Probability, Uncertainty and Quantitative Risk https://www.aimsciences.org/journal/2095-9672 Vol. 6, No. 1, 2021, 1-22 DOI:10.3934/puqr.2021001
G-Lévy processes under sublinear expectations#
Mingshang Hu1, Shige Peng2, *
1Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, Shandong, China 2School of Mathematics, Shandong University, Jinan 250100, Shandong, China
Email: [email protected], [email protected]
Abstract We introduce G-Lévy processes which develop the theory of processes with independent and stationary increments under the framework of sublinear expectations. We then obtain the Lévy–Khintchine formula and the existence for G-Lévy processes. We also introduce G-Poisson processes.
Keywords Sublinear expectation, G-normal distribution, G-Brownian motion, G- expectation, Lévy process, G-Lévy process, G-Poisson process, Lévy-Khintchine formula, Lévy-Itô decomposition
2020 Mathematics Subject Classification 60H10; 60J60; 60J65
1. Introduction Distribution and independence are two important notions in the theory of probability and statistics. These two notions were introduced in [15,14] under the framework of sublinear expectations. Recently, a new central limit theorem (CLT) under sublinear expectations was obtained in [19] based on a new i.i.d. assumption. The corresponding limit distribution of the CLT is a G-normal distribution. This new type of sublinear distribution was first introduced in [16] (see also [17−20]) for a new type of G-Brownian motion and the related calculus of Itô's type. G-Brownian motion has a very interesting structure which non-trivially generalizes the classical one. Briefly speaking, a G-Brownian motion is a continuous process with independent and stationary increments under a given sublinear expectation. An interesting phenomenon of G-Brownian motion is that its quadratic process is also a continuous process with independent and stationary increments and thus can still be regarded as a G- Brownian motion. A natural problem is how to develop the theory of Lévy processes, i.e., processes with independent and stationary increments but not necessarily continuous,
Received 22 June 2020; Accepted 14 December 2020 #This paper was originally published on arXiv (math.PR/0911.3533v1). *Corresponding author
© Shandong University and AIMS, LLC 2 Mingshang Hu, Shige Peng under sublinear expectations. In particular, how to define Poisson processes under sublinear expectations. The purpose of this paper is to study the distribution property, i.e., Lévy–Khintchine formula, of a Lévy process under sublinear expectations. The corresponding Lévy –Itô decomposition will be discussed in our forthcoming work. We consider types of Lévy processes under sublinear expectations, called G-Lévy processes, and obtain that the corresponding distributions satisfy a new type of nonlinear parabolic integro-partial differential equation. Conversely, we can directly construct G-Lévy processes from these types of equations. A specific case is G-Poisson processes. By comparison with classical methods, our methods are more simple and direct. Books on Lévy processes, e.g., [5,13,21], are recommended for understanding the present results. This paper is organized as follows: in section 2, we recall some important notions and results of sublinear expectations and G-Brownian motions. In section 3, we introduce G- Lévy processes. We discuss the characterization of G-Lévy processes in section 4. In section 5, we obtain the Lévy–Khintchine formula for G-Lévy processes. The existence of G-Lévy processes is given in section 6. For convenience, we present some basic results of
this new type of nonlinear parabolic integro-partial differential equation in the appendix. 2. Basic settings We present some preliminaries in the theory of sublinear expectations and the related
G -Brownian motions. More detail can be found in [14−20]. 2.1. Sublinear expectation
Let Ω be a given set and let H be a linear space of real functions defined on Ω such n n that if X1,...,Xn ∈ H , then φ(X1, ··· ,Xn) ∈ H for each φ ∈ CLip(R ) , where CLip(R ) denotes the space of Lipschitz functions.
− Remark 1 In particular, all constants belong to H and |X|, X+, X ∈ H if X ∈ H .
n Here, we use CLip(R ) in our framework only for convenience. In fact, our essential requirement is that H contains all constants and, moreover, X ∈ H implies |X| ∈ H . In n general, CLip(R ) can be replaced with other spaces for specific problems. We list two other spaces used in this paper.
n ● Cb.Lip(R ) : the space of bounded and Lipschitz functions; k Rn ● Cb ( ) : the space of bounded and k-time continuously differentiable functions with bounded derivatives of all orders less than or equal to k.
Definition 2 A sublinear expectation Eˆ on H is a functional Eˆ : H → R satisfying the following properties: for all X, Y ∈ H , we have (a) Monotonicity: Eˆ[X] ≥ Eˆ[Y ] if X ≥ Y . (b) Constant preserving: Eˆ[c] = c for c ∈ R . (c) Sub-additivity: Eˆ[X] − Eˆ[Y ] ≤ Eˆ[X − Y ] . (d) Positive homogeneity: Eˆ[λX] = λEˆ[X] for λ ≥ 0 . Probability, Uncertainty and Quantitative Risk 3
The triplet (Ω, H, Eˆ) is called a sublinear expectation space (compare with a probability space (Ω, F,P ) ).
Remark 3 If (c) results in equality, then Eˆ is a linear expectation on H . We recall that the notion of the above sublinear expectations was systematically introduced by Artzner, Delbaen, Eber, and Heath [2,3] in the case where Ω is a finite set, and by Delbaen [8] for the general situation with the notation of risk measure: ρ(X) := Eˆ[−X] . See also Huber ∗ [10] for even earlier study of this notion Eˆ (called the upper expectation E in Chapter 10 of [10]).
n Let X = (X1,...,Xn) , Xi ∈ H , denoted by X ∈ H , be a given n-dimensional random vector on a sublinear expectation space (Ω, H, Eˆ) . We define a functional on n CLip(R ) by ˆ ˆ n FX [φ] := E[φ(X)] for all φ ∈ CLip(R ). n n ˆ ˆ The triplet (R ,CLip(R ), FX [·]) forms a sublinear expectation space. FX is called the distribution of X.
ˆ Remark 4 If the distribution FX of X ∈ H is not a linear expectation, then X is said to have distributional uncertainty. The distribution of X has the following four typical parameters: µ¯ := Eˆ[X], µ := −Eˆ[−X], σ¯2 := Eˆ[X2], σ2 := −Eˆ[−X2].
The intervals [µ, µ¯] and [σ2, σ¯2] characterize the mean-uncertainty and the variance- uncertainty of X.
The following simple properties are very useful in sublinear analysis.
Proposition 5 Let X, Y ∈ H be such that Eˆ[Y ] = −Eˆ[−Y ] , i.e., Y has no mean uncertainty. Then we have Eˆ[X + Y ] = Eˆ[X] + Eˆ[Y ].
In particular, if Eˆ[Y ] = Eˆ[−Y ] = 0 , then Eˆ[X + Y ] = Eˆ[X] .
Proof It is simply because we have Eˆ[X + Y ] ≤ Eˆ[X] + Eˆ[Y ] and Eˆ[X + Y ] ≥ Eˆ[X] − Eˆ[−Y ] = Eˆ[X] + Eˆ[Y ]. □ Noting that Eˆ[c] = −Eˆ[−c] = c for all c ∈ R , we immediately have the following corollary.
Corollary 6 For each X ∈ H , we have Eˆ[X + c] = Eˆ[X] + c for all c ∈ R .
Proposition 7 For each X, Y ∈ H , we have |Eˆ[X] − Eˆ[Y ]| ≤ Eˆ[X − Y ] ∨ Eˆ[Y − X].
In particular, |Eˆ[X] − Eˆ[Y ]| ≤ Eˆ[|X − Y |] . 4 Mingshang Hu, Shige Peng
Proof By sub-additivity and monotonicity of Eˆ[·] , it is easy to prove the inequalities. □
We recall some important notions under sublinear expectations.
Definition 8 Let X1 and X2 be two n-dimensional random vectors defined, respectively, ˆ ˆ on sublinear expectation spaces (Ω1, H1, E1) and (Ω2, H2, E2) . They are called identically d distributed, denoted by X1 = X2 , if ˆ ˆ n E1[φ(X1)] = E2[φ(X2)] for all φ ∈ CLip(R ). d It is clear that X1 = X2 if and only if their distributions coincide.
Definition 9 Let (Ω, H, Eˆ) be a sublinear expectation space. A random vector Y = n (Y1, ··· ,Yn) ∈ H is said to be independent from another random vector X = m ˆ m n (X1, ··· ,Xm) ∈ H under E[·] if for each test function φ ∈ CLip(R × R ) we have ˆ ˆ ˆ E[φ(X,Y )] = E[E[φ(x, Y )]x=X ]. m d X¯ = (X¯1, ··· , X¯m) ∈ H is said to be an independent copy of X if X¯ = X and X¯ is independent from X.
Remark 10 Under a sublinear expectation space (Ω, H, Eˆ) . Y is independent from X means that the distributional uncertainty of Y does not change after the realization of X = x . Or, in other words, the “conditional sublinear expectation” of Y knowing X is ˆ E[φ(x, Y )]x=X . In the case of linear expectation, this notion of independence is just the classical one.
It is important to note that under sublinear expectations the condition “Y is independent from X” does not automatically imply that “X is independent from Y.” See the following example:
Example 11 We consider a case where Eˆ is a sublinear expectation and X,Y ∈ H are identically distributed with Eˆ[X] = Eˆ[−X] = 0 and σ¯2 = Eˆ[X2] > σ2 = −Eˆ[−X2] . We also Eˆ | | Eˆ + − Eˆ + 1 Eˆ | | 1 Eˆ | | assume that [ X ] = [X + X ] > 0 , thus [X ] = 2 [ X + X] = 2 [ X ] > 0 . In the case where Y is independent from X, we have Eˆ[XY 2] = Eˆ[X+σ¯2 − X−σ2] = (¯σ2 − σ2)Eˆ[X+] > 0. But if X is independent from Y, we have Eˆ[XY 2] = 0.
2.2. G-Brownian motion
For a given positive integer n, we denote by ⟨x, y⟩ the scalar product of x, y ∈ Rn and by |x| = ⟨x, x⟩1/2 the Euclidean norm of x. We also denote by S(d) the space of all d × d × symmetric matrices and by Rn d the space of all n × d matrices. For A, B ∈ S(d) , A ≥ B means that A − B is non-negative.
Definition 12 (G-normal distribution with zero mean) A d-dimensional random vector Probability, Uncertainty and Quantitative Risk 5
ˆ X = (X1, ··· ,Xd) on a sublinear expectation space (Ω, H, E) is said to be G-normally distributed if for each a , b ≥ 0 we have √ aX + bX¯ =d a2 + b2X, where X¯ is an independent copy of X. Here, the letter G denotes the function 1 G(A) := Eˆ[⟨AX, X⟩]for A ∈ S(d). 2 ˆ ˆ It is easy to prove that E[Xi] = E[−Xi] = 0 for i = 1, . . . , d and the function G is a monotonic and sublinear function. ˆ Let (Ω, H, E) be a sublinear expectation space, (Xt)t≥0 is called a d-dimensional d process if Xt ∈ H for each t ≥ 0 .
Definition 13 (G-Brownian motion) Let G : S(d) → R be a given monotonic and sublinear function. A d-dimensional process (Bt)t≥0 on a sublinear expectation space (Ω, H, Eˆ) is called a G-Brownian motion if the following properties are satisfied:
(i) B0 = 0 ; ≥ − ∈ N (ii) For each t, s 0 , Bt+s Bt is independent from (Bt1 ,Bt2 ,...,Btn ) for each n and 0 ≤ t1 ≤ · · · ≤ tn ≤ t ; d √ (iii) Bt+s − Bt = sX for t, s ≥ 0 , where X is G-normally distributed.
Remark 14 If Eˆ is a linear expectation in the above two definitions, then the function G is a linear function, X is classically normal, and (Bt)t≥0 is a classical Brownian motion.
The above two definitions can be non-trivially generalized to the following situations.
Definition 15 (G-normal distribution with mean uncertainty) A pair of d-dimensional random vectors (X, η) on a sublinear expectation space (Ω, H, Eˆ) is called G-distributed if for each a , b ≥ 0 we have √ (aX + bX,¯ a2η + b2η¯) =d ( a2 + b2X, (a2 + b2)η), where (X,¯ η¯) is an independent copy of (X, η) . Here, the letter G denotes the function 1 G(p, A) := Eˆ[ ⟨AX, X⟩ + ⟨p, η⟩] for (p, A) ∈ Rd × S(d). 2
Obviously, X is G¯ -normally distributed with G¯(A) = G(0,A) . The distribution of η can be seen as the pure uncertainty of mean (see [17–20]). It is easy to prove that G is a sublinear function monotonic in A ∈ S(d) .
Definition 16 (Generalized G-Brownian motion) Let G : Rd × S(d) → R be a given sublinear function monotonic in A ∈ S(d) . A d-dimensional process (Bt)t≥0 on a sublinear expectation space (Ω, H, Eˆ) is called a generalized G-Brownian motion if the following properties are satisfied:
(i) B0 = 0 ; ≥ − ∈ (ii) For each t, s 0 , Bt+s Bt is independent from (Bt1 ,Bt2 ,...,Btn ) for each n N and 0 ≤ t1 ≤ · · · ≤ tn ≤ t ; 6 Mingshang Hu, Shige Peng
d √ (iii) Bt+s − Bt = sX + sη for t, s ≥ 0 , where (X, η) is G-distributed.
The construction of G-Brownian motion was first given in [16,17] and G-distributed random vector was given in [19]. Moreover, we have the characterization of the generalized G-Brownian motion (see [18,20]).
Theorem 17 Let (Xt)t≥0 be a d-dimensional process defined on a sublinear expectation space (Ω, H, Eˆ) such that
(i) X0 = 0 ; (ii) For each t, s ≥ 0 , Xt+s − Xt and Xs are identically distributed and Xt+s − Xt is ∈ N ≤ ≤ · · · ≤ ≤ independent from (Xt1 ,Xt2 ,...,Xtn ) for each n and 0 t1 tn t ; ˆ 3 −1 (iii) limt↓0 E[|Xt| ]t = 0 .
Then (Xt)t≥0 is a generalized G-Brownian motion, where
ˆ 1 −1 d G(p, A) = lim E[ ⟨AXt,Xt⟩ + ⟨p, Xt⟩]t for(p, A) ∈ R × S(d). t↓0 2
Remark 18 In fact, paths of (Xt)t≥0 in the above theorem are continuous due to condition (iii) (see [9,11]). In the following sections, we consider Lévy processes without
condition (iii) which contain jumps. 3. G-Lévy processes ˆ A process {Xt(ω): ω ∈ Ω, t ≥ 0} defined on a sublinear expectation space (Ω, H, E) is called càdlàg if for each ω ∈ Ω , limδ↓0 Xt+δ(ω) = Xt(ω) and Xt−(ω) := limδ↓0 Xt−δ(ω) exists for all t ≥ 0 . We now give the definition of Lévy processes under sublinear expectations.
Definition 19 A d-dimensional càdlàg process (Xt)t≥0 defined on a sublinear expec- tation space (Ω, H, Eˆ) is called a Lévy process if the following properties are satisfied:
(i) X0 = 0 ; (ii) Independent increments: for each t, s > 0 , the increment Xt+s − Xt is independent ∈ N ≤ ≤ · · · ≤ ≤ from (Xt1 ,Xt2 ,...,Xtn ) , for each n and 0 t1 tn t ; (iii) Stationary increments: the distribution of Xt+s − Xt does not depend on t.
Remark 20 If (Xt)t≥0 is a Lévy process, then the finite-dimensional distribution of
(Xt)t≥0 is uniquely determined by the distribution of Xt for each t ≥ 0 .
Proposition 21 Let (Xt)t≥0 be a d-dimensional Lévy process defined on a sublinear ˆ n×d expectation space (Ω, H, E) . Then, for each A ∈ R , (AXt)t≥0 is an n-dimensional Lévy process.
Proof By the definition of distribution and independence, it is easy to prove the result.□
Let (Xt)t≥0 be a d-dimensional Lévy process defined on a sublinear expectation space (Ω, H, Eˆ) . In this paper, we suppose that there exists a 2d -dimensional Lévy process Probability, Uncertainty and Quantitative Risk 7
c d ˜ H˜ E˜ (Xt ,Xt )t≥0 defined on a sublinear expectation space (Ω, , ) such that the distri- c d butions of (Xt + Xt )t≥0 and (Xt)t≥0 are the same. In this paper, we only consider the c d distribution property of (Xt)t≥0 . Hence, we can suppose Xt = Xt + Xt on the same sublinear expectation space (Ω, H, Eˆ) .
Remark 22 In the classical linear expectation case, by the Lévy–Itô decomposition, the c above assumption of (Xt)t≥0 obviously holds, where (Xt )t≥0 is the continuous part and d (Xt )t≥0 is the jump part.
c d Furthermore, we suppose (Xt + Xt )t≥0 satisfying the following assumption: Eˆ | c|3 −1 Eˆ | d| ≤ ≥ lim [ Xt ]t = 0; [ Xt ] Ct for t 0, (1) t↓0 where C is a constant.
c c Remark 23 By the assumption on (Xt )t≥0 , we know that (Xt )t≥0 is a generalized G- d Brownian motion. The assumption on the jump part (Xt )t≥0 implies that it is of finite variation. The more complicated situation will be discussed in our forthcoming work.
d Example 24 Suppose (Xt )t≥0 is a 1 -dimensional positive Lévy process, i.e., jumps are positive. Note that Eˆ d Eˆ d Eˆ d − d Eˆ d Eˆ d [Xt+s] = [Xt ] + [Xt+s Xt ] = [Xt ] + [Xs ] Eˆ d Eˆ d Eˆ d and [Xt ] is increasing in t, then we obtain [Xt ] = [X1 ]t . Obviously, it satisfies (1).
Definition 25 A d-dimensional Lévy process (Xt)t≥0 is called a G-Lévy process if there c d ≥ c d exists a decomposition Xt = Xt + Xt for each t 0 , where (Xt ,Xt )t≥0 is a 2d - dimensional Lévy process satisfying (1).
By Proposition 21, we immediately have:
Proposition 26 Let (Xt)t≥0 be a d-dimensional G-Lévy process defined on a sublinear ˆ n×d expectation space (Ω, H, E) . Then, for each A ∈ R , (AXt)t≥0 is an n-dimensional G-
Lévy process. 4. Characterization of G-Lévy processes
c d Let (Xt)t≥0 be a d-dimensional G-Lévy process with the decomposition Xt = Xt + Xt . d ˆ In this section, we show that for each given φ ∈ Cb.Lip(R ), u(t, x) := E[φ(x + Xt)] is a viscosity solution of the following equation:
∂tu(t, x) − GX [u(t, x + ·) − u(t, x)] = 0, u(0, x) = φ(x), (2) where GX [f(·)] is a nonlocal operator defined by · Eˆ −1 ∈ 3 Rd GX [f( )] := lim [f(Xδ)]δ for f Cb ( ) with f(0) = 0. (3) δ↓0
We first show that the definition of GX [f(·)] is meaningful. For this, we need the 8 Mingshang Hu, Shige Peng following lemmas.
Lemma 27 For each δ ≤ 1 , we have Eˆ | c|p ≤ p/2 [ Xδ ] Cpδ for each p > 0, where Cp is a constant depending only on p.
Proof This is a direct consequence of Theorem 17. □
∈ 3 Rd Lemma 28 For each given f Cb ( ) with f(0) = 0 , we have 1 Eˆ[f(X )] = Eˆ[f(Xd) + ⟨Df(0),Xc⟩ + ⟨D2f(0)Xc,Xc⟩] + o(δ). δ δ δ 2 δ δ Proof It is easy to check that I1 :=f(X ) − f(Xc) − f(Xd) δ ∫ δ δ δ 1 ⟨ c d c − c ⟩ = Xδ , Df(Xδ + αXδ ) Df(αXδ ) dα 0 and 2 c − ⟨ c⟩ − 1⟨ 2 c c⟩ Iδ :=f(Xδ ) Df(0),Xδ D f(0)Xδ ,Xδ ∫ ∫ 2 1 1 ⟨ 2 c − 2 c c⟩ = (D f(αβXδ ) D f(0))Xδ ,Xδ αdβdα. 0 0
Note that Df is bounded. Then we get ∫ 1 1 3 2 Eˆ | 1| ≤ Eˆ | c|3 3 Eˆ | d c − c | 2 3 [ Iδ ] ( [ Xδ ]) ( [ Df(Xδ + αXδ ) Df(αXδ ) dα]) 0∫ 1 1 2 ≤ Eˆ | c|3 3 Eˆ | d c − c | 3 C( [ Xδ ]) ( [ Df(Xδ + αXδ ) Df(αXδ ) dα]) 0 1 2 ≤ Eˆ | c|3 3 Eˆ | d| 3 C1( [ Xδ ]) ( [ Xδ ]) 7/6 ≤ C2δ = o(δ). Eˆ | 2| ≤ Eˆ | c|3 It is easy to obtain [ Iδ ] C [ Xδ ] = o(δ). Noting that 1 |Eˆ[f(X )] − Eˆ[f(Xd) + ⟨Df(0),Xc⟩ + ⟨D2f(0)Xc,Xc⟩]| ≤ Eˆ[|I1 + I2|], δ δ δ 2 δ δ δ δ we conclude the result. □
∈ Rd × S ∈ 2 Rd Lemma 29 Let (p, A) (d) and f Cb ( ) with f(0) = 0 be given. Then Eˆ d ⟨ c⟩ 1 ⟨ c c⟩ −1 limδ↓0 [f(Xδ ) + p, Xδ + 2 AXδ ,Xδ ]δ exists.
Proof We define 1 g(t) = Eˆ[f(Xd) + ⟨p, Xc⟩ + ⟨AXc,Xc⟩]. t t 2 t t
Obviously, g(0) = 0 . For each t, s ∈ [0, 1] , ˆ ˆ |g(t + s) − g(t)| ≤ Cs + E[Y ] ∨ E[−Y ] ≤ C1s, Probability, Uncertainty and Quantitative Risk 9
1 c c c c where Y =⟨p+ AXc,Xc −Xc⟩+ ⟨A(X −X ),X −X ⟩ . Thus, g(·) is differentiable t t+s t 2 t+s t t+s t ′ almost everywhere on [0, 1] . For each fixed t0 < 1 such that g (t0) exists, we have g(δ) g(t + δ) − g(t ) = 0 0 − Λ , δ δ δ where Λ =δ−1(g(t + δ) − Eˆ[f(Xd ) + f(Xd − Xd ) + ⟨p, Xc ⟩ δ 0 t0 t0+δ t0 t0+δ 1 1 + ⟨AXc ,Xc ⟩ + ⟨A(Xc − Xc ),Xc − Xc ⟩]). 2 t0 t0 2 t0+δ t0 t0+δ t0 √ Similar to the above estimate, it is not difficult to prove that |Λδ| ≤ C t0 , where C is a constant independent of δ and t . Thus, 0 √ −1 −1 | lim sup g(δ)δ − lim inf g(δ)δ | ≤ 2C t0. δ↓0 δ↓0
Letting t0 ↓ 0 , we get the result. □ By the above two lemmas, we know that the definition of GX [f(·)] is meaningful. It is · ∈ 3 Rd easy to check that GX [f( )] satisfies the following properties: for each f, g Cb ( ) with f(0) = 0, g(0) = 0,
1) Monotonicity: GX [f(·)] ≥ GX [g(·)] if f ≥ g . 2) Sub-additivity: GX [f(·) + g(·)] ≤ GX [f(·)] + GX [g(·)]. 3) Positive homogeneity: GX [λf(·)] = λGX [f(·)] for all λ ≥ 0. Now, we give the definition of viscosity solution for equation (2).
Definition 30 A bounded upper semicontinuous (lower semicontinuous) function u is called a viscosity subsolution (viscosity supersolution) of the equation (2) if u(0, x) ≤ φ(x) ≥ ∈ ∞ × Rd ∈ 2,3 ≥ ≤ ( φ(x) ) and for each (t, x) (0, ) and for each ψ Cb such that ψ u ( u ) and ψ(t, x) = u(t, x) , we have
∂tψ(t, x) − GX [ψ(t, x + ·) − ψ(t, x)] ≤ 0 ( ≥ 0). A bounded continuous function u is called a viscosity solution of equation (2) if it is both a viscosity subsolution and a viscosity supersolution.
We now give the characterization of G-Lévy processes.
d Theorem 31 Let (Xt)t≥0 be a d-dimensional G-Lévy process. For each φ ∈ Cb.Lip(R ) , ˆ define u(t, x) = E[φ(x + Xt)] . Then u is a viscosity solution of equation (2).
Proof We first show that u is a continuous function. Obviously, |u(t, x) − u(t, y)| ≤ C|x − y| . Note that ˆ ˆ u(t + s, x) = E[φ(x + Xs + Xt+s − Xs)] = E[u(t, x + Xs)], √ ≤ | − | ≤ Eˆ | c d| ≤ then for s 1 , u(t + s, x) u(t, x) C [ Xs + Xs ] C1 s . Thus, u is continuous. ∈ ∞ × Rd ∈ 2,3 ≥ For each fixed (t, x) (0, ) and ψ Cb such that ψ u and ψ(t, x) = u(t, x) , we have ˆ ˆ ψ(t, x) = u(t, x) = E[u(t − δ, x + Xδ)] ≤ E[ψ(t − δ, x + Xδ)]. 10 Mingshang Hu, Shige Peng
Therefore, ˆ 0 ≤ E[ψ(t − δ, x + Xδ) − ψ(t, x)] ˆ = −∂tψ(t, x)δ + E[ψ(t, x + Xδ) − ψ(t, x) + Iδ] ˆ ˆ ≤ −∂tψ(t, x)δ + E[ψ(t, x + Xδ) − ψ(t, x)] + E[|Iδ|], ∫ 1 − − where Iδ = δ 0 [∂tψ(t, x) ∂tψ(t βδ, x + Xδ)]dβ . It is easy to show that ˆ ˆ E[|Iδ|] ≤ CE[δ(δ + |Xδ|)] = o(δ).
By the definition of GX , we get
∂tψ(t, x) − GX [ψ(t, x + ·) − ψ(t, x)] ≤ 0. Hence, u is a viscosity subsolution of (2). Similarly, we prove that u is a viscosity supersolution of (2). Thus, u is a viscosity solution of (2). □
Remark 32 We do not know the uniqueness of viscosity solution for (2). For this, we need the representation of GX presented in the following section.
5. Lévy–Khintchine representation of GX
In this section, we give a representation of the infinitesimal generator GX , which can be seen as the Lévy–Khintchine formula for G-Lévy processes. We first give some lemmas.
d d Lemma 33 Let (p, A) ∈ R × S(d) and f ∈ Cb.Lip(R ) with f(0) = 0 and f(x) = o(|x|) Eˆ d ⟨ c⟩ 1 ⟨ c c⟩ −1 be given. Then limδ↓0 [f(Xδ ) + p, Xδ + 2 AXδ ,Xδ ]δ exists.
Proof Since f(x) = o(|x|) , there exists a sequence {δn : n ≥ 1} such that δn ↓ 0 and 1 |f(x)| ≤ |x| on |x| ≤ δ . For each fixed δ , we choose f ε ∈ C2(Rd) with f ε(0) = 0 n n n n b n such that | − ε | ≤ 4Lε| | 1 | | f(x) fn(x) x + x , δn n where L is the Lipschitz constant of f. Thus, Eˆ[|f(Xd) − f ε(Xd)|] 4Lε 1 Eˆ[|Xd|] 4Lε 1 δ n δ ≤ ( + ) δ ≤ C( + ). δ δn n δ δn n By Lemma 29 and the above estimate, we obtain Eˆ d ⟨ c⟩ 1⟨ c c⟩ −1 lim sup [f(Xδ ) + p, Xδ + AXδ ,Xδ ]δ δ↓0 2 ≤ Eˆ d ⟨ c⟩ 1⟨ c c⟩ −1 4Lε 1 lim inf [f(Xδ ) + p, Xδ + AXδ ,Xδ ]δ + 2C( + ), δ↓0 2 δn n which implies the result by first letting ε ↓ 0 and then n → ∞ . □ We denote by d L0 = {f ∈ Cb.Lip(R ): f(0) = 0 and f(x) = o(|x|)} and d d L = {(f, p, q, A): f ∈ L0 and (p, q, A) ∈ R × R × S(d)}. Probability, Uncertainty and Quantitative Risk 11
ˆ It is clear that L0 and L are both linear spaces. Now, we define a functional F[·] on L by ⟨p, Xd⟩ 1 Fˆ Eˆ d δ ⟨ c⟩ ⟨ c c⟩ −1 [(f, p, q, A)] := lim [f(Xδ ) + + q, Xδ + AXδ ,Xδ ]δ . δ↓0 | d|2 1 + Xδ 2
Similar to the proof of the above lemma, we know that the definition of Fˆ[·] is meaningful.
∈ 3 Rd Remark 34 For g Cb ( ) with g(0) = 0 , define ⟨Dg(0), x⟩ f(x) = g(x) − for x ∈ Rd. 1 + |x|2
d One can verify that f ∈ Cb.Lip(R ) with f(0) = 0 and f(x) = o(|x|) . So we add the term ⟨ d⟩ | d|2 −1 Fˆ · p, Xδ (1 + Xδ ) in the definition of [ ] .
Lemma 35 The functional Fˆ : L → R satisfies the following properties: ˆ ˆ (1) F[(f1, p, q, A1)] ≥ F[(f2, p, q, A2)] if f1 ≥ f2 and A1 ≥ A2 . ˆ ˆ ˆ (2) F[(f1 + f2, p1 + p2, q1 + q2,A1 + A2)] ≤ F[(f1, p1, q1,A1)] + F[(f2, p2, q2,A2)]. (3) Fˆ[λ(f, p, q, A)] = λFˆ[(f, p, q, A)] for all λ ≥ 0. ˆ (4) If fn ∈ L0 satisfies fn ↓ 0 , then F[(fn, 0, 0, 0)] ↓ 0 .
Proof It is easy to prove (1), (2), and (3). We now prove (4). For each fixed 0 < η1 < η2 < ∞ , it is easy to check | | | | ≤ f1(y) | | x x fn(x) ( sup | | ) x + ( sup fn(y)) + ( sup f1(y)) . 0<|y|≤η1 y η1≤|y|≤η2 η1 |y|≥η2 η2 Thus,
sup fn(y) sup f1(y) ≤| |≤ ∈Rd Fˆ ≤ f1(y) η1 y η2 y [(fn, 0, 0, 0)] C( sup | | + + ). 0<|y|≤η1 y η1 η2
Noting that f (y) ↓ 0 , we then have supη1≤|y|≤η2 n
sup d f (y) ˆ f1(y) y∈R 1 lim F[(fn, 0, 0, 0)] ≤ C( sup + ). n→∞ | | 0<|y|≤η1 y η2
First letting η1 → 0 and then η2 → ∞ , we obtain (4). □ By (2) and (3) of the above lemma, we immediately obtain that there exists a family of linear functionals {Fu : u ∈ U0} defined on L such that ˆ F[(f, p, q, A)] = sup Fu[(f, p, q, A)]. u∈U0
The proof is found in [19]. Note that (1) and (4) of the above lemma, then for each Fu , ′ ′ × by the Daniell –Stone theorem, there exist (p , q ,Q) ∈ Rd × Rd × Rd d and a unique measure v on (Rd\{0}, B(Rd\{0})) such that ∫ ′ ′ 1 T Fu[(f, p, q, A)] = f(z)v(dz) + ⟨p, p ⟩ + ⟨q, q ⟩ + tr[AQQ ]. Rd\{0} 2 Thus, 12 Mingshang Hu, Shige Peng ∫ Fˆ { ⟨ ′⟩ ⟨ ′⟩ 1 T } [(f, p, q, A)] = sup Rd\{ } f(z)v(dz) + p, p + q, q + tr[AQQ ] . ′ ′ 0 2 (4) (v,p ,q ,Q)∈U0 In particular, ∫ Eˆ d −1 lim [f(Xδ )]δ = sup Rd\{0} f(z)v(dz), (5) δ↓0 v∈V where ′ ′ ′ ′ V = {v : ∃(p , q ,Q) such that (v, p , q ,Q) ∈ U0}. (6) Taking a specific f, we easily prove that { | | ≥ } ∞ (1) For each ε > 0 , supv∈V v( z : z ε ) < . V { | | ≥ } (2) For each∫ ε > 0 , the restriction of on the set z : z ε is tight. | | ∞ (3) supv∈V Rd z v(dz) < .
In fact, it is easy to deduce that (3) implies (1) and (2). Similarly, it is also easy to ′ ′ show that all (p , q ,Q) in U0 are bounded. Now, we give the representation of GX . For each f ∈ C3(Rd) with f(0) = 0 , by Lemma 28 and the above analysis, we have b ∫ ⟨ ⟩ · { − Df(0), z ⟨ ′ ′⟩ GX [f( )] = sup (f(z) 2 )v(dz) + Df(0), p + q ′ ′ d | | (v,p ,q ,Q)∈U0 R \{0} 1 + z 1 + tr[D2f(0)QQT ]}. 2 ∫ | | ∞ Noting that supv∈V Rd z v(dz) < , we then have the following Lévy –Khintchine representation of G : X ∫ G [f(·)] = { f(z)v(dz) + ⟨Df(0), q⟩ + 1 [D2f(0)QQT ]}, X sup Rd\{0} 2 tr (7) (v,q,Q)∈U where ∫ z U= {(v, p′ + q′ − v(dz),Q): ∀(v, p′, q′,Q) ∈ U }. | |2 0 Rd\{0} 1 + z We summarize the above discussions as a theorem.
· ∈ 3 Rd Theorem 36 Let (Xt)t≥0 be a d-dimensional G-Lévy process. Then GX [f( )] , f Cb ( ) , × has the Lévy–Khintchine representation (7), where (q, Q) ∈ Rd × Rd d and v is a measure on (Rd\{0}, B(Rd\{0})) satisfying ∫ { | | | | T } ∞ sup Rd z v(dz) + q + tr[QQ ] < . (8) (v,q,Q)∈U
Remark 37 In the above theorem, the set U represents the uncertainty of the Lévy measure, the drift, and the covariance matrix of (Xt)t≥0 .
We then have the following theorem.
d Theorem 38 Let (Xt)t≥0 be a d-dimensional G-Lévy process. For each φ ∈ Cb.Lip(R ) , ˆ define u(t, x) = E[φ(x + Xt)] . Then u is the unique viscosity solution of the following integro-partial differential equation: Probability, Uncertainty and Quantitative Risk 13 ∫
∂tu(t, x) − sup { (u(t, x + z) − u(t, x))v(dz) + ⟨Du(t, x), q⟩ ∈U Rd\{ } (v,q,Q) 0 (9) 1 + tr[D2u(t, x)QQT ]} = 0, 2 where U represents GX .
Proof By Theorems 5.4 and 4.5, u is a viscosity solution of (9). For the uniqueness, see appendix. □
Remark 39 The definition of the viscosity solution for (9) is the same as Definition 30. 6. Existence of G-Lévy processes
+ d d We denote by Ω = D0(R , R ) the space of all R -valued càdlàg functions (ωt)t∈R+ , with ω0 = 0 , equipped with the Skorokhod topology. The corresponding canonical process is Bt(ω) = ωt for ω ∈ Ω , t ≥ 0 . We define ∨ Ft := σ{Bs : s ≤ t} and F = Ft. t≥0
Following [15,16,17], for each fixed T ∈ [0, ∞) , we set F { ∈ N ∈ ∞ ∈ Rd×n } Lip( T ) := φ(Bt1∧T ,...,Btn∧T ): n , t1, . . . tn [0, ), φ Cb.Lip( ) .
It is clear that Lip(Ft) ⊂ Lip(FT ) for t ≤ T . We also set ∪∞ Lip(F) := Lip(Fn). n=1
Let U be given and satisfy (8). We consider the corresponding integro-partial d differential equation (9). For each given initial condition φ ∈ Cb.Lip(R ) , the viscosity solution uφ for (9) exists (see appendix). Furthermore, we have the following theorem.
Theorem 40 Let uφ denote the viscosity solution of (9) with the initial condition d φ ∈ Cb.Lip(R ) . Then we have 1) uφ ≥ uψ if φ ≥ ψ . 2) uφ+ψ ≤ uφ + uψ . 3) uφ+c = uφ + c for c ∈ R . 4) uλφ = λuφ for all λ ≥ 0 . φ · 5) uφ(t + s, x) = uu (t,x+ )(s, 0) .
Proof It is easy to check 3)–5). 1) and 2) are proved in the appendix. □
ˆ We now introduce a sublinear expectation E on Lip(F) by the following two steps:
d Step 1. For each ξ ∈ Lip(F) of the form ξ = φ(Bt+s − Bt) , t, s ≥ 0 and φ ∈ Cb.Lip(R ) , we define Eˆ[ξ] = u(s, 0) , where u is a viscosity solution of (9) with the initial condition u(0, x) = φ(x) . 14 Mingshang Hu, Shige Peng
d×m Step 2. For each ξ ∈ Lip(F) , we can find a ϕ ∈ Cb.Lip(R ) such that ξ = − − ··· Eˆ ϕ(Bt1 ,Bt2 Bt1 ,...,Btm Btm−1 ) , t1 < t2 < < tm . Then we define [ξ] = ϕm , where
ϕm ∈ R is obtained via the following procedure: Eˆ − ϕ1(x1, . . . , xm−1) = [ϕ(x1, . . . , xm−1,Btm Btm−1 )]; Eˆ − ϕ2(x1, . . . , xm−2) = [ϕ1(x1, . . . , xm−2,Btm−1 Btm−2 )]; . . Eˆ − ϕm−1(x1) = [ϕm−2(x1,Bt2 Bt1 )]; Eˆ ϕm = [ϕm−1(Bt1 )]. F The related conditional expectation of ξ under tj is defined by Eˆ |F Eˆ − − |F [ξ tj ] = [ϕ(Bt1 ,Bt2 Bt1 ,...,Btm Btm−1 ) tj ] − =ϕm−j(Bt1 ,...,Btj Btj−1 ).
By the above theorem, it is easy to prove that Eˆ[·] consistently defines a sublinear expectation on Lip(F) . Then Lip(F) can be extended to a Banach space under the norm || || Eˆ | | 1 F |Eˆ − Eˆ | ≤ Eˆ | − | X := [ X ] . We denote this space by LG( ) . Note that [X] [Y ] [ X Y ] . Eˆ · 1 F Then [ ] can be extended as a continuous mapping on LG( ) which is still a sublinear ˆ expectation. Similarly, it is easy to check that the conditional expectation E[·|Ft] can be 1 F → 1 F also extended as a continuous mapping LG( ) LG( t) . We now prove that the canonical process (Bt)t≥0 is a G-Lévy process. For this, we need the following lemma.
{ }∞ { }∞ Lemma 41 Let Xn n=1 and Yn n=1 be two sequences of d-dimensional random ˆ vectors on a sublinear expectation space (Ω, H, E) . We assume that Yn is independent d ˆ from Xn for n = 1, 2,... . If there exist X, Y ∈ H such that E[|Xn − X|] → 0 and ˆ E[|Yn − Y |] → 0 , then Y is independent from X.
2d Proof For each fixed φ ∈ CLip(R ) , we define ˆ ˆ φ¯n(x) = E[φ(x, Yn)] and φ¯(x) = E[φ(x, Y )]. ˆ ˆ ˆ It is clear that |φ¯(x) − φ¯n(¯x)| ≤ C(|x − x¯| + E[|Yn − Y |]) . Thus, |E[φ ¯n(Xn)] − E[φ ¯(X)]| ≤ ˆ ˆ ˆ ˆ C(E[|Xn − X|] + E[|Yn − Y |]) . Note that E[φ ¯n(Xn)] = E[φ(Xn,Yn)] . Then we obtain Eˆ[φ ¯(X)] = Eˆ[φ(X,Y )] , which implies that Y is independent from X.
Theorem 42 The canonical process (Bt)t≥0 is a G-Lévy process.
+ 2d Proof Consider D0(R , R ) and the canonical process (B¯t, B˜t)t≥0 . Similarly as above, ˆ we construct a sublinear expectation, still denoted by E[·] , on Lip(F) via the following integro-partial differential equation: ∫
∂tw(t, x, y) − sup { (w(t, x, y + z) − w(t, x, y))v(dz) + ⟨Dxw(t, x, y), q⟩ (v,q,Q)∈U Rd\{0} 1 + tr[D2w(t, x, y)QQT ]} = 0. 2 x
It is easy to check that the distribution of (B¯t)t≥0 satisfies the following equation: Probability, Uncertainty and Quantitative Risk 15
1 2 T ∂tu(t, x) − sup {⟨Du(t, x), q⟩ + tr[D u(t, x)QQ ]} = 0. (q,Q)∈U 2
Thus, (B¯t)t≥0 is the generalized G-Brownian motion. The distribution of (B˜t)t≥0 satisfies the following equation: ∫
∂tu(t, y) − sup (u(t, y + z) − u(t, y))v(dz) = 0. v∈V Rd\{0}
We now show that B˜ belongs to L1 (F) and satisfies (1). Consider the function ∑ t G d N ϕ0(y) = d + yi(arctan yi + π/2) . Define ϕ (y) = ϕ0(y − N) for N > 0 and i=1 ∫ 0 N N − N ϕi+1(y) = sup (ϕ0 (y + z) ϕ0 (y))v(dz) for i = 1, 2,.... v∈V Rd\{0} ∑ ∞ ti Then, it is not difficult to check that uN (t, y) := ϕN (y) is the solution of the i=0 i! i ∑ N ≥ d + above equation with the initial condition ϕ0 . It is easy to check that ϕ0(y) i=1 yi N → → ∞ | ˜ | ∈ 1 F and u (t, 0) 0 as N∫ . Therefore, we conclude that Bt LG( ) . Noting that | | | | u(t, y) := y + t supv∈V Rd z v(dz) is a viscosity supersolution of the above equation, then (1) holds. It is also easy to check that the distribution of (B¯t + B˜t)t≥0 satisfies (9). By the above lemma, (Bt)t≥0 is a G-Lévy process. □
Example 43 We consider the following 1-dimensional equation:
∂tu(t, x) − Gλ(u(t, x + 1) − u(t, x)) = 0, u(0, x) = φ(x), + − where Gλ(a) = a − λa , λ ∈ [0, 1] . This equation is a special case of the above equation with V = {lδ1 : l ∈ [λ, 1]} . Thus, we can construct the corresponding sublinear expectation ˆ E[·] . The canonical process (Bt)t≥0 is called the G-Poisson process under this sublinear expectation Eˆ[·] . We also have ∑ ∞ ti − ● Eˆ t If φ is increasing, then [φ(x + Bt)] = i=0 i! φ(x + i)e . ∑ ∞ (λt)i − ● Eˆ λt If φ is decreasing, then [φ(x + Bt)] = i=0 i! φ(x + i)e . ˆ ˆ In particular, E[Bt] = t and −E[−Bt] = λt . Thus, it characterizes a Poisson process with intensity uncertainty in [λ, 1] .
Remark 44 We consider the following integro-partial differential equation: ∫ ⟨ ⟩ − { − − Du(t, x), z ∂tu(t, x) sup (u(t, x + z) u(t, x) 2 )v(dz) d 1 + |z| (v,q,Q)∈U R \{0} (10) 1 +⟨Du(t, x), q⟩ + tr[D2u(t, x)QQT ]} = 0, 2 where ∫ 2 T sup { (|z|I[|z|≥1] + |z| I[|z|<1])v(dz) + |q| + tr[QQ ]} < ∞ (v,q,Q)∈U Rd and 16 Mingshang Hu, Shige Peng ∫ lim sup |z|2v(dz) = 0. ↓ κ 0 v∈V |z|≤κ
d φ For each given initial condition φ ∈ Cb.Lip(R ) , the viscosity solution u for (10) exists (see appendix). Thus, we can construct the corresponding sublinear expectation. Obviously, c d the canonical process (Bt)t≥0 is a Lévy∫ process and has the decomposition Bt = Bt + Bt . | | ∞ But it does not satisfy (1) if supv∈V |z|≤1 z v(dz) = . 7. Appendix In the appendix, we mainly consider the domination of viscosity solutions for (9) and (10). We refer to [1,4,6,7,12,19] and the references therein. For simplicity, we consider the following type of integro-partial differential equation:
2 d ∂tu(t, x) − G(Du(t, x),D u(t, x), u(t, x + ·)) = 0, u(0, x) = φ ∈ Cb.Lip(R ), (11) Rd × S × 1,2 → R ∈ × Rd where G : (d) Cb , (t, x) QT := (0,T ) . We suppose G satisfies the following conditions:
(A1) If pk → p , Xk → X and for (tk, xk) → (t, x), ϕk(tk, xk + ·) → ϕ(t, x + ·) locally d n n uniform on R , ϕk uniformly bounded and D ϕk → D ϕ , n = 1, 2 , locally uniform on QT , then
G(pk,Xk, ϕk(tk, xk + ·)) → G(p, X, ϕ(t, x + ·)).
(A2) If X ≥ Y and (ϕ − ψ)(t, ·) has a global minimum at x, then G(p, X, ϕ(t, x + ·)) ≥ G(p, Y, ψ(t, x + ·)).
(A3) For each constant c ∈ R , G(p, X, ϕ(t, x + ·) + c) = G(p, X, ϕ(t, x + ·)) .
For equation (9), the corresponding G is ∫ 1 G(p, X, u(t, x + ·)) = sup { (u(t, x + z) − u(t, x))v(dz) + ⟨p, q⟩ + tr[XQQT ]}. (v,q,Q)∈U Rd\{0} 2 Obviously, it satisfies all of the above assumptions. The above assumptions also hold for the equation (10). The definition of viscosity solution for (11) is the same as Definition 30. We also suppose that for each given κ ∈ (0, 1) , there exists Gκ : Rd × S(d) 1,2 ×SCb(QT ) × C (QT ) → R , where SCb(QT ) denotes the set of bounded upper or lower semicontinuous functions, satisfying the following assumptions: for p ∈ Rd , X, Y ∈ S(d) , − ∈ ∈ ∈ 1,2 u, v USCb(QT ) , w SCb(QT ) , ϕ , ψ , ψk Cb (QT ) ,
(B1) Gκ(p, X, ϕ(t, x + ·), ϕ(t, x + ·)) = G(p, X, ϕ(t, x + ·)) .
(B2) If X ≥ Y , (v − u)(t, ·) and (ϕ − ψ)(t, ·) have a global minimum at x, then Gκ(p, X, v(t, x + ·), ϕ(t, x + ·)) ≥ Gκ(p, Y, u(t, x + ·), ψ(t, x + ·)).
(B3) For c1, c2 ∈ R , κ κ G (p, X, w(t, x + ·) + c1, ϕ(t, x + ·) + c2) = G (p, X, w(t, x + ·), ϕ(t, x + ·)). Probability, Uncertainty and Quantitative Risk 17
d (B4) If ψk(t, ·) → w(t, ·) locally uniform on R and ψk(t, ·) uniformly bounded, then κ κ G (p, X, ψk(t, x + ·), ϕ(t, x + ·)) → G (p, X, w(t, x + ·), ϕ(t, x + ·)) .
For equation (9), the corresponding Gκ is ∫ Gκ(p, X, u(t, x + ·), ϕ(t, x + ·)) = sup { (u(t, x + z) − u(t, x))v(dz) ∈U | | (v,q,Q∫ ) z >κ 1 + (ϕ(t, x + z) − ϕ(t, x))v(dz) + ⟨p, q⟩ + tr[XQQT ]}. |z|≤κ 2
It is easy to check that this Gκ satisfies the above assumptions. The above assumptions also hold for equation (10).
Remark 45 Our assumptions (A1) and (B4) are different than those in [4,12]. This is because the measures in U may be singular, which makes the problem more difficult when taking the limit. See the following example.
Example 46 Consider V = {δx : x ∈ (1, 2]} and an upper semicontinuous function → f(x) = I[0,1](x) . Let fn be a sequence of continuous∫ functions such that fn f pointwise.∫ Then it is easy to show that supv∈V fn(z)v(dz) does not tend to supv∈V f(z)v(dz) .
Proposition 47 Suppose u ∈ USCb(QT ) ( u ∈ LSCb(QT ) ) is a viscosity subsolution 1,2 (viscosity supersolution) of (11). If u is continuous in x and for ϕ ∈ C (QT ) , (t, x) ∈ QT is a global maximum point (minimum point) of u − ϕ , then for each κ ∈ (0, 1) we have κ 2 ∂tϕ(t, x) − G (Dϕ(t, x),D ϕ(t, x), u(t, x + ·), ϕ(t, x + ·)) ≤ 0(≥ 0). The proof is found in [12] and the references therein. In the following, we first extend the matrix lemma in [6].
∈ S ≤ 1 Theorem 48 Suppose that X,Y (N) satisfy X Y < γ I for some γ > 0 . Define γ − −1 γ − −1 γ ≥ γ ≥ γ ≥ − 1 X = X(I γX) and Y = Y (I γY ) . Then Y X X and X γ I .
∈ S 1 ∈ RN Proof For each A (N) with A < γ I , it is easy to check that for each fixed y , {⟨ ⟩ − 1 | − |2} ⟨ − −1 ⟩ max Ax, x γ x y = A(I γA) y, y . (12) x∈RN ≤ 1 From the condition X Y < γ I , we have 1 1 ⟨Xx, x⟩ − |x − y|2 ≤ ⟨Y x, x⟩ − |x − y|2 forx, y ∈ RN . γ γ
Thus, for each fixed y ∈ RN , 1 1 max {⟨Xx, x⟩ − |x − y|2} ≤ max {⟨Y x, x⟩ − |x − y|2}. x∈RN γ x∈RN γ By (12), we obtain ⟨Xγ y, y⟩ ≤ ⟨Y γ y, y⟩ for all y ∈ RN , 18 Mingshang Hu, Shige Peng
γ ≥ γ γ ≥ γ ≥ − 1 which yields Y X . It is easy to check X X and X γ I . The proof is complete. □ In particular, we consider (n − 1)I −I · · · −I . . . −I .. .. . J := . nd . . . . .. .. −I −I · · · −I (n − 1)I nd×nd 1 1 It is easy to prove that J 2 = nJ and for each given γ ∈ (0, ) , J < I and nd nd n nd γ γ 1 J = J . In the following, we always use J for convenience. Then we nd 1 − nγ nd nd immediately get the following corollary.
Corollary 49 Let X ∈ S(d) , i = 1, . . . , n , satisfy i X1 0 ··· 0 . . . 0 .. .. . ≤ J . . . . nd . .. .. 0 0 ··· 0 Xn
1 − γ Then, for each given γ ∈ (0, ) , (I − γX ) 1 exists for i = 1, . . . , n , and X := n i i −1 γ Xi(I − γXi) , i = 1, . . . , n , satisfy X ≥ Xi and i Xγ 0 ··· 0 1 .. .. . 1 0 . . . 1 − Ind ≤ ≤ Jnd. γ . . . 1 − nγ . .. .. 0 ··· γ 0 0 Xn
We now give the main lemma (see Lemma 7.8 in [12]).
Lemma 50 Let ui ∈ USCb(QT ) be viscosity subsolutions of 2 ∂tu(t, x) − Gi(Du(t, x),D u(t, x), u(t, x + ·)) = 0, i = 1, . . . , n, 1,2 on Q , where G , i = 1, . . . , n , satisfy (A1) –(A3). Let ϕ ∈ C satisfy that T i ∑ b ¯ ∈ × Rnd n − (t, x¯1,..., x¯n) (0,T ) is a global maximum point of i=1 ui(t, xi) ϕ(t, x1, . . . , xn) . Moreover, suppose that there exist continuous functions g > 0 , g , . . . , g such that 0 1 n g1(t, x1) 0 ··· 0 . . . 0 .. .. . D2ϕ ≤ g (t, x)J + . 0 nd . . . . .. .. 0 0 ··· 0 gn(t, xn) ∈ 1 ∈ R ∈ S Then, for each γ (0, n ) , there exist bi and Xi (d) , i = 1, . . . , n , such that (i) b1 + ··· + bn = ∂tϕ(t,¯ x¯1,..., x¯n) ; (ii) Probability, Uncertainty and Quantitative Risk 19 X1 0 ··· 0 .. .. . g0(t,¯ x¯) 0 . . . g0(t,¯ x¯) − Ind ≤ − Knd ≤ Jnd, γ . . . 1 − nγ . .. .. 0 0 ··· 0 Xn where x¯ = (¯x ,..., x¯ ) and 1 n g1(t,¯ x¯1) 0 ··· 0 . . . 0 .. .. . K = ; nd . . . . .. .. 0 0 ··· 0 gn(t,¯ x¯n) (iii) for each i = 1, . . . , n , − ¯ ¯ · ≤ bi Gi(Dxi ϕ(t, x¯),Xi, ϕ(t, x¯1,..., x¯i−1, x¯i + , x¯i+1,..., x¯n)) 0.
Remark 51 Applying the above matrix inequalities, the proof in [12] still holds.
Remark 52 If ui is continuous in x, we further get that for each κ ∈ (0, 1) , − κ ¯ ¯ · ¯ · ≤ bi Gi (Dxi ϕ(t, x¯),Xi, ui(t, x¯i + ), ϕ(t, x¯1,..., x¯i−1, x¯i + , x¯i+1,..., x¯n)) 0.
We now give the main theorem, which combines the methods in [19] and [4].
d Theorem 53 (Domination theorem) Let ui ∈ USCb([0,T ) × R ) be viscosity subsolutions of 2 ∂tu(t, x) − Gi(Du(t, x),D u(t, x), u(t, x + ·)) = 0, i = 1, . . . , n, κ on QT , where Gi and Gi , i = 1, . . . , n , satisfy (A1)–(A3) and (B1)–(B4). We suppose also that (i) There exists a constant C > 0 such that for each κ ∈ (0, 1) , p, q ∈ Rd , X, Y ∈ S(d) , ∈ ∈ 1,2 ∈ 1 Rd ∈ 2 Rd u SCb(QT ) , φ C (QT ) , ψ1 Cb ( ) and ψ2 Cb ( ) , | κ · · · · − κ · · | G1 (p, X, u(t, ) + ψ1( ), φ(t, ) + ψ2( )) G1 (q, Y, u(t, ), φ(t, )) 2 ≤C(|p − q| + ||X − Y || + sup |Dψ1(x)| + sup (|Dψ2(x)| + |D ψ2(x)|)); x∈Rd x∈Rd
(ii) For given constants βi > 0 , i = 1, . . . , n , the following domination condition holds for ∈ × Rnd ∈ Rd × S ∈ 1,2 Gi : for∑ each (t, x1, .∑ . . , xn) (0,T ) ∑ , (pi,Xi) ∑ (d) and ϕi C ∑(QT ) such n n ≤ n · ≤ n n that i=1 βipi = 0 , i=1 βiXi 0 , i=1 βiui(t, xi + ) i=1 βiui(t, xi) and i=1 βiDϕi (t, xi) = 0 , ∑n κ · · ≤ → βiGi (pi,Xi, ui(t, xi + ), ϕi(t, xi + )) oκ(1) as κ 0. i=1 · ∈ Rd ∑Then a similar domination∑ also holds for the solutions: if ui(0, ) Cb.Lip( ) and n · ≤ n · ≤ i=1 βiui(0, ) 0 , then i=1 βiui(t, ) 0 for all t > 0 . ¯ Proof For each given δ > 0 , it is easy to check that for each 1 ≤ i ≤ n , u˜i := ¯ ui − δ/(T − t) is a viscosity solution of 2 ¯ 2 ∂tu˜(t, x) − Gi(Du˜(t, x),D u˜(t, x), u˜(t, x + ·)) ≤ −c, c := δ/T . (13)
For each λ > 0 , the sup-convolution of u˜i is defined by 20 Mingshang Hu, Shige Peng
| − |2 | − |2 ¯ λ { − x y } { − x y } − δ u˜i (t, x) := sup u˜i(t, y) = sup ui(t, y) − . y∈Rd λ y∈Rd λ T t
λ λ The function u˜i is upper semicontinuous in (t, x) and continuous in x. Moreover, u˜i is · ∈ Rd still a viscosity solution of (13) (see Lemma 7.3 in [12]). Noting that∑ ui(0, ) Cb.Lip( ) , n we can then choose a small λ > 0 such that for each λ ≤ λ , β u˜λ(0, ·) ≤ 0 . Since ∑ ∑0 0 i=1∑ i i ∑ n ≤ n ≤ ¯ ↓ n λ ↓ n i=1 βiui 0 follows from i=1 βiu˜i 0 in the limit δ 0 and i=1 βiu˜i i=1 βiu˜i as λ ↓ 0 , it suffices to prove the theorem under the additional assumptions: uˆi is a viscosity solution of (13), uˆi is continuous in x, and limt→T uˆi(t, x) = −∞ uniformly in [0,T ) × Rd . To prove the theorem, we assume to the contrary that ∑n sup βiuˆi(t, x) = m0 > 0. ∈ ×Rd (t,x) [0,T ) i=1 nd We define for x = (x1, x2, . . . , xn) ∈ R 1 ∑ ϕ (x) = |x − x |2 + ψ (x ), ε,β 2ε i j β 1 1≤i By the construction of ψβ , for small ε , β > 0 , the maximum of the above function is∑ achieved at some (t,¯ x¯) = (t,¯ x¯ ,..., x¯ ) with |x¯ | ≤ 2/β , |x¯ − x¯ |2 ≤ ∑ 1 n 1 1≤i Remark 54 The above theorem still holds for general G, which may contain (t, x, u, Du, D2u, u(t, ·)) . We have the following corollaries which are important in this paper. d Corollary 55 (Comparison theorem) Let the functions u ∈ USCb([0,T ) × R ) and d v ∈ LSCb([0,T ) × R ) be, respectively, a viscosity subsolution and a viscosity supersolution of (11). Suppose G and Gκ satisfy (A1) –(A3) and (B1) –(B4) and the condition (i) of Theorem 53. Furthermore, we suppose that d 1,2 (iii) For each (p, X) ∈ R × S(d) , u ∈ SCb(QT ) , ϕ 1 , ϕ 2 ∈ C (QT ) with Dϕ1(t, x) = Dϕ2(t, x) , as κ → 0 , κ κ |G (p, X, u(t, x + ·), ϕ1(t, x + ·)) − G (p, X, u(t, x + ·), ϕ2(t, x + ·))| → 0. · · ∈ Rd · ≤ · · ≤ · If u(0, ), v(0, ) Cb.Lip( ) and u(0, ) v(0 ), then u(t ) v(t, ) for all t > 0. Corollary 56 (Sub-additivity) Let uφ denote the viscosity solution of (11) with initial condition φ . Suppose G and Gκ satisfy all conditions of Corollary 55 and the following condition: d 1,2 (iv) For each (pi,Xi) ∈ R × S(d) , ui ∈ SCb(QT ) , ϕi ∈ C (QT ) , i = 1, 2 , κ G (p1 + p2,X1 + X2, (u1 + u2)(t, ·), (ϕ1 + ϕ2)(t, ·)) κ κ ≤ G (p1,X1, u1(t, ·), ϕ1(t, ·)) + G (p2,X2, u2(t, ·), ϕ2(t, ·)). φ+ψ φ ψ d Then u ≤ u + u for each φ , ψ ∈ Cb.Lip(R ) . Corollary 57 (Convexity) Let uφ denote the viscosity solution of (11) with initial condition φ . Suppose G and Gκ satisfy all conditions of Corollary 55 and the following condition: d 1,2 (v) For each λ ∈ (0, 1) , (pi,Xi) ∈ R × S(d) , ui ∈ SCb(QT ) , ϕi ∈ C (QT ) , i = 1, 2 , κ G (λ(p1,X1, u1(t, ·), ϕ1(t, ·)) + (1 − λ)(p2,X2, u2(t, ·), ϕ2(t, ·))) κ κ ≤ λG (p1,X1, u1(t, ·), ϕ1(t, ·)) + (1 − λ)G (p2,X2, u2(t, ·), ϕ2(t, ·)). λφ+(1−λ)ψ φ ψ d Then u ≤ λu + (1 − λ)u for each φ , ψ ∈ Cb.Lip(R ) . For our main equations (9) and (10), it is easy to check that all assumptions (i)–(v) 22 Mingshang Hu, Shige Peng hold. Perron's method for (11) still holds (the proof is similar to Proposition 1 in [1]). ∈ 2 Rd For each φ Cb ( ) , it is easy to find constants M1 and M2 such that u(t, x) := M1t + φ and v(t, x) := M2t + φ are, respectively, the viscosity subsolution and supersolution of (9) and (10). By Perron's method and approximation, the solutions of (9) d and (10) exist for each φ ∈ Cb.Lip(R ) . Acknowledgements This work was supported by National Key R&D Program of China (Grant No. 2018YFA0703900), National Natural Science Foundation of China (Grant No. 11671231) , Tian Yuan Fund of the National Natural Science Foundation of China (Grant Nos. 11526205 and 11626247) and National Basic Research Program of China (973 Program) (Grant No. 2007CB814900). References [ 1 ] Alvarez, O. and Tourin, A., Viscosity solutions of nonlinear integrodifferential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1996, 13(3): 293-317. [ 2 ] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., Thinking Coherently, RISK, 1997, 10: 68-71. [ 3 ] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., Coherent measures of risk, Mathematical Finance, 1999, 9(3): 203-228. [ 4 ] Barles, G. and Imbert, C., Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited, Ann. I. H. Poincaré-AN, 2008, 25: 567-585. 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