We obtain a representation theorem for the generators of BSDEs driven by G-Brownian motions and then we use the representation theorem to get a converse comparison theorem for G-BSDEs and some equivalent results for nonlinear expectations generated by G-BSDEs.

1. Introduction

Let (Ω, ℱ, P) be a probability space, and, for fixed T ∈ [0, +∞), let (B_t) _0≤t≤T be a standard Brownian motion and let ℱ_t be the augmentation of σ{B_s, 0 ≤ s ≤ t}. Then Pardoux and Peng [1] introduced the backward stochastic differential equations (BSDEs) and proved the existence and uniqueness result of the BSDEs. In 1997, Peng [2] promoted g-expectations based on BSDEs. One of the important properties of g-expectations is comparison theorem or monotonicity. Chen [3] first considers a converse result of BSDEs under equal case. After that, Briand et al. [4] obtained a converse comparison theorem for BSDEs under general case. They also derived a representation theorem for the generator g. Following this paper, Jiang [5] discussed a more general representation theorem then, in his another paper [6], showed a more general converse comparison theorem. Here the representation theorem is an important method in solving the converse comparison problem and other problems (see Jiang [7]).

Peng [8–13] defined the G-expectations and G-Brownian motions (G-BMs) and proved the representation theorem of G-expectation by a set of singular probabilities, which differs from nonlinear g-expectations because g-expectations are equivalent with a group of absolutely continuous probabilities with respect to the probability measure P. Soner et al. [14] obtained an existence and uniqueness result of 2 BSDEs. Recently, Hu et al. [15] proved another existence and uniqueness result on BSDEs driven by G-Brownian motions (G-BSDEs).

An important advantage of G-BSDEs is the easiness to define the nonlinear expectations. Hu et al. in [16] gave a comparison theorem for G-BSDEs and talked about the properties of corresponding nonlinear expectations. In this paper, we consider the representation theorem for generators of G-BSDEs and then consider the converse comparison theorem of G-BSDEs and some equivalent results for nonlinear expectations generated by G-BSDEs. In the following, in Section 2, we review some basic concepts and results about G-expectations. We give the representation theorem of G-BSDEs in Section 3. In Section 4, we consider the applications of representation theorem of G-BSDEs, which contain the converse comparison theorem and some equivalent results for nonlinear expectations generated by G-BSDEs.

2. Preliminaries

We review some basic notions and results of G-expectation, the related spaces of random variables, and the backward stochastic differential equations driven by a G-Brownian motion. The readers may refer to [10, 13, 15, 17–19] for more details.

Definition 1. Let Ω be a given set and let ℋ be a vector lattice of real valued functions defined on Ω, namely, c ∈ ℋ for each constant c and |X | ∈ ℋ if X ∈ ℋ. ℋ is considered as the space of random variables. A sublinear expectation on ℋ is a functional satisfying the following properties: for all X, Y ∈ ℋ, one has

(a)
monotonicity: if X ≥ Y, then ;
(b)
constant preservation: ;
(c)
subadditivity: ;
(d)
positive homogeneity: for each λ ≥ 0. is called a sublinear expectation space.

Definition 2. Let X₁ and X₂ be two n-dimensional random vectors defined, respectively, in sublinear expectation spaces and . They are called identically distributed, denoted by , if , for all φ ∈ C_b·Lip(ℝⁿ), where C_b·Lip(ℝⁿ) denotes the space of bounded and Lipschitz functions on ℝⁿ.

Definition 3. In a sublinear expectation space , a random vector Y = (Y₁, …, Y_n), Y_i ∈ ℋ, is said to be independent of another random vector X = (X₁, …, X_m), X_i ∈ ℋ under , denoted by Y⊥X, if for every test function φ ∈ C_b·Lip(ℝ^m × ℝⁿ) one has .

Definition 4 (G-normal distribution). A d-dimensional random vector X = (X₁, …, X_d) in a sublinear expectation space is called G-normally distributed if for each a, b ≥ 0 one has

()

where

is an independent copy of X; that is,

and

. Here, the letter G denotes the function

()

where 𝕊_d denotes the collection of d × d symmetric matrices.

Peng [13] showed that X = (X₁, …, X_d) is G-normally distributed if and only if for each φ ∈ C_b·Lip(ℝ^d),

, (t, x) ∈ [0, ∞) × ℝ^d, is the solution of the following G-heat equation:

()

The function G(·) : 𝕊_d → ℝ is a monotonic, sublinear mapping on 𝕊_d and

implies that there exists a bounded, convex, and closed subset

such that

()

where

denotes the collection of nonnegative elements in 𝕊_d.

In this paper, we only consider nondegenerate G-normal distribution; that is, there exists some such that for any A ≥ B.

Definition 5. (i) Let denote the space of ℝ^d-valued continuous functions on [0, ∞) with ω₀ = 0 and let B_t(ω) = ω_t be the canonical process. Set

()

Let G : 𝕊_d → ℝ be a given monotonic and sublinear function. G-expectation is a sublinear expectation defined by

()

for all

, where ξ₁, …, ξ_n are identically distributed d-dimensional G-normally distributed random vectors in a sublinear expectation space

such that ξ_i+1 is independent of (ξ₁, …, ξ_i) for every i = 1, …, m − 1. The corresponding canonical process

is called a G-Brownian motion.

(ii) For each fixed t ∈ [0, ∞), the conditional G-expectation for , where without loss of generality we suppose t_i = t, is defined by

()

where

()

For each fixed T > 0, we set

()

For each p ≥ 1, we denote by

(resp.,

) the completion of L_ip(Ω) (resp., L_ip(Ω_T)) under the norm

. It is easy to check that

for 1 ≤ p ≤ q and

can be extended continuously to

For each fixed a ∈ ℝ^d,

is a 1-dimensional G_a-Brownian motion, where

and

. Let

, N = 1,2, …, be a sequence of partitions of [0, t] such that

; the quadratic variation process of B^a is defined by

()

For each fixed

, the mutual variation process of B^a and

is defined by

()

Definition 6. For fixed T > 0, let be the collection of processes in the following form: for a given partition {t₀, …, t_N} = π_T of [0, T],

()

where

, j = 0,1, 2, …, N − 1. For p ≥ 1, one denotes by

the completion of

under the norms

, respectively.

For each , we can define the integrals and for each a, . For each with p ≥ 1, we can define Itô′s integral .

Let . For p ≥ 1 and , set . Denote by the completion of under the norm .

We consider the following type of G-BSDEs (in this paper, we always use Einstein convention):

()

where

()

satisfy the following properties.

(H1)
There exists some β > 1 such that for any .
(H2)
There exists some L > 0 such that
()

For simplicity, we denote by the collection of processes (Y, Z, K) such that , , K is a decreasing G-martingale with K₀ = 0 and .

Definition 7. Let and f and g_ij satisfy (H1) and (H2) for some β > 1. A triplet of processes (Y, Z, K) is called a solution of (13) if for some 1 < α ≤ β the following properties hold:

(a)
;
(b)
.

Theorem 8 (see [15].)Assume that and f and g_ij satisfy (H1) and (H2) for some β > 1. Then, (13) has a unique solution (Y, Z, K). Moreover, for any 1 < α < β, one has , , and .

We have the following estimates.

Proposition 9 (see [15].)Let and f, g_ij satisfy (H1) and (H2) for some β > 1. Assume that for some 1 < α < β is a solution of (13). Then, there exists a constant C_α > 0 depending on α, T, G, L such that

()

where

Proposition 10 (see [15], [20].)Let α ≥ 1 and δ > 0 be fixed. Then, there exists a constant C depending on α and δ such that

()

Theorem 11 (see [16].)Let (Y^l, Z^l, K^l), l = 1,2, be the solutions of the following G-BSDEs:

()

where

, f and g_ij satisfy (H1) and (H2) for some β > 1 and

are RCLL processes in

such that

. If

is an increasing process, then

for t ∈ [0, T].

In this paper, we also need the following assumptions for G-BSDE (13).

(H3)
For each fixed (ω, y, z) ∈ Ω_T × ℝ × ℝ^d, t → f(t, ω, y, z) and t → g_ij(t, ω, y, z) are continuous.
(H4)
For each fixed (t, y, z) ∈ [0, T) × ℝ × ℝ^d, f(t, y, z), , and
()
(H5)
For each (t, ω, y) ∈ [0, T] × Ω_T × ℝ, f(t, ω, y, 0) = g_ij(t, ω, y, 0) = 0.

Assume that

; f and g_ij satisfy (H1), (H2), and (H5) for some β > 1. Let (Y^T,ξ, Z^T,ξ, K^T,ξ) be the solution of G-BSDE (13) corresponding to ξ, f, and g_ij on [0, T]. It is easy to check that

on [0, T] for T^′ > T. Following [16], we can define consistent nonlinear expectation

()

and set

3. Representation Theorem of Generators of G-BSDEs

We consider the following type of G-FBSDEs:

()

where h_ij = h_ji and g_ij = g_ji, 1 ≤ i, j ≤ d.

We now give the main result in this section.

Theorem 12. Let b : ℝⁿ → ℝⁿ, h_ij : ℝⁿ → ℝⁿ, and σ : ℝⁿ → ℝ^n×d be Lipschitz functions and let f and g_ij satisfy (H1), (H2), (H3), and (H4) for some β > 1. Then, for each (t, x, y, p) ∈ [0, T) × ℝⁿ × ℝ × ℝⁿ and α ∈ (1, β), one has

()

Proof. For each fixed (t, x, y, p) ∈ [0, T) × ℝⁿ × ℝ × ℝⁿ, we write (Y^ε, Z^ε, K^ε) instead of for simplicity. We have for each γ ≥ 1 (see [16, 19]). Thus, by Theorem 8, G-BSDE (22) has a unique solution (Y^ε, Z^ε, K^ε) and . We set, for s ∈ [t, t + ε],

()

Applying Itô’s formula to

on [t, t + ε], it is easy to verify that

solves the following G-BSDE:

()

From Proposition 9,

()

hold for some constant C_α > 0, only depending on α, T, G, and L. By Proposition 10 and the Lipschitz assumption, we obtain

()

where C₁ is a constant depending on x, y, p, α, β, T, G, and L. Noting that

(see [16, 19]), where C₂ depends on T and L, and the following inequality holds:

()

Together with assumption (H4), we get

()

where C₃ depends on x, y, p, α, β, T, G, and L. Now, we prove (23). Let us consider

()

where

()

It is easy to check that

, where C₄ depends on G, L, and T. Thus, by (29), we get

()

which implies

. We set

()

By the Lipschitz condition, we can get

, where C₅ depends on p, G, L, and T. Noting that

(see [16, 19]), where C₆ depends on L, G, and α, we obtain

()

which implies

. Now, we set

()

It is easy to deduce that

, where C₇ depends on G. Then,

()

Take limit on both sides of the above inequality and use assumption (H4); then, we have

()

On the other hand,

()

Then, we have

()

The proof is complete.

4. Some Applications

4.1. Converse Comparison Theorem for G-BSDEs

We consider the following G-BSDEs:

()

where

We first generalized the comparison theorem in [16].

Proposition 13. Let f^l and satisfy (H1) and (H2) for some β > 1, l = 1,2. If , then, for each , one has for t ∈ [0, T].

Proof. From the above G-BSDEs, we have

()

where

()

By the assumption, it is easy to check that (V_t) _t≤T is a decreasing process. Thus, using Theorem 11, we obtain

for t ∈ [0, T].

Remark 14. Suppose d = 1 and let f¹ = 10 | z|, f² = |z|, g¹ = |z|, and g² = 2 | z|. It is easy to check that f² − f¹ + 2G(g² − g¹) ≤ 0. Thus, does not imply f² ≤ f¹ and .

Now, we give the converse comparison theorem.

Theorem 15. Let f^l and satisfy (H1), (H2), (H3), (H4), and (H5) for some β > 1, l = 1,2. If for each t ∈ [0, T] and , then q.s..

Proof. For simplicity, we take the notation , l = 1,2. For each fixed (t, y, z) ∈ [0, T) × ℝ × ℝ^d, let us consider

()

where h_ij = h_ji ∈ ℝ^d. By Theorem 12, we have, for each α ∈ (1, β),

()

Since

()

Take a h_ij such that

. Therefore,

q.s. By the assumptions (H2) and (H3), it is easy to deduce that

q.s.

In the following, we use the notation , l = 1,2.

Corollary 16. Let f^l and be deterministic functions and satisfy (H1), (H2), (H3), and (H5) for some β > 1, l = 1,2. If for each , then .

Proof. Taking η_ε as in Theorem 15, since f^l and are deterministic, we could get , for l = 1, 2. And the proof in Theorem 15 still holds true.

4.2. Some Equivalent Relations

We consider the following G-BSDE:

()

where g_ij = g_ji. We use the notation

Proposition 17. Let f and g_ij satisfy (H1), (H2), (H3), (H4), and (H5) for some β > 1 and fix α ∈ (1, β). Then, one has

(1)
for t ∈ [0, T], , and if and only if for each t ∈ [0, T], y, y^′ ∈ ℝ, z ∈ ℝ^d,
()
(2)
for t ∈ [0, T], , and if and only if for each t ∈ [0, T], y, y^′ ∈ ℝ, z, z^′ ∈ ℝ^d,
()
(3)
for t ∈ [0, T], λ ∈ [0,1], , and if and only if for each t ∈ [0, T], y, y^′ ∈ ℝ, z, z^′ ∈ ℝ^d, λ ∈ [0,1],
()
(4)
for t ∈ [0, T], λ ≥ 0, and if and only if for each t ∈ [0, T], y ∈ ℝ, z ∈ ℝ^d, λ ≥ 0,
()

Proof. (1) “⇒” part. For each fixed t ∈ [0, T), y, y^′ ∈ ℝ, z ∈ ℝ^d, we take

()

where h_ij = h_ji ∈ ℝ^d. Then, by Theorem 12 and

, we can obtain

()

We choose h_ij such that g_ij(t, y^′, z)+〈z, h_ij〉 = 0, which implies (47).

“⇐” part. Let (Y, Z, K) be the solution of G-BSDE (46) corresponding to terminal condition ξ. We claim that (Y_s + η, Z_s, K_s) _s∈[t,T] is the solution of G-BSDE (46) corresponding to terminal condition ξ + η on [t, T]. For this, we only need to check that, for s ∈ [t, T],

()

By (47) we can get

()

which implies (53). The proof of (1) is complete.

(2) “⇒” part. For each fixed t ∈ [0, T), y, y^′ ∈ ℝ, z, z^′ ∈ ℝ^d, we consider ξ_ε = y + 〈z, h_ij〉(〈Bⁱ, B^j〉 _t+ε − 〈Bⁱ, B^j〉 _t)+〈z, B_t+ε − B_t〉 and , where h_ij = h_ji ∈ ℝ^d and . Then, by Theorem 12 and , we obtain

()

We choose h_ij,

such that g_ij(t, y, z)+〈z, h_ij〉 = 0 and

, which implies (48).

“⇐” part. Let (Y, Z, K) and (Y^′, Z^′, K^′) be the solutions of G-BSDE (46) corresponding to terminal condition ξ and η, respectively. Then, (Y + Y^′, Z + Z^′, K) solves the following G-BSDE:

()

where

()

By (48), it is easy to check that V_t is an increasing process. Then, by Theorem 11, we can get

. The proof of (2) is complete.

Finally, we could prove (3) as in (2) and (4) as in (1).

Proposition 18. One has the following.

(1)
If G(A) + G(−A) > 0 for any A ∈ 𝕊_d and A ≠ 0, then (47) holds if and only if f and g_ij are independent of y.
(2)
If there exists an A ∈ 𝕊_d with A ≠ 0 such that G(A) + G(−A) = 0 and G(A) ≠ 0, then, for any fixed g(t, y, z) satisfying (H1)–(H5), one has f(t, y, z) = −2G(A)g(t, y, z) and satisfying (47).

Proof. It is easy to verify (2), and we only need to prove (1). If (47) holds, it is easy to check that holds. Then, from the assumption, we get g_ij(t, y, z) = g_ij(t, 0, z). Therefore, by (47), we have f(t, y, z) = f(t, 0, z), which implies that f and g_ij are independent of y. The converse part is obvious.

Acknowledgments

K. He acknowledges the financial support from the National Natural Science Foundation of China (Grant nos. 11301068 and 11171062) and the Innovation Program of Shanghai Municipal Education Commission (Grant no. 12ZZ063). M. Hu acknowledges the financial support from the National Natural Science Foundation of China (Grant nos. 11201262 and 11101242) and the Scientific Research Foundation for the Excellent Middle-Aged and Young Scientists of Shandong Province of China (Grant no. BS2013SF020).

References

1 Pardoux É. and Peng S. G., Adapted solution of a backward stochastic differential equation, Systems & Control Letters. (1990) 14, no. 1, 55–61, https://doi.org/10.1016/0167-6911(90)90082-6, MR1037747, ZBL0692.93064.
10.1016/0167-6911(90)90082-6
Web of Science® Google Scholar
2 Peng S., N. El Karoui and L. Mazliak, Backward SDE and related g-expectation, Backward Stochastic Differential Equations, 1997, 364, 141–159, Pitman Research Notes in Mathematics Series, MR1752680.
Google Scholar
3 Chen Z. J., A property of backward stochastic differential equations, Comptes Rendus de l′Académie des Sciences. Série I. (1998) 326, no. 4, 483–488, https://doi.org/10.1016/S0764-4442(97)89796-0, MR1648973, ZBL0914.60025.
10.1016/S0764-4442(97)89796-0
Google Scholar
4 Briand P., Coquet F., Hu Y., Mémin J., and Peng S. G., A converse comparison theorem for BSDEs and related properties of g-expectation, Electronic Communications in Probability. (2000) 5, 101–117, https://doi.org/10.1214/ECP.v5-1025, MR1781845.
10.1214/ECP.v5-1025
Web of Science® Google Scholar
5 Jiang L., Representation theorems for generators of backward stochastic differential equations, Comptes Rendus Mathématique. Académie des Sciences. Paris I. (2005) 340, no. 2, 161–166, https://doi.org/10.1016/j.crma.2004.10.023, MR2116776, ZBL1078.60043.
10.1016/j.crma.2004.10.023
Web of Science® Google Scholar
6 Jiang L., Converse comparison theorems for backward stochastic differential equations, Statistics & Probability Letters. (2005) 71, no. 2, 173–183, https://doi.org/10.1016/j.spl.2004.10.032, MR2126773, ZBL1079.60054.
10.1016/j.spl.2004.10.032
Web of Science® Google Scholar
7 Jiang L., Convexity, translation invariance and subadditivity for g-expectations and related risk measures, The Annals of Applied Probability. (2008) 18, no. 1, 245–258, https://doi.org/10.1214/105051607000000294, MR2380898.
10.1214/105051607000000294
Web of Science® Google Scholar
8 Peng S., Filtration consistent nonlinear expectations and evaluations of contingent claims, Acta Mathematicae Applicatae Sinica. (2004) 20, no. 2, 191–214, https://doi.org/10.1007/s10255-004-0161-3, MR2064000, ZBL1061.60063.
10.1007/s10255-004-0161-3
Google Scholar
9 Peng S., Nonlinear expectations and nonlinear Markov chains, Chinese Annals of Mathematics B. (2005) 26, no. 2, 159–184, https://doi.org/10.1142/S0252959905000154, MR2143645, ZBL1077.60045.
10.1142/S0252959905000154
Web of Science® Google Scholar
10 Peng S., G-expectation, G-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications, 2007, 2, Springer, Berlin, Germany, 541–567, The Abel Symposium, https://doi.org/10.1007/978-3-540-70847-6_25, MR2397805.
10.1007/978-3-540-70847-6_25
Google Scholar
11 Peng S., G-Brownian motion and dynamic risk measure under volatility uncertainty, http://arxiv.org/abs/0711.2834v1.
Google Scholar
12 Peng S., Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Processes and Their Applications. (2008) 118, no. 12, 2223–2253, https://doi.org/10.1016/j.spa.2007.10.015, MR2474349.
10.1016/j.spa.2007.10.015
Web of Science® Google Scholar
13 Peng S., A new central limit theorem under sublinear expectations, http://arxiv.org/abs/0803.2656v1.
Google Scholar
14 Soner H. M., Touzi N., and Zhang J., Wellposedness of second order backward SDEs, Probability Theory and Related Fields. (2012) 153, no. 1-2, 149–190, https://doi.org/10.1007/s00440-011-0342-y, MR2925572, ZBL1252.60056.
10.1007/s00440-011-0342-y
Web of Science® Google Scholar
15 Hu M. S., Ji S. L., Peng S. G., and Song Y. S., Backward stochastic differential equations driven by G-brownian motion, http://arxiv.org/abs/1206.5889v1.
Google Scholar
16 Hu M. S., Ji S. L., Peng S. G., and Song Y. S., Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion, http://arxiv.org/abs/1212.5403v1.
Google Scholar
17 Peng S., G.G-Brownian motion and dynamic risk measure under volatility uncertainty, http://arxiv.org/abs/0711.2834v1.
Google Scholar
18 Peng S. G., Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Processes and their Applications. (2008) 118, no. 12, 2223–2253, https://doi.org/10.1016/j.spa.2007.10.015, MR2474349.
10.1016/j.spa.2007.10.015
Web of Science® Google Scholar
19 Peng S. G., Nonlinear expectations and stochastic calculus under uncertainty, http://arxiv.org/abs/1002.4546v1.
Google Scholar
20 Song Y. S., Some properties on G-evaluation and its applications to G-martingale decomposition, Science China Mathematics. (2011) 54, no. 2, 287–300, https://doi.org/10.1007/s11425-010-4162-9, MR2771205.
10.1007/s11425-010-4162-9
Web of Science® Google Scholar

Citing Literature

All articles

Representation Theorem for Generators of BSDEs Driven by G-Brownian Motion and Its Applications

Abstract

1. Introduction

2. Preliminaries

3. Representation Theorem of Generators of G-BSDEs

4. Some Applications

4.1. Converse Comparison Theorem for G-BSDEs

4.2. Some Equivalent Relations

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Representation Theorem for Generators of BSDEs Driven by G-Brownian Motion and Its Applications

Abstract

1. Introduction

2. Preliminaries

3. Representation Theorem of Generators of G-BSDEs

4. Some Applications

4.1. Converse Comparison Theorem for G-BSDEs

4.2. Some Equivalent Relations

Acknowledgments

References

Citing Literature

References

Related

Information