties of BSDEs have beenstudied in [3,6,10,11]. The following two propositions will be used frequently in this paper, onecan find their proofs in [3,10,11].
Proposition 2.1 (Comparison Theorem). Let both g and g′ satisfy (A1) and (A2) and(ξ, ξ′) ∈ L2(FT )×L2(FT ). Suppose (yu, zu)u∈[0,T ] and (y′
u, z′u)u∈[0,T ] respectively be the uniquesolution to the BSDE in (1.1) and the following BSDE
y′u = ξ′ +
∫ T
u
g′(s, y′s, z
′s)ds −
∫ T
u
z′s · dBs, u ∈ [0, T ].
If P − a.s., ξ ≥ ξ′ and a.s., a.e., g(s, y′s, z
′s) ≥ g′(s, y′
s, z′s), then for each t ∈ [0, T ],
yt ≥ y′t, P− a.s..
Furthermore, if P(ξ > ξ′) > 0, then for each t ∈ [0, T ], we have yt > y′t, P − a.s..
Remark 2.1. From the Comparison Theorem, we know that if g(t, 0, 0) ≥ 0 a.s., a.e., thenfor each 0 ≤ s ≤ t ≤ T and each constant c > 0, Eg
s,t[c] > Egs,t[0] ≥ 0, P − a.s., Thus, (4.1),
(4.2) and (4.3) in Section 4 are all well defined.The following Continuity Property on solutions of BSDEs comes from Theorem 1 in [3].
Proposition 2.2 (Continuity Property). Let the generator g satisfy the assumptions (A1)and (A2), let t ∈ [0, T ] and ξn ∈ L2(Ft), n ∈ N. If
limn→∞ ξn = ξ, P − a.s., |ξn| ≤ η, P− a.s.
and Eη2 < +∞, then for all s ∈ [0, t], we have
limn→∞ Eg
s,t[ξn] = Eg
s,t[ limn→∞ ξn] = Eg
s,t[ξ], P − a.s..
3 Technical Results
In this section, we will give three technical results, which play a key role in the proof of ourmain results. Before that, let us first introduce the following natural and reasonable definition.
Definition 3.1. Let the generator g satisfy the assumptions (A1) and (A2). We say that g
is positively homogeneous in (y, z), if g also satisfies that for each (y, z, λ) ∈ R × Rd × R+,
g(t, λy, λz) = λg(t, y, z), a.s., a.e.. (3.1)
We say that g is sub-additive (resp. super-additive) in (y, z), if g also satisfies that for each(yi, zi) ∈ R × Rd (i = 1, 2),
g(t, y1 + y2, z1 + z2) ≤ g(t, y1, z2) + g(t, y1, z2), a.s., a.e., (resp. ≥). (3.2)
We say g is decreasing (resp. increasing) in y, if g also satisfy that for each (y, z, c) ∈R × Rd × R−, we have
g(t, y + c, z) ≥ g(t, y, z), a.s., a.e., (resp. ≤). (3.3)
14 S.J. Fan
By virtue of the positive homogeneity of the generator and the existence and uniquenesstheorem of BSDEs, the following Lemma 3.1 is obvious.
Lemma 3.1. Let the generator g satisfy the assumptions (A1) and (A2). If g is positivelyhomogeneous in (y, z), then for each 0 ≤ s ≤ t ≤ T and each (ξ, η) ∈ L2(Ft) × L2(Fs) withη ≥ 0 and bounded, we have
Egs,t[ηξ] = ηEg
s,t[ξ], P − a.s.. (3.4)
Making use of the existence and uniqueness theorem of BSDEs and the Comparison Theo-rem, one can also prove the following two lemmas. We only prove Lemma 3.3 here, the proofof Lemma 3.2 is similar.
Lemma 3.2. Let the generator g satisfy the assumptions (A1) and (A2). If g is sub-additive(resp. super-additive) in (y, z), then for each 0 ≤ s ≤ t ≤ T and each (ξ, η) ∈ L2(Ft)×L2(Ft),we have
Egs,t[ξ + η] ≤ Eg
s,t[ξ] + Egs,t[η], P − a.s., (resp. ≥). (3.5)
Lemma 3.3. Let the generator g satisfy the assumptions (A1) and (A2). If g is decreasing(resp. increasing) in y, then for each 0 ≤ s ≤ t ≤ T and each (ξ, η) ∈ L2(Ft) × L2(Fs) withη ≤ 0, we have
Egs,t[ξ + η] ≥ Eg
s,t[ξ] + η, P − a.s., (resp. ≤). (3.6)
Proof. We shall prove only the decreasing case, the increasing case is similiar. Suppose thatg is decreasing in y. Given 0 ≤ s ≤ t ≤ T and (ξ, η) ∈ L2(Ft) × L2(Fs) with η ≤ 0 andbounded, let (yu, zu)u∈[s,t] and (y′
u, z′u)u∈[s,t] be the unique adapted solutions to the followingBSDEs respectively:
yu = ξ +∫ t
u
g(r, yr, zr)dr −∫ t
u
zr · dBr, u ∈ [s, t],
y′u = ξ + η +
∫ t
u
g(r, y′r, z
′r)dr −
∫ t
u
z′r · dBr, u ∈ [s, t]. (3.7)
Then we have,
yu + η =ξ + η +∫ t
u
g(r, yr, zr)dr −∫ t
u
zr · dBr,
=ξ + η +∫ t
u
g(r, yr + η, zr)dr −∫ t
u
zr · dBr, u ∈ [s, t]. (3.8)
where, ∀(r, y, z) ∈ [s, t] × R1+d, the process g(r, y, z) := g(r, y − η, z), P − a.s.. Obviously,g satisfy the assumptions (A1) and (A2), then according to the existence and uniqueness ofsolution to the following BSDE (3.9) along with η ∈ L2(Fs) and (3.8), we can conclude that(yu + η, zu)u∈[s,t] is the unique adapted solution (yu, zu)u∈[s,t] to BSDE (3.9):
yu = ξ + η +∫ t
u
g(r, yr, zr)dr −∫ t
u
zr · dBr, u ∈ [s, t]. (3.9)
Since g is decreasing in y and η ≤ 0, from (A1) and the definition of g, we can get
g(r, yr, zr) = g(r, yr + η, zr) ≥ g(r, yr, zr) = g(r, yr, zr), a.s., a.e. in [s, t] × Ω.
Moment Inequality and Holder Inequality for BSDEs 15
Thus bythe Comparison Theorem, comparing the solutions to BSDE (3.7) and BSDE (3.9), wecan conclude that
Egs,t[ξ + η] = y′
s ≥ ys = ys + η = Egs,t[ξ] + η, P − a.s..
Thus, we have (3.6) and the proof is complete.
Remark 3.3. Furthermore, we can prove that the converse propositions of Lemma 3.1,Lemma 3.2 and Lemma 3.3 are also true.
4 Main Results
In this section, we will state and prove our main results. First, let us introduce the followingdefinition.
Definition 4.1. Suppose the generator g satisfies (A1) and (A2). We say that the Momentinequality for BSDEs with generator g holds in general, if for each 1 ≤ p ≤ q, 0 ≤ s ≤ t ≤ T
and each |ξ|q ∈ L2(Ft), then(Eg
s,t[|ξ|p]) 1
p ≤ (Egs,t[|ξ|q]
) 1q , P− a.s.. (4.1)
We say that the Holder inequality for BSDEs with generator g holds in general, if for eachp ≥ 1, q ≥ 1 such that 1/p + 1/q = 1, 0 ≤ s ≤ t ≤ T and each (|ξ|p, |η|q) ∈ L2(Ft) × L2(Ft),then we have
Egs,t[|ξη|] ≤
(Egs,t[|ξ|p]
) 1p · (Eg
s,t[|η|q]) 1
q , P − a.s.. (4.2)
We say that the Minkowski inequality for BSDEs with generator g holds in general, if foreach p ≥ 1, 0 ≤ s ≤ t ≤ T and each (|ξ|p, |η|p) ∈ L2(Ft) × L2(Ft), we have
(Egs,t[|ξ + η|p]) 1
p ≤ (Egs,t[|ξ|p]
) 1p +
(Egs,t[|η|p]
) 1p , P − a.s.. (4.3)
The following Theorem 4.1 gives a sufficient condition on g under which the Moment in-equality for BSDEs with generator g holds in general, which is the first main result of thispaper.
Theorem 4.1. Let the generator g satisfy (A1) and (A2). If g is positively homogeneous in(y, z) and is decreasing in y, then the Moment inequality for BSDEs with generator g holds ingeneral.
Proof. Applying Taylor’s formula to the function ϕ(x) = xk with k ≥ 1, we have
∀ x, y ≥ 0, xk ≥ yk + kyk−1(x − y) = kyk−1x + (1 − k)yk. (4.4)
For each 0 ≤ s ≤ t ≤ T and η ∈ L2(Ft) with |η|k ∈ L2(Ft), since g is positively homogeneousin (y, z), from the Comparison Theorem (Proposition 2.1), we know that Eg
s,t[|η|] ≥ Egs,t[0] = 0.
Thus take x = |η|, y = Egs,t[|η|] in (4.4), we have
|η|k ≥ k(Egs,t[|η|])k−1 · |η| + (1 − k)(Eg
s,t[|η|])k, P − a.s..
For each n ∈ N, we define Ωn = {ω : Egs,t[|η|] ≤ n}, and denote the indicator function of Ωn by
1Ωn . From the Comparison Theorem (Proposition 2.1), we know that Egs,t[·] is nondecreasing,
so we have
Egs,t[1Ωn |η|k] ≥ Eg
s,t[1Ωnk(Egs,t[|η|])k−1 · |η| + 1Ωn(1 − k)(Eg
s,t[|η|])k], P − a.s.. (4.5)
16 S.J. Fan
Since g is decreasing in y, from (4.5) and Lemma 3.3, in view of 1Ωn(1− k)(Egs,t[|η|])k ∈ L2(Fs)
with valued in R−, we know that
Egs,t[1Ωn |η|k] ≥ Eg
s,t[1Ωnk(Egs,t[|η|])k−1 · |η|] + 1Ωn(1 − k)(Eg
s,t[|η|])k, P − a.s.. (4.6)
Furthermore, since g is positively homogeneous in (y, z), from Lemma 3.1, in view of 1Ωn
k(Egs,t[|η|])k−1 ∈ L2(Fs) with bounded and valued in R+, we know that, P − a.s.,
Egs,t[1Ωnk(Eg
s,t[|η|])k−1 · |η|] = 1Ωnk(Egs,t[|η|])k−1 · Eg
s,t[|η|] = 1Ωnk(Egs,t[|η|])k.
Thus, from (4.6), we have
Egs,t[1Ωn |η|k] ≥ 1Ωn(Eg
s,t[|η|])k, P − a.s.. (4.7)
Finally, due to the Continuity Property of the solutions of BSDEs (Proposition 2.2), in view ofthe definition of 1Ωn and |η|k ∈ L2(Ft), by letting n → ∞ in (4.7), we can obtain that, for each0 ≤ s ≤ t ≤ T and each η ∈ L2(Ft) such that |η|k ∈ L2(Ft),
Egs,t[|η|k] ≥ (Eg
s,t[|η|])k
, P − a.s.. (4.8)
For each 1 ≤ p ≤ q, let k = q/p and η = |ξ|p in (4.8), we have
Egs,t[|ξ|q] ≥
(Egs,t[|ξ|p]
) qp , P− a.s. (4.9)
With the inequality given in (4.9) in hand, we know that the Moment inequality (4.1) for BSDEswith the prescribed generator g does hold in general. The proof is complete.
Example 4.1. Let the generator g(t, y, z) = k(t) · |z| for each bounded adapted process k(t)or let
g(t, y, z) ={
ay + c|z|, y ≤ 0;by + c|z|, y > 0,
∀ (a, b, c) ∈ R− × R− × R.
According to Proposition 5.1, we know that the Moment inequality for BSDEs with the pre-scribed generator g holds in general.
Remark 4.1. The condition that g is decreasing in y in Theorem 4.1 cannot be lifted. Forexample, for the generator g(t, y, z) ≡ y which is positively homogeneous in (y, z) but is notdecreasing in y, making use of Ito’s formula, we can obtain that, for each 0 ≤ s ≤ t ≤ T andeach ξ ∈ L2(Ft),
Egs,t[ξ] = e(t−s)E[ξ|Fs].
Thus, let p = 1, q = 2 and ξ ≡ 1, we have
∀ 0 ≤ s < t ≤ T, (Egs,t[|ξ|p])
1p = e(t−s) > e
(t−s)2 = (Eg
s,t[|ξ|q])1q .
It follows from the above inequality that the Moment inequality for BSDEs with the prescribedgenerator g cannot hold in general.
Remark 4.2. The condition that g is positively homogeneous in (y, z) in Theorem 4.1cannot be eliminated in general. For example, for the generator g(t, y, z) ≡ −y + 2 which isdecreasing in y but is not positively homogeneous in (y, z), making use of Ito’s formula, we canobtain that for each 0 ≤ s ≤ t ≤ T and each ξ ∈ L2(Ft),
Egs,t[ξ] = e−(t−s)(E[ξ|Fs] − 2) + 2.
Moment Inequality and Holder Inequality for BSDEs 17
Thus, let p = 1, q = 2, t − s = 2 and ξ ≡ 3, we have
∀ 0 ≤ s < t ≤ T, (Egs,t[|ξ|p])
1p = e−2 + 2 > (7e−2 + 2)
12 = (Eg
s,t[|ξ|q])1q .
It follows from the above inequality that the Moment inequality for BSDEs with the prescribedgenerator g cannot hold in general.
The following Theorem 4.2 gives a sufficient condition on g under which the Holder inequalityand the Minkowski inequality for BSDEs with generator g hold in general, which is the secondmain result of this paper.
Theorem 4.2. Let the generator g satisfy (A1) and (A2). If g is positively homogeneousand sub-additive in (y, z), then the Holder inequality and the Minkowski inequality for BSDEswith generator g hold in general.
Proof. The proof is divided into three steps.First step: we will prove that (4.2) holds true when P−a.s., Eg
s,t[|ξ|p] > 0 and Egs,t[|η|q] > 0.
Applying Taylor’s formula to the bivariate function ϕ(x, y) = xαy1−α with α ∈ [0, 1], wehave, for each (x, y) ∈ R+ × R+ and each x0 > 0 and y0 > 0,
xαy1−α ≤xα0 y1−α
0 + αxα−10 y1−α
0 (x − x0) + (1 − α)xα0 y−α
0 (y − y0)
=αxα−10 y1−α
0 x + (1 − α)xα0 y−α
0 y. (4.10)
For each 0 ≤ s ≤ t ≤ T and (X, Y ) ∈ L2(Ft) × L2(Ft) such that Egs,t[|X |] > 0 and
Egs,t[|Y |] > 0, P− a.s., let x = |X |, y = |Y | and x0 = Eg
s,t[|X |], y0 = Egs,t[|Y |] in (4.4), we have,
P − a.s.,
|X |α|Y |1−α ≤ α(Egs,t[|X |])α−1(Eg
s,t[|Y |])1−α|X | + (1 − α)(Egs,t[|X |])α(Eg
s,t[|Y |])−α|Y |.
For each n ∈ N, we define
Ωn = {ω : 1/n ≤ Egs,t[|X |] + Eg
s,t[|Y |] ≤ n},
and denote the indicator function of Ωn by 1Ωn . From the Comparison Theorem (Proposition2.1), we know that Eg
s,t[·] is non-decreasing. Thus, noting that |X |α|Y |1−α ≤ α|X |+(1−α)|Y | ∈L2(Ft), we have, P− a.s.,
Egs,t[1Ωn |X |α|Y |1−α] ≤Eg
s,t[1Ωnα(Egs,t[|X |])α−1(Eg
s,t[|Y |])1−α|X | (4.11)
+ 1Ωn(1 − α)(Egs,t[|X |])α(Eg
s,t[|Y |])−α|Y |].
Since g is sub-additive in (y, z), from (4.11) and Lemma 3.2, in view of the definition of 1Ωn ,we have, P− a.s.,
Egs,t[1Ωn |X |α|Y |1−α] ≤Eg
s,t[1Ωnα(Egs,t[|X |])α−1(Eg
s,t[|Y |])1−α|X |]+ Eg
s,t[1Ωn(1 − α)(Egs,t[|X |])α(Eg
s,t[|Y |])−α|Y |]. (4.12)
Furthermore, since g is positively homogeneous in (y, z), from (4.12) and Lemma 3.1, in viewof the definition of 1Ωn with Eg
s,t[|X |] > 0 and Egs,t[|Y |] > 0, P − a.s., we know that, P− a.s.,
Egs,t[1Ωn |X |α|Y |1−α] ≤1Ωnα(Eg
s,t[|X |])α−1(Egs,t[|Y |])1−α · Eg
s,t[|X |]+ 1Ωn(1 − α)(Eg
s,t[|X |])α(Egs,t[|Y |])−α · Eg
s,t[|Y |]=1Ωn(Eg
s,t[|X |])α(Egs,t[|Y |])1−α. (4.13)
18 S.J. Fan
Due to the Continuous Property of the solutions of BSDEs (Proposition 2.2), in view of thedefinition of 1Ωn with Eg
s,t[|X |] > 0 and Egs,t[|Y |] > 0, P − a.s. again as well as |X |α|Y |1−α ∈
L2(Ft). Taking limits by letting n → ∞ in (4.13), we can obtain that for each 0 ≤ s ≤ t ≤ T
and (X, Y ) ∈ L2(Ft) × L2(Ft) such that Egs,t[|X |] > 0 and Eg
s,t[|Y |] > 0, P − a.s.,
Egs,t[|X |α|Y |1−α] ≤ (Eg
s,t[|X |])α(Egs,t[|Y |])1−α, P − a.s.. (4.14)
Thus, for each 1 ≤ p ≤ q such that 1/p+ 1/q = 1, let α = 1/p, X = |ξ|p and Y = |η|q in (4.14),we have, for each 0 ≤ s ≤ t ≤ T and each (|ξ|p, |η|q) ∈ L2(Ft) × L2(Ft) such that Eg
s,t[|ξ|p] > 0and Eg
s,t[|η|q] > 0, P − a.s.,
Egs,t[|ξη|] ≤ (Eg
s,t[|ξ|p])1p · (Eg
s,t[|η|q])1q , P − a.s.. (4.15)
Second step: We prove that (4.2) holds in general. In fact, for each 0 ≤ s ≤ t ≤ T , each(|ξ|p, |η|q) ∈ L2(Ft) × L2(Ft) and n ∈ N, since for each (x, y) ∈ R+ × R+,
(x + y)p ≤ 2p−1(xp + yp) and (x + y)q ≤ 2q−1(xq + yq), (4.16)
we have (|ξ+1/n|p, |η+1/n|q) ∈ L2(Ft)×L2(Ft). Furthermore, by Remark 2.1, since g(t, 0, 0) =0, a.s., a.e., we can also infer that
Egs,t[(|ξ| + 1/n)p] ≥ Eg
s,t[(1/n)p] > Egs,t[0] = 0, P − a.s.
andEg
s,t[(|η| + 1/n)q] ≥ Egs,t[(1/n)q] > Eg
s,t[0] = 0, P − a.s..
Thus according to (4.15), we have, P − a.s.,
Egs,t[(|ξ| + 1/n) · (|η| + 1/n)] ≤ (Eg
s,t[(|ξ| + 1/n)p]) 1
p · (Egs,t[(|η| + 1/n)q]
) 1q . (4.17)
Finally, due again to the Continuity Property of the solutions of BSDEs (Proposition 2.2), byletting n → ∞ in (4.17), we know that the Holder inequality (4.2) for BSDEs with this kind ofgenerator g does hold in general.
Third step: In the sequel, we shall prove that the Minkowski inequality in (4.3) for BSDEswith the prescribed generator g holds also in general. In fact, for each p ≥ 1, 0 ≤ s ≤ t ≤ T
and each (|ξ|p, |η|p) ∈ L2(Ft) × L2(Ft), by (4.16), we know that |ξ + η|p ∈ L2(Ft). Thus, let1/p + 1/q = 1, making use of Lemma 3.2 and the Holder inequality (4.2) for BSDEs with thiskind of generator g, we can get, P − a.s.,
Egs,t[|ξ + η|p] ≤ Eg
s,t[|ξ||ξ + η|p−1] + Egs,t[|η||ξ + η|p−1]
≤(Egs,t[|ξ|p])1/p(Eg
s,t[|ξ + η|p])1/q + (Egs,t[|η|p])1/p(Eg
s,t[|ξ + η|p])1/q. (4.18)
Thus we have(Eg
s,t[|ξ + η|p]) 1p ≤ (Eg
s,t[|ξ|p])1p + (Eg
s,t[|η|p])1p , P− a.s.. (4.19)
The proof is complete.
Remark 4.3. We can prove directly that (4.2) holds in general without considering the firststep in which P − a.s., Eg
s,t[|ξ|p] > 0 and Egs,t[|η|q] > 0.
Moment Inequality and Holder Inequality for BSDEs 19
In fact, according to the classical results of BSDEs, we know that for each A ∈ Fs and eachζ ∈ Ft valued in R+ , if 1AEg
s,t[ζ] = 0, 1Aζ = 0. Then we have, P − a.s.,
limn→∞ 1Ωn(Eg
s,t[|X |])α(Egs,t[|Y |])1−α =1{Eg
s,t[|X|]>0}∪{Egs,t[|Y |]>0}(Eg
s,t[|X |])α(Egs,t[|Y |])1−α
=(Egs,t[|X |])α(Eg
s,t[|Y |])1−α, (4.20)
and
limn→∞ 1Ωn |X |α|Y |1−α =1{Eg
s,t[|X|]>0}∪{Egs,t[|Y |]>0}|X |α|Y |1−α
=|X |α|Y |1−α. (4.21)
Thus, with (4.20) and (4.21) in hand, from (4.13), making use of the Continuous Property ofsolutions of BSDEs, we know that (4.2) holds in general.
Example 4.2. Let the generator g(t, y, z) = k(t) · |z| for each bounded adapted process k(t)with k(t) ≥ 0 or let
g(t, y, z) ={
ay + c|z|, y ≤ 0;by + c|z|, y > 0,
∀ (a, b, c) ∈ R × R × R+.
According to Theorem 4.2, we know that the Holder inequality and the Minkowski inequalityfor BSDEs with this kind of generator g hold in general.
Remark 4.4. The conditions that g is positively homogeneous and sub-additive in (y, z)in Theorem 4.2 cannot be lifted. In fact, for BSDE with the generator g(t, y, z) ≡ y + 2and g(t, y, z) ≡ −|z| respectively, similar to Remark 4.1 and Remark 4.2, we can constructthe corresponding counterexamples which show that the Holder inequality and the Minkowskiinequality for BSDEs with this g do not hold in general.
Remark 4.5. For classical mathematical expectation, the Holder inequality imply the mo-ment inequality. But for BSDEs, the corresponding result is no longer true. In fact, BSDEwith the generator g(t, y, z) ≡ y in Remark 4.1 is just a counterexample. The main reason isthat Eg
s,t[1] is not necessarily to equal to 1.
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