- Email: [email protected]
Stochastic differential equations driven by G -Brownian motion and ordinary differential equations
Available online at www.sciencedirect.com
ScienceDirect Stochastic Processes and their Applications 124 (2014) 3869–3885 www.elsevier.com/locate/spa
Stochastic differential equations driven by G-Brownian motion and ordinary differential equations Peng Luo a,b,c , Falei Wang a,∗ a School of Mathematics, Shandong University, China b Qilu Securities Institute for Financial Studies, Shandong University, China c Department of Mathematics and Statistics, University of Konstanz, Germany
Received 29 November 2013; received in revised form 2 April 2014; accepted 1 July 2014 Available online 8 July 2014
Abstract In this paper, we show that the integration of a stochastic differential equation driven by G-Brownian motion (G-SDE for short) in R can be reduced to the integration of an ordinary differential equation (ODE for short) parameterized by a variable in (Ω , F). By this result, we obtain a comparison theorem for G-SDEs and its applications. c 2014 Elsevier B.V. All rights reserved. ⃝
MSC: 60H30; 60H10 Keywords: G-Brownian motion; G-Itˆo’s formula; G-SDE; Comparison theorem
1. Introduction Motivated by uncertainty problems, risk measures and the superhedging in finance, Peng systemically established a time-consistent fully nonlinear expectation theory (see [13–15]). As a typical and important case, Peng introduced the G-expectation theory (see [16,17] and the references therein) in 2006. In the G-expectation framework (G-framework for short), the notion of G-Brownian motion and the corresponding stochastic calculus of Itˆo’s type were established. On ∗ Corresponding author. Tel.: +86 13573775301.
E-mail addresses: [email protected] (P. Luo), [email protected] (F. Wang). http://dx.doi.org/10.1016/j.spa.2014.07.004 c 2014 Elsevier B.V. All rights reserved. 0304-4149/⃝
3870
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
that basis, Gao [4] and Peng [17] studied the existence and uniqueness of the solution of G-SDE under a standard Lipschitz condition. Moreover, Lin [12] obtained the existence and uniqueness of the solution of G-SDE with reflecting boundary. For a recent account and development of this theory we refer the reader to [1,7–11,19]. Under the classical framework, Doss [3] and Huang, Xu and Hu [6] studied the sample solutions of stochastic differential equations, which enables us to transfer a stochastic differential equation into a set of ordinary differential equations for each sample path. Using the method of sample solutions to SDEs, Huang [5] established a comparison theorem of SDEs. The aim of this paper is to study the sample solutions of G-SDEs by ODEs parameterized by a variable in basis probability space. Since G-SDE admits a unique solution in the space MG2 (0, T ), the main difficulty is how to prove that the sample solution belongs to this space. We overcome this problem through some G-stochastic calculus techniques. Then we show that the solution of G-SDE can be represented as a function of both G-Brownian motion and a finite variation process. Since we can use the existing results in the theory of ordinary differential equations directly, this approach provides a powerful tool both in the theoretical analysis and in the practical computation of G-SDEs. In particular, we get a new kind of comparison theorem for G-SDEs. Moreover, a necessary and sufficient condition for comparison theorem of G-SDEs is also obtained. This paper is organized as follows: In Section 2, we recall some notations and results that we will use in this paper. In Section 3, we study the sample solution of G-SDE under some strong conditions, then, in Section 4, we extend this result to a more general case. Finally in Section 5, we establish a new kind of comparison theorem and give its applications. 2. Preliminaries The main purpose of this section is to recall some preliminary results in G-framework which are needed in the sequel. More details can be found in Denis et al. [2], Li and Peng [7], Lin [9,12] and Peng [17]. 2.1. Sublinear expectation Definition 2.1. Given a set Ω and a linear space H of real valued functions defined on Ω . Moreover, if X i ∈ H, i = 1, . . . , d, then ϕ(X 1 , . . . , X d ) ∈ H for all ϕ ∈ Cb,Lip (Rd ), where Cb,Lip (Rd ) is the space of all bounded real-valued Lipschitz continuous functions. A sublinear expectation Eˆ on H is a functional Eˆ : H → R satisfying the following properties: for all X, Y ∈ H, ˆ ] ≥ E[Y ˆ ]; (a) Monotonicity: if X ≥ Y , then E[X ˆ (b) Constant preserving: E[c] = c, ∀ c ∈ R; ˆ ˆ ] + E[Y ˆ ]; (c) Sub-additivity: E[X + Y ] ≤ E[X ˆ ˆ ], ∀ λ ≥ 0. (d) Positive homogeneity: E[λX ] = λE[X ˆ is called a sublinear expectation space. X ∈ H is called a random The triple (Ω , H, E) ˆ variable in (Ω , H, E). We often call Y = (Y1 , . . . , Yd ), Yi ∈ H a d-dimensional random vector ˆ in (Ω , H, E). ˆ a n-dimensional random vector Y = Definition 2.2. In a sublinear expectation space (Ω , H, E), (Y1 , . . . , Yn ) is said to be independent from an m-dimensional random vector X = (X 1 , . . . , X m )
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3871
under Eˆ if for any test function ϕ ∈ Cb,Lip (Rm+n ) ˆ ˆ E[ϕ(x, ˆ E[ϕ(X, Y )] = E[ Y )]x=X ]. Definition 2.3. Let X 1 and X 2 be two n-dimensional random vectors defined on sublinear expectation spaces (Ω1 , H1 , Eˆ 1 ) and (Ω2 , H2 , Eˆ 2 ), respectively. They are called identically d
distributed, denoted by X 1 = X 2 , if Eˆ 1 [ϕ(X 1 )] = Eˆ 2 [ϕ(X 2 )],
∀ϕ ∈ Cb,Lip (Rn ).
d X¯ is said to be an independent copy of X if X¯ = X and X¯ is independent from X .
Definition 2.4 (G-Normal Distribution). A random variable X on a sublinear expectation space ˆ is called (centralized) G-normal distributed if for any a, b ≥ 0 (Ω , H, E) d a X + b X¯ = a 2 + b2 X, where X¯ is an independent copy of X . The letter G denotes the function 1 2 + (σ a − σ 2 a − ) 2 2 ] ≤ E[X ˆ ˆ 2 ] =: σ 2 . with σ 2 := −E[−X G(a) =
2.2. G-Brownian motion Definition 2.5 (G-Brownian Motion). A process (Bt ∈ H)t≥0 on a sublinear expectation space ˆ is called a G-Brownian motion if the following properties are satisfied: (Ω , H, E) (a) B0 = 0. d √ (b) For each t, s ≥ 0 the increment Bt+s − Bt = s X and independent from (Bt1 , Bt2 , . . . , Btn ) for each n ∈ N, 0 ≤ t1 ≤ t2 ≤ · · · ≤ tn ≤ t, where X is G-normal distributed. Denote by Ω = C0 (R+ ) the space of all R-valued continuous paths (ωt )t∈R+ , with ω0 = 0, equipped with the distance ρ(ω1 , ω2 ) :=
∞ i=1
2−i [ max |ωt1 − ωt2 | ∧ 1]. t∈[0,i]
B(Ω ) is the Borel σ -algebra of Ω . For each t ∈ [0, ∞), we introduce the following spaces. • • • • •
Ωt := {ω(· ∧ t) : ω ∈ Ω }, Ft := B(Ωt ), L 0 (Ω ): the space of all B(Ω )-measurable real functions, L 0 (Ωt ): the space of all Ft -measurable real functions, Bb (Ω ): all bounded elements in L 0 (Ω ), Bb (Ωt ) := Bb (Ω ) ∩ L 0 (Ωt ), Cb (Ω ): all continuous elements in Bb (Ω ), Cb (Ωt ) := Cb (Ω ) ∩ L 0 (Ωt ).
In Peng [17], a G-Brownian motion is constructed on a sublinear expectation space ˆ (Eˆ t )t≥0 ), where L p (Ω ) is a Banach space under the natural norm ∥X ∥ p = (Ω , L1G , E, G ˆ E[|X | p ]1/ p . In this space the corresponding canonical process Bt (ω) = ωt is a G-Brownian p p motion. Denote by L b (Ω ) the completion of Bb (Ω ). Denis et al. [2] proved that L b (Ω ) ⊃ p LG (Ω ) ⊃ Cb (Ω ), and there exists a weakly compact family P of probability measures defined
3872
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
on (Ω , B(Ω )) such that ˆ ] = sup E P [X ], E[X P∈P
X ∈ L1G (Ω ).
ˆ Consider a 1-dimensional Remark 2.6. Denis et al. [2] gave a concrete set P M that represents E. Brownian motion Bt on (Ω , F, P), then t −1 2 2 2 P M := Pθ : Pθ = P ◦ X , X t = θs d Bs , θ ∈ L F ([0, T ]; [σ , σ ]) 0
ˆ where L 2 ([0, T ]; [σ 2 , σ 2 ]) is the collection of all F-adapted measuris a set that represents E, F 2 able processes with σ ≤ |θ (s)|2 ≤ σ 2 . Now we introduce the natural Choquet capacity c(A) := sup P(A), P∈P
A ∈ B(Ω ).
Definition 2.7. A set A ⊂ B(Ω ) is polar if c(A) = 0. A property holds “quasi-surely” (q.s.) if it holds outside a polar set. Definition 2.8. A real function X on Ω is said to be quasi-continuous if for each ε > 0, there exists an open set O with c(O) < ε such that X | O c is continuous. Definition 2.9. We say that X : Ω → R has a quasi-continuous version if there exists a quasicontinuous function Y : Ω → R such that X = Y , q.s. p
p
Then L b (Ω ) and LG (Ω ) can be characterized as follows: p ˆ | p I|X |≥N ] = 0} L b (Ω ) = {X ∈ L 0 (Ω )| lim E[|X N →∞
and p
p
LG (Ω ) = {X ∈ L b (Ω ) | X has a quasi-continuous version}. 2.3. G-stochastic calculus Peng [17] also introduced the related stochastic calculus of Itˆo’s type with respect to GBrownian motion (see Li and Peng [7], Lin [12] for more general and systematic research). Let T ∈ R+ be fixed. Definition 2.10. For each p ≥ 1, consider the following simple type of processes: N −1 0, p MG (0, T ) = η := ηt (ω) = ξ j (ω)I[t j ,t j+1 ) (t) ∀ N > 0, j=0
0 = t0 < · · · < t N = T, ξ j ∈ 0, p
p
p LG (Ωt j ),
j = 0, 1, 2, . . . , N − 1 .
Denote by MG (0, T ) the completion of MG (0, T ) under the norm T 1/ p p ˆ ∥η∥ M p (0,T ) = E[|η(t)| ]dt . G
0
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3873
Definition 2.11. For each η ∈ MG0,2 (0, T ) with the form N −1
ηt (ω) =
ξk (ω)I[tk ,tk+1 ) (t),
k=0
define I (η) =
T
ηs d Bs :=
0
N −1
ξk (Bt N − Bt N ). k
k+1
k=0
The mapping I : MG0,2 (0, T ) → L2G (ΩT ) can be continuously extended to I : MG2 (0, T ) → L2G (ΩT ). For each η ∈ MG2 (0, T ), the stochastic integral is defined by T I (η) := ηs d Bs , η ∈ MG2 (0, T ). 0
Unlike the classical theory, the quadratic variation process of G-Brownian motion B is not always a deterministic process and it can be formulated in L2G (Ωt ) by ⟨B⟩t := lim
N −1
N →∞
where tiN =
iT N
i=0
(Bt N − Bt N ) = 2
i
i+1
Bt2
t
−2
Bs d Bs ,
0
for each integer N ≥ 1.
Definition 2.12. Define a mapping MG0,1 (0, T ) → L1G (ΩT ): T
Q(η) = 0
ηs d⟨B⟩s :=
N −1
ξk [⟨B⟩t N − ⟨B⟩t N ]. k
k+1
k=0
Then Q can be uniquely extended to MG1 (0, T ) → L1G (ΩT ). We also denote this mapping by T Q(η) := ηs d⟨B⟩s , η ∈ MG1 (0, T ). 0
In view of the dual formulation of G-expectation as well as the properties of the quadratic variation process ⟨B⟩ in G-framework, Gao [4] obtained the following BDG type inequalities. p
Lemma 2.13. For each p ≥ 1 and η ∈ MG (0, T ), t p T ˆE sup ηs d⟨B⟩s ≤ σ¯ 2 p T p−1 ˆ s | p ]ds. E[|η 0≤t≤T
0
0
p
Lemma 2.14. Let p ≥ 2 and η ∈ MG (0, T ). Then there exists some constant C p depending only on p and T such that t p T 2p Eˆ sup ηs d Bs ≤ C p Eˆ |ηs |2 ds . 0≤t≤T
0
0
3874
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3. G-Stochastic differential equation Let us first recall some notations, • C n (Rd ): the space of all functions of class C n from Rd into R, n (Rd ): the space of all bounded functions of class C n (Rd ) whose partial derivatives of • Cb,lip order less than or equal to n are bounded Lipschitz continuous functions, • C n ([0, T ] × Rd ): the space of all functions of class C n from [0, T ] × Rd into R, n ([0, T ] × Rd ): the space of all bounded functions of class C n ([0, T ] × Rd ) whose partial • Cb,lip derivatives of order less than or equal to n are bounded Lipschitz continuous functions. Consider the following SDE driven by a 1-dimensional G-Brownian motion: t t t Xt = X0 + b(s, X s )ds + h(s, X s )d⟨B⟩s + σ (s, X s )d Bs , t ∈ [0, T ], 0
0
(1)
0
where the initial condition X 0 ∈ R is a given constant. We recall the following assumption. (H) b, h, σ : Ω × [0, T ] × R → R are given functions satisfying b(·, x), h(·, x), σ (·, x) ∈ MG2 (0, T ) for each x ∈ R. Moreover, there exists some constant K such that |ϕ(t, x) − ϕ(t, y)| ≤ K |x − y| for each t ∈ [0, T ], x, y ∈ R, ϕ = b, h and σ , respectively. From Peng [17], we have the following. Theorem 3.1. Under the assumption (H), there exists a unique solution X ∈ MG2 (0, T ) to the stochastic differential equation (1). Remark 3.2. We remark that there is a potential to extend our results to a much more general setting. However, in order to focus on the main ideas, in this paper we content ourselves with the case that the coefficients are 1-dimensional satisfying bounded conditions. In particular, by slightly more involved estimates, we can extend our results to the multi-dimensional case without bounded conditions. 3.1. A simple case In order to explain the main ideas, we first consider a simple G-SDE, t t 1 t σ (X s )∂x σ (X s )d⟨B⟩s + σ (X s )d Bs , Xt = X0 + b(X s )ds + 2 0 0 0
t ∈ [0, T ], (2)
1 (R) and b(x) ∈ C where σ (x) ∈ Cb,lip b,lip (R). By Theorem 3.1, G-SDE (2) admits a unique 2 solution X ∈ MG (0, T ). Now consider the following ODE
dy = σ (y), dx
y(0) = v ∈ R.
The above ODE has a unique solution y = ϕ(x, v) ∈ C(R2 ). Then, ∂x ϕ = σ (ϕ),
ϕ(0, v) = v.
(3)
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3875
Consequently, ∂v ϕ(x, v) = exp
x
∂x σ (ϕ(y, v))dy ,
0
∂x2x ϕ(x, v) = (∂x σ σ )(ϕ(x, v)).
Next we introduce the following ODE with parameter ω: Bt (ω) d V = exp − ∂x σ (ϕ(y, Vt ))dy b(ϕ(Bt (ω), Vt ))dt, t 0 V0 = X 0 .
(4)
For every fixed ω, recalling the Cauchy–Lipschitz theorem, Eq. (4) has a unique solution Vt = Vt (ω) and Vt is a continuous finite variation process. Moreover, Vt (ω) is a continuous function on (Ω , ρ). The following result is important in our future discussion. Lemma 3.3. For any p ≥ 0, there exists a constant C p depending only on p such that, ˆ sup e p|Bt | ] ≤ C p . E[ 0≤t≤T
Proof. For any p ≥ 0, we have n | p B | t p|B | t ˆ sup e Eˆ sup E[ ]≤ . n! 0≤t≤T 0≤t≤T n=0 Applying Doob’s maximal inequality yields that n ˆE[ sup | p Bt |n ] ≤ 1 + 1 ˆ p BT |n ]. E[| n−1 0≤t≤T By Exercise 1.7 in Chapter 3 of Peng [17], one can show that for some constant C ′p depending only on p, | p BT |n Eˆ ≤ C ′p . n! n=0 Since limn 1 +
1 n−1
n
= e, we can find some constant C p depending only on p such that,
ˆ sup e p|Bt | ] ≤ C p , E[ 0≤t≤T
which is the desired result.
p
Lemma 3.4. For each p ≥ 1, Vt ∈ LG (Ωt ). Moreover, there exists some constant C p depending only on p such that for each s ≤ t ∈ [0, T ], ˆ tV − TsV | p ] ≤ C p |t − s| p , E[|T where T V is the total variation process of V .
3876
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
Proof. By Eq. (4), t Vt = V0 + exp − 0
Bu (ω)
∂x σ (ϕ(y, Vu ))dy b(ϕ(Bu (ω), Vu ))du.
0
Denote by C p a constant depending only on p, which is allowed to change from line to line. Then applying Lemma 3.3, we conclude T Bu (ω) ˆE[ sup |Vt | p ] ≤ C p Eˆ |V0 | p + exp − p ∂x σ (ϕ(y, Vu ))dy du 0
0≤t≤T
0
ˆ sup eC p|Bt | ]) ≤ C p . ≤ C p (|V0 | p + C E[ 0≤t≤T p
Since Vt (ω) is a continuous function on (Ω , ρ), recalling the pathwise description of LG (Ωt ), p Vt ∈ LG (Ωt ) for each p ≥ 1. Note that Bu t TtV = exp − ∂x σ (ϕ(y, Vu ))dy |b(ϕ(Bu , Vu ))|du, 0
0
applying Lemma 3.3 again, we obtain for each s ≤ t ∈ [0, T ], ˆ tV − TsV | p ] ≤ C p |t − s| p , E[|T which completes the proof.
By Lemma 3.4, we deduce that ϕ(Bt , Vt ) ∈ MG2 (0, T ). Since ϕ satisfies the conditions of Theorem A.1, applying the G-Itˆo formula, we get 1 dϕ(Bt , Vt ) = ∂x ϕ(Bt , Vt )d Bt + ∂v ϕ(Bt , Vt )d Vt + ∂x2x ϕ(Bt , Vt )d⟨B⟩t 2 1 = b(ϕ(Bt , Vt ))dt + ∂x σ (ϕ(Bt , Vt ))σ (ϕ(Bt , Vt ))d⟨B⟩t + σ (ϕ(Bt , Vt ))d Bt . 2 Consequently, X t = ϕ(Bt , Vt ) is the unique MG2 (0, T )-solution of G-SDE (2). 3.2. The general case In this section, we will extend the above result to a more general case, where all the coefficients are functions in t, Bt and x. Assume b(t, x, y), h(t, x, y) ∈ Cb,lip ([0, T ] × R2 ) 1 ([0, T ] × R2 ). It is obvious G-SDE and σ (t, x, y) ∈ Cb,lip t t t Xt = X0 + b(s, Bs , X s )ds + h(s, Bs , X s )d⟨B⟩s + σ (s, Bs , X s )d Bs , 0
0
0
(5)
t ∈ [0, T ] has a unique solution X ∈ MG2 (0, T ). Then the following ODE dy = σ (t, x, y), dx
y(t, 0) = v ∈ R
(6)
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3877
admits a unique solution y = ϕ(t, x, v) ∈ C([0, T ] × R2 ). Moreover, we can get x ∂ y σ (t, u, ϕ(t, u, v))du ∂v ϕ(t, x, v) = exp 0
and
∂t ϕ(t, x, v) = exp ×
0 x
x
∂ y σ (t, z, ϕ(t, z, v))dz
∂t σ (t, u, ϕ(t, u, v))e
−
u 0
∂ y σ (t,z,ϕ(t,z,v))dz
du .
0
Set g(t, x, v) := ∂v ϕ −1 (t, x, v)(b(t, x, ϕ(t, x, v)) − ∂t ϕ(t, x, v)), 1 f (t, x, v) := ∂v ϕ −1 (t, x, v) h(t, x, ϕ(t, x, v)) − (∂x σ + ∂ y σ σ )(t, x, ϕ(t, x, v)) . 2 Then consider the following initial value problem with parameter ω: d Vt = g(t, Bt (ω), Vt )dt + f (t, Bt (ω), Vt )d⟨B⟩t (ω), V0 = X 0 .
(7)
Note that ⟨B⟩t is a continuous finite variation process, then the ODE (7) has a unique solution V = Vt (ω) and Vt is a continuous finite variation process. Since ⟨B⟩t (ω) is not always a deterministic process, in general we cannot get Vt (ω) is a continuous function on (Ω , ρ) as the above section. However, we also have the following result. Lemma 3.5. For each p ≥ 1, there exists some constant C p depending only on p such that, for each s ≤ t ∈ [0, T ], ˆ tV − TsV | p ] ≤ C p |t − s| p , E[|T where T V is the total variation process of V . Proof. The proof is immediate in light of Lemma 3.4.
Now we shall give the main result of this section. Theorem 3.6. Assume b(t, x, y), h(t, x, y) ∈ Cb,lip ([0, T ] × R2 ) and σ (t, x, y) ∈ 1 ([0, T ] × R2 ), then for each p ≥ 1, V ∈ L p (Ω ) and ϕ(t, B , V ) is the unique M 2 (0, T )Cb,lip t t t t G G solution of G-SDE (5). p
Proof. It is obvious Vt ∈ L b (Ωt ). Then applying Theorem A.1, we obtain q.s. dϕ(t, Bt , Vt ) = ∂t ϕ(t, Bt , Vt )dt + ∂x ϕ(t, Bt , Vt )d Bt + ∂v ϕ(t, Bt , Vt )d Vt 1 + ∂x2x ϕ(t, Bt , Vt )d⟨B⟩t 2 = b(t, Bt , ϕ(t, Bt , Vt ))dt + h(t, Bt , ϕ(t, Bt , Vt ))d⟨B⟩t + σ (t, Bt , ϕ(t, Bt , Vt ))d Bt . By a standard argument, there exists some constant C such that, t ˆE[|ϕ(t, Bt , Vt ) − X t |2 ] ≤ C ˆ E[|ϕ(s, Bs , Vs ) − X s |2 ]ds. 0
Applying Gronwall’s lemma, we obtain ϕ(t, Bt , Vt ) = X t , q.s.
3878
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
By the uniqueness of solution of ODE (6), v = ϕ(t, −x, ϕ(t, x, v)), p
thus, Vt = ϕ(t, −Bt , X t ) q.s. In particular, Vt has a quasi-continuous version and Vt ∈ LG (Ωt ). The proof is completed. 4. G-diffusion process The objective of this section is to remove the condition that σ is continuously differentiable and to obtain a more general result on this topic. By an approximation approach, we can also represent the solution of G-SDE as a function of Bt and a continuous finite variation process Vt as the above section. Theorem 4.1. If b, σ, h ∈ Cb,lip (R), then there exists a unique continuous finite variation p process Vt ∈ LG (Ωt ) for each p ≥ 1 such that, X t = ϕ(Bt , Vt ), where ϕ is the solution of the ODE: ∂x ϕ(x, v) = σ (ϕ(x, v)),
ϕ(0, v) = v.
1 (R), then for q.s. ω, V (ω) is the solution of the following ODE: Moreover if σ ∈ Cb,lip t
Bt d Vt = exp − ∂x σ (ϕ(y, Vt ))dy b(ϕ(Bt , Vt ))dt + (h(ϕ(Bt ), Vt )) 0 1 − ∂x σ σ (ϕ(Bt , Vt ))d⟨B⟩t , 2 V0 = X 0 .
(8)
1 (R), then the theorem holds true. If σ ∈ C Proof. If σ ∈ Cb,lip b,lip (R), one can define σ n (x) := σ (y)ρn (y − x)dy = σ (y + x)ρn (y)dy, R
R
where ρn is a nonnegative function defined on {x : |x| ≤ n1 } with R ρn (y)dy = 1. From this definition, we conclude that K n |σ (x) − σ (x)| ≤ |σ (y + x) − σ (x)|ρn (y)dy ≤ K |y|ρn (y)dy ≤ , n R R C∞
where K is the Lipschitz coefficient of b, h and σ . 1 (R). Thus, X n := ϕ n (B , V n ) is the solution of G-SDE: For each n, it is obvious σ n ∈ Cb,lip t t t t t t X tn = X 0 + b(X sn )ds + h(X sn )d⟨B⟩s + σ n (X sn )d Bs , t ∈ [0, T ], 0
0
0
where ϕ n satisfies ∂x ϕ n (x, v) = σ n (ϕ n (x, v)),
ϕ n (0, v) = v
3879
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
and Bt n n n n n n d Vt = exp − ∂x σ (ϕ (y, Vt ))dy b(ϕ (Bt , Vt ))dt + h(ϕ n (Bt , Vtn )) 0 1 n n n n − ∂x σ σ (ϕ (Bt , Vt ))d⟨B⟩t , 2 n V0 = X 0 . For each n, there exists some constant C depending only on T and K such that, ˆ sup |X tn − X t |2 ] ≤ C . E[ n2 t∈[0,T ] Indeed, applying BDG inequalities, we obtain for some constant C, which is allowed to change from line to line, t ˆE[ sup |X tn − X t |2 ] ≤ Eˆ sup b(X sn ) − b(X s )ds t∈[0,T ]
0
t∈[0,T ] t
+ 0
≤ C Eˆ
K |X tn
− X t |ds
K |X tn
K − Xt | + n
0 T
+ 0
1 + n2
1 + n2
≤C ≤C
T
0
n
0
2
(X sn ) − σ (X s )d Bs
2
T
+ 0
K |X tn
− X t |d⟨B⟩s
2
d Bs
n 2 ˆ E[|X t − X t | ]dt
T
ˆ sup E[ 0
σ
+
2
T
t
h(X sn ) − h(X s )d⟨B⟩s
s∈[0,t]
|X sn
− X s | ]dt . 2
By Gronwall’s lemma, we can get the desired result. Moreover, choosing a subsequence if necessary, X n → X uniformly in [0, T ] q.s. For each v1 , v2 , x ∈ R, |ϕ n (x, v1 ) − ϕ(x, v2 )| ≤ |ϕ n (x, v1 ) − ϕ n (x, v2 )| + |ϕ n (x, v2 ) − ϕ(x, v2 )|. Applying Taylor’s formula yields that |ϕ n (x, v1 ) − ϕ n (x, v2 )| ≤ |∂v ϕ n (x, v ∗ )| |v1 − v2 | ≤ |v1 − v2 |eC|x| . By the definitions of ϕ n and ϕ, we obtain x n n n |ϕ (x, v2 ) − ϕ(x, v2 )| ≤ σ (ϕ (s, v2 )) − σ (ϕ(s, v2 ))ds
0 |x|
≤ 0
1 ds. K |ϕ n (s, v2 ) − ϕ(s, v2 )| + n
3880
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
From Gronwall’s lemma, we conclude for some constant C |x| C|x| e . |ϕ n (x, v1 ) − ϕ(x, v2 )| ≤ C |v1 − v2 | + n Define Vt := ϕ(−Bt , X t ), thus X t = ϕ(Bt , Vt ) and Vt has a quasi-continuous version. Moreover, lim
sup |ϕ n (−Bt , X tn ) − ϕ(−Bt , X t )|
sup |Vtn − Vt | = lim
n→∞ t∈[0,T ]
n→∞ t∈[0,T ]
≤ C lim
n→∞
sup t∈[0,T ]
|X tn
sup |Bt | |Bt | C t∈[0,T ] − X t | + sup e = 0. t∈[0,T ] n
Since for each n and t, s ∈ [0, T ], there exists some constant C such that |Vtn − Vsn | ≤ C sup eC|Bt | |t − s|. t∈[0,T ]
Thus |Vt − Vs | ≤ C sup eC|Bt | |t − s|. t∈[0,T ] p
p
By the pathwise description of LG (Ωt ), Vt ∈ LG (Ωt ) for each p ≥ 1 and the proof is completed. In general, we can also get Theorem 4.2. If b, σ, h ∈ Cb,lip ([0, T ] × R2 ), then there exists a unique continuous finite p variation process Vt ∈ LG (Ωt ) for each p ≥ 1 such that X t = ϕ(t, Bt , Vt ), 1 ([0, T ]×R2 ), then for q.s. ω, V is the solution where ϕ is given by Eq. (6). Moreover if σ ∈ Cb,lip t of ODE (7).
5. Comparison theorem for G-SDEs In the above sections, we establish the relations between G-SDEs and ODEs. From these results, we shall study the comparison theorem for G-SDEs. We refer to Lin [8] for some sufficient condition under which a comparison theorem for G-SDEs is also obtained by virtue of a stochastic calculus approach. We begin with a lemma, which is from [5]. Lemma 5.1. Assume that two functions f (t, x) and f˜(t, x) are defined on R2 , satisfying the Carath´eodory conditions, that is, they are measurable in t, continuous in x and dominated by a locally integrable function m t in R2 . Let (t0 , x0 ), (t0 , x˜0 ) be two points in R2 such that x0 ≤ x˜0 . Moreover, xt is a solution to the initial value problem d xt = f (t, xt )dt,
x t0 = x 0 ,
and x˜t is the maximal solution to the problem d x˜t = f˜(t, x˜t )dt,
x˜t0 = x˜0 .
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3881
If the inequality (t − t0 ) f (t, x) ≤ (t − t0 ) f˜(t, x) holds a.e. in R2 , then x(t) ≤ x(t) ˜ for every t in the common interval of existence of the solutions xt and x˜t . Then we have the following comparison theorem. 1 ([0, T ]×R2 ) Theorem 5.2. Let b(t, x, y), h(t, x, y) ∈ Cb,lip ([0, T ]×R2 ) and σ (t, x, y) ∈ Cb,lip be given. If there exists three functions σ˜ , f˜ and g˜ satisfying the Carath´eodory conditions and the inequalities
xσ (t, x, y) ≤ x σ˜ (t, x, y), 2G( f (t, x, y) − f˜(t, x, y)) + g(t, x, y) − g(t, ˜ x, y) ≤ 0
in [0, T ] × R2 .
Then for the unique solution X t of SDE (5) X t ≤ ϕ(t, ˜ Bt , V˜t ) holds for q.s. ω and every t in the common interval where both sides are defined. Here ϕ˜ and V˜ are the maximal solutions to the problems d ϕ˜ = σ˜ (t, x, ϕ), ˜ dx
ϕ(t, ˜ 0, v) = v ∈ R
and d V˜t = g(t, ˜ Bt , V˜t )dt + f˜(t, Bt (ω), V˜t )d⟨B⟩t ,
V˜0 = X˜ 0
with X 0 ≤ X˜ 0 , respectively. Proof. According to Lemma 5.1, we get ϕ(t, x, v) ≤ ϕ(t, ˜ x, v) ˜ provided v ≤ v. ˜ From [2] or [18], d⟨B⟩t = αˆ t (ω)dt, where αˆ is well defined for each ω and q.s. takes value in [σ 2 , σ 2 ]. Since 2G( f (t, x, v) − f˜(t, x, v)) + g(t, x, v) − g(t, ˜ x, v) ≤ 0, we obtain ( f (t, x, v) − f˜(t, x, v))αˆ t + g(t, x, v) − g(t, ˜ x, v) ≤ 0. Then applying Lemma 5.1 again, we also have Vt ≤ V˜t in the common interval where both sides are defined, which is the desired result. Now we consider some examples of its applications. Example 5.3. Consider two G-SDEs with the same diffusion coefficient σ : i d X t = bi (t, Bt , X ti )dt + h i (t, Bt , X ti )d⟨B⟩t + σ (t, Bt , X ti )d Bt , X 0i = X 0i (i = 1, 2), where σ, b1 , b2 , h 1 , h 2 satisfy the conditions in Theorem 5.2 and X 01 ≤ X 02 , b1 − b2 + 2G(h 1 − h 2 ) ≤ 0. Denote: g i (t, x, v) = ∂v ϕ −1 (t, x, v)(bi (t, x, ϕ(t, x, v)) − ∂t ϕ(t, x, v)), 1 f i (t, x, v) = ∂v ϕ −1 (t, x, v) h i (t, x, ϕ(t, x, v)) − (∂x σ + ∂ y σ σ )(t, x, ϕ(t, x, v)) , 2 (i = 1, 2).
3882
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
One can easily show that g 1 − g 2 + 2G( f 1 − f 2 ) ≤ 0. Applying Theorem 5.2, we obtain X t1 ≤ X t2 q.s. Remark 5.4. In Example 5.3, we can also assume that σ ∈ Cb,lip ([0, T ] × R2 ). Indeed, applying Theorem 4.1, there exists a sequence X i,n → X i uniformly in [0, T ]. Then we conclude X t1 ≤ X t2 from X t1,n ≤ X t2,n for each t ∈ [0, T ]. In particular, we obtain a necessary and sufficient condition for comparison theorem of 1dimensional G-SDEs. Theorem 5.5. Consider two G-SDEs: d X ti,xi = bi (X ti,xi )dt + h i (X ti,xi )d⟨B⟩t + σ i (X ti,xi )d Bt , X 0i,xi = xi ,
(9)
where σ i , bi , h i ∈ Cb,lip (R) and i ∈ {1, 2}, then for each x1 ≤ x2 , X t1,x1 ≤ X t2,x2 if and only if b1 (x) − b2 (x) + 2G(h 1 (x) − h 2 (x)) ≤ 0,
σ 1 (x) = σ 2 (x),
∀x ∈ R.
Proof. We shall only have to prove that from X t1,x ≤ X t2,x for each x ∈ R, we infer that b1 (x) − b2 (x) + 2G(h 1 (x) − h 2 (x)) ≤ 0,
σ 1 (x) = σ 2 (x).
By X t1,x ≤ X t2,x , we get t t t b1 (X s1,x )ds + h 1 (X s1,x )d⟨B⟩s + σ 1 (X s1,x )d Bs 0 0 0 t t t 2 2,x 2 2,x ≤ b (X s )ds + h (X s )d⟨B⟩s + σ 2 (X s2,x )d Bs . 0
0
bi (X si,x ) − bi (x),
αsi
0
h i (X si,x ) − h i (x)
βsi
Set = = there exists some constant C such that,
and γsi = σ i (X si,x ) − σ i (x). From Peng [17],
ˆ sup (|αsi |2 + |βsi |2 + |γsi |2 )] ≤ Ct. E[ s∈[0,t]
For each t ∈ [0, T ], we have (b1 (x) − b2 (x))t + (h 1 (x) − h 2 (x))⟨B⟩t + (σ 1 (x) − σ 2 (x))Bt t t t 1 2 1 2 (γs2 − γs1 )d Bs . ≤ (αs − αs )ds + (βs − βs )d⟨B⟩s + 0
0
0
Applying BDG inequalities, we can find some constant C so that 2 t t i ˆ si |2 ]ds ≤ Ct 2 . Eˆ γs d Bs ≤ C E[|γ 0
0
Thus q.s. lim t↓0
0
t
γi √s d Bs = 0. t
(10)
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3883
In a similar way we can also obtain q.s. t t 1 1 i lim √ αs ds = 0, lim √ βsi d⟨B⟩s = 0. t↓0 t↓0 t 0 t 0 Recalling that Bt Bt c lim sup √ = +∞ = 1, c lim inf √ = −∞ = 1, t↓0 t t t↓0 then there exists a subset Ω0 ⊂ Ω with c(Ω0 ) = 1, such that for each ω ∈ Ω0 , we can find a B sequence (rn := rn (ω)) so that limrn ↓0 √rrnn = +∞. By Eq. (10), we derive that Br (σ 1 (x) − σ 2 (x)) lim √ n ≤ 0. rn ↓0 rn Consequently, σ 1 (x) ≤ σ 2 (x). Similarly we can prove σ 1 (x) ≥ σ 2 (x), then, σ 1 (x) = σ 2 (x). Finally, taking expectation on both sides of Eq. (10) yields b1 (x) − b2 (x) + 2G(h 1 (x) − h 2 (x)) t t 1 ≤ lim Eˆ (αs2 − αs1 )ds + (βs2 − βs1 )d⟨B⟩s = 0, t→0 t 0 0 which completes the proof.
Remark 5.6. Let σ 1 = σ 2 = b1 = h 2 = 0, b2 = σ 2 , h 1 = 1, one can show that b1 (x) − b2 (x) + 2G(h 1 (x) − h 2 (x)) ≤ 0 and X t1.x = x + ⟨B⟩t ≤ x + σ 2 t = X t2,x q.s. Thus b1 (x) − b2 (x) + 2G(h 1 (x) − h 2 (x)) ≤ 0 does not imply b1 (x) ≤ b2 (x) and h 1 (x) ≤ h 2 (x). Example 5.7. Consider two G-SDEs with different diffusion coefficients: d X i = σ (X i )d B + 1 σ (X i )σ ′ (X i )d⟨B⟩ , i t i t t t t i t 2 i i X 0 = X 0 (i = 1, 2), 1 (R). Consider the following initial value problems: where σi > 0 and σi ∈ Cb,lip
dϕi = σi (ϕi ), ϕi (0) = vi (i = 1, 2). dx Clearly, the solutions ϕi (x, vi ) satisfy the equalities ϕ1 (x,v1 ) ϕ2 (x,v2 ) ds ds =x= . σ (s) σ (s) 1 2 v1 v2 Note that bi ≡ 0, we can obtain Vi ≡ X 0i by Eq. (4). Hence, if for every x ∈ R the inequality x x dy dy ≥ 1 2 σ (y) σ X0 1 X 0 2 (y) holds, then ϕ1 (x, X 01 ) ≤ ϕ2 (x, X 02 ) and therefore q.s. X t1 = ϕ1 (Bt , X 01 ) ≤ ϕ2 (Bt , X 02 ) = X t2 .
3884
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
Example 5.8. Consider the following G-SDE: d X t = b(X t )dt + h(X t )d⟨B⟩t + σ (X t )d Bt , X 0 = X 0, 1 (R). Then we have for some constant C where b, h ∈ Cb,lip (R) and σ ∈ Cb,lip
|σ (x)| ≤ C,
|g(x, v)| ≤ CeC|x| ,
| f (x, v)| ≤ CeC|x| .
Let σ˜ (x) = Csgn(x), g(x, ˜ v) = CeC|x| and f˜(x, v) = CeC|x| , combining these three inequalities and using Theorem 5.2, we obtain an asymptotic estimation for the paths of G-Itˆo diffusion process X t , for q.s. ω, t X t ≤ C|Bt | + C eC|Bs | ds + X 0 . 0
A symmetric argument shows that, for q.s. ω, t X t ≥ −C|Bt | − C eC|Bs | ds + X 0 . 0
Acknowledgments The authors would like to thank Prof. Peng, S. for his helpful discussions and suggestions. The authors also thank the editor and two anonymous referees for their careful reading, helpful suggestions. The second author was partially supported by Graduate Independent Innovation Foundation of Shandong University (No. YZC12062). Appendix The G-Itˆo formula for a G-Itˆo process was obtained by Peng [17] and improved by Gao [4], p Zhang et al. [19] in LG (Ω ). Li and Peng [7] significantly improved the previous ones for a general p p C 1,2 -function in a larger space L b (Ω ) instead of LG (Ω ) (see also Lin [9,12]). For reader’s convenience, we give the following G-Itˆo formula. Indeed, it can be viewed as a special case of Theorem 2.33 of Lin [9]. For each 0 ≤ t ≤ T , consider a G-Itˆo process: t t t Xt = X0 + f u du + h u d⟨B⟩u + gu d Bu . 0
0
0
Theorem A.1. Suppose ϕ ∈ C([0, T ] × R2 ) satisfies for each t1 , t2 ∈ [0, T ], x1 , x2 , v1 , v2 ∈ R, |ψ(t1 , x1 , v1 ) − ψ(t2 , x2 , v2 )| ≤ C(1 + |x1 | + |x2 |)eC(|x1 |+|x2 |) × (|t1 − t2 | + |x1 − x2 | + |v1 − v2 |), where ψ = ∂t ϕ, ∂x ϕ, ∂x2x ϕ and ∂v ϕ. Let f , h and g be bounded processes in MG2 (0, T ). If for p each p ≥ 1, the continuous finite variation process Vt ∈ L b (Ωt ) and there exists some constant
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3885
C p such that for each s ≤ t ∈ [0, T ]: ˆ tV − TsV | p ] ≤ C p |t − s| p , E[|T where T V is the total variation process of V . Then in L 2b (Ωt ), t t ϕ(t, X t , Vt ) = ϕ(0, X 0 , V0 ) + ∂u ϕ(u, X u , Vu )du + ∂x ϕ(u, X u , Vu ) f u du 0 0 t t + ∂x ϕ(u, X u , Vu )h u d⟨B⟩u + ∂x ϕ(u, X u , Vu )gu d Bu 0 0 t 1 t 2 ∂v ϕ(u, X u , Vu )d Vu + + ∂x x ϕ(u, X u , Vu )gu2 d⟨B⟩u . 2 0 0 Proof. The proof is immediate in light of Lemma 3.3, Theorem 5.4 of Li and Peng [7] and Theorem 2.33 of Lin [9]. References [1] X. Bai, Y. Lin, On the existence and uniqueness of solutions to stochastic differential equations driven by GBrownian motion with integral-Lipschitz coefficients, 2010. arXiv:1002.1046. [2] L. Denis, M. Hu, S. Peng, Function spaces and capacity related to a sublinear expectation: application to GBrownian motion paths, Potential Anal. 34 (2) (2011) 139–161. [3] H. Doss, Liens entre e´ quations diff´erentielles stochastiques et ordinaires, Ann. Inst. H. Poincar´e Sect. A (N.S.) 13 (1977) 99–125. [4] F. Gao, Pathwise properties and homomorphic flows for stochastic differential equations driven by G-Brownian motion, Stochastic Process. Appl. 119 (2009) 3356–3382. [5] Z. Huang, A comparison theorem for solutions of stochastic differential equations and applications, Proc. Amer. Math. Soc. 91 (1984) 611–617. [6] Z. Huang, M. Xu, Z. Hu, On the generalized sample solutions of stochastic differential equations., Wuhan Univ. J. (Natuaral Sci. Ed.) 2 (1981) 11–21 (Chinese). [7] X. Li, S. Peng, Stopping times and related Itˆo calculus with G-Brownian motion, Stochastic Process. Appl. 121 (2009) 1492–1508. [8] Q. Lin, Differentialbility of stochastic differential equations driven by G-Brownian motion, Sci. China Math. 56 (2013) 1087–1107. ´ [9] Y. Lin, Equations diff´erentielles stochastiques sous les esp´erances math´ematiques non-lin´eaire et applications (Ph.D. thesis), Universit´e de Rennes 1, 2013, www.tel.archives-ouvertes.fr/tel-00955814. [10] Q. Lin, Local time and Tanaka formula for the G-Brownian motion, J. Math. Anal. Appl. 398 (2013) 315–334. [11] Q. Lin, Some properties of stochastic differential equations driven by G-Brownian motion, Acta Math. Sin. (Engl. Ser.) 29 (2013) 923–942. [12] Y. Lin, Stochastic differential eqations driven by G-Brownian motion with reflecting boundary, Electron. J. Probab. 18 (9) (2013) 1–23. [13] S. Peng, Filtration consistent nonlinear expectations and evaluations of contingent claims, Acta Math. Appl. Sin. Engl. Ser. 20 (2) (2004) 1–24. [14] S. Peng, Nonlinear expectations and nonlinear Markov chains, Chinese Ann. Math. 26B (2) (2005) 159–184. [15] S. Peng, Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Sci. China Ser. A 52 (7) (2009) 1391–1411. [16] S. Peng, Backward stochastic differential equation, nonlinear expectation and their applications, in: Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010. [17] S. Peng, Nolinear expectations and stochastic calculus under uncertainty, 2010. arxiv:1002.4546v1. [18] H.M. Soner, N. Touzi, J. Zhang, Wellposedness of second order Backward SDEs, Probab. Theory Related Fields 153 (2012) 149–190. [19] B. Zhang, J. Xu, D. Kannan, Extension and application of Itˆo’s formula under G-framework, Stoch. Anal. Appl. 28 (2010) 322–349.
ScienceDirect Stochastic Processes and their Applications 124 (2014) 3869–3885 www.elsevier.com/locate/spa
Stochastic differential equations driven by G-Brownian motion and ordinary differential equations Peng Luo a,b,c , Falei Wang a,∗ a School of Mathematics, Shandong University, China b Qilu Securities Institute for Financial Studies, Shandong University, China c Department of Mathematics and Statistics, University of Konstanz, Germany
Received 29 November 2013; received in revised form 2 April 2014; accepted 1 July 2014 Available online 8 July 2014
Abstract In this paper, we show that the integration of a stochastic differential equation driven by G-Brownian motion (G-SDE for short) in R can be reduced to the integration of an ordinary differential equation (ODE for short) parameterized by a variable in (Ω , F). By this result, we obtain a comparison theorem for G-SDEs and its applications. c 2014 Elsevier B.V. All rights reserved. ⃝
MSC: 60H30; 60H10 Keywords: G-Brownian motion; G-Itˆo’s formula; G-SDE; Comparison theorem
1. Introduction Motivated by uncertainty problems, risk measures and the superhedging in finance, Peng systemically established a time-consistent fully nonlinear expectation theory (see [13–15]). As a typical and important case, Peng introduced the G-expectation theory (see [16,17] and the references therein) in 2006. In the G-expectation framework (G-framework for short), the notion of G-Brownian motion and the corresponding stochastic calculus of Itˆo’s type were established. On ∗ Corresponding author. Tel.: +86 13573775301.
E-mail addresses: [email protected] (P. Luo), [email protected] (F. Wang). http://dx.doi.org/10.1016/j.spa.2014.07.004 c 2014 Elsevier B.V. All rights reserved. 0304-4149/⃝
3870
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
that basis, Gao [4] and Peng [17] studied the existence and uniqueness of the solution of G-SDE under a standard Lipschitz condition. Moreover, Lin [12] obtained the existence and uniqueness of the solution of G-SDE with reflecting boundary. For a recent account and development of this theory we refer the reader to [1,7–11,19]. Under the classical framework, Doss [3] and Huang, Xu and Hu [6] studied the sample solutions of stochastic differential equations, which enables us to transfer a stochastic differential equation into a set of ordinary differential equations for each sample path. Using the method of sample solutions to SDEs, Huang [5] established a comparison theorem of SDEs. The aim of this paper is to study the sample solutions of G-SDEs by ODEs parameterized by a variable in basis probability space. Since G-SDE admits a unique solution in the space MG2 (0, T ), the main difficulty is how to prove that the sample solution belongs to this space. We overcome this problem through some G-stochastic calculus techniques. Then we show that the solution of G-SDE can be represented as a function of both G-Brownian motion and a finite variation process. Since we can use the existing results in the theory of ordinary differential equations directly, this approach provides a powerful tool both in the theoretical analysis and in the practical computation of G-SDEs. In particular, we get a new kind of comparison theorem for G-SDEs. Moreover, a necessary and sufficient condition for comparison theorem of G-SDEs is also obtained. This paper is organized as follows: In Section 2, we recall some notations and results that we will use in this paper. In Section 3, we study the sample solution of G-SDE under some strong conditions, then, in Section 4, we extend this result to a more general case. Finally in Section 5, we establish a new kind of comparison theorem and give its applications. 2. Preliminaries The main purpose of this section is to recall some preliminary results in G-framework which are needed in the sequel. More details can be found in Denis et al. [2], Li and Peng [7], Lin [9,12] and Peng [17]. 2.1. Sublinear expectation Definition 2.1. Given a set Ω and a linear space H of real valued functions defined on Ω . Moreover, if X i ∈ H, i = 1, . . . , d, then ϕ(X 1 , . . . , X d ) ∈ H for all ϕ ∈ Cb,Lip (Rd ), where Cb,Lip (Rd ) is the space of all bounded real-valued Lipschitz continuous functions. A sublinear expectation Eˆ on H is a functional Eˆ : H → R satisfying the following properties: for all X, Y ∈ H, ˆ ] ≥ E[Y ˆ ]; (a) Monotonicity: if X ≥ Y , then E[X ˆ (b) Constant preserving: E[c] = c, ∀ c ∈ R; ˆ ˆ ] + E[Y ˆ ]; (c) Sub-additivity: E[X + Y ] ≤ E[X ˆ ˆ ], ∀ λ ≥ 0. (d) Positive homogeneity: E[λX ] = λE[X ˆ is called a sublinear expectation space. X ∈ H is called a random The triple (Ω , H, E) ˆ variable in (Ω , H, E). We often call Y = (Y1 , . . . , Yd ), Yi ∈ H a d-dimensional random vector ˆ in (Ω , H, E). ˆ a n-dimensional random vector Y = Definition 2.2. In a sublinear expectation space (Ω , H, E), (Y1 , . . . , Yn ) is said to be independent from an m-dimensional random vector X = (X 1 , . . . , X m )
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3871
under Eˆ if for any test function ϕ ∈ Cb,Lip (Rm+n ) ˆ ˆ E[ϕ(x, ˆ E[ϕ(X, Y )] = E[ Y )]x=X ]. Definition 2.3. Let X 1 and X 2 be two n-dimensional random vectors defined on sublinear expectation spaces (Ω1 , H1 , Eˆ 1 ) and (Ω2 , H2 , Eˆ 2 ), respectively. They are called identically d
distributed, denoted by X 1 = X 2 , if Eˆ 1 [ϕ(X 1 )] = Eˆ 2 [ϕ(X 2 )],
∀ϕ ∈ Cb,Lip (Rn ).
d X¯ is said to be an independent copy of X if X¯ = X and X¯ is independent from X .
Definition 2.4 (G-Normal Distribution). A random variable X on a sublinear expectation space ˆ is called (centralized) G-normal distributed if for any a, b ≥ 0 (Ω , H, E) d a X + b X¯ = a 2 + b2 X, where X¯ is an independent copy of X . The letter G denotes the function 1 2 + (σ a − σ 2 a − ) 2 2 ] ≤ E[X ˆ ˆ 2 ] =: σ 2 . with σ 2 := −E[−X G(a) =
2.2. G-Brownian motion Definition 2.5 (G-Brownian Motion). A process (Bt ∈ H)t≥0 on a sublinear expectation space ˆ is called a G-Brownian motion if the following properties are satisfied: (Ω , H, E) (a) B0 = 0. d √ (b) For each t, s ≥ 0 the increment Bt+s − Bt = s X and independent from (Bt1 , Bt2 , . . . , Btn ) for each n ∈ N, 0 ≤ t1 ≤ t2 ≤ · · · ≤ tn ≤ t, where X is G-normal distributed. Denote by Ω = C0 (R+ ) the space of all R-valued continuous paths (ωt )t∈R+ , with ω0 = 0, equipped with the distance ρ(ω1 , ω2 ) :=
∞ i=1
2−i [ max |ωt1 − ωt2 | ∧ 1]. t∈[0,i]
B(Ω ) is the Borel σ -algebra of Ω . For each t ∈ [0, ∞), we introduce the following spaces. • • • • •
Ωt := {ω(· ∧ t) : ω ∈ Ω }, Ft := B(Ωt ), L 0 (Ω ): the space of all B(Ω )-measurable real functions, L 0 (Ωt ): the space of all Ft -measurable real functions, Bb (Ω ): all bounded elements in L 0 (Ω ), Bb (Ωt ) := Bb (Ω ) ∩ L 0 (Ωt ), Cb (Ω ): all continuous elements in Bb (Ω ), Cb (Ωt ) := Cb (Ω ) ∩ L 0 (Ωt ).
In Peng [17], a G-Brownian motion is constructed on a sublinear expectation space ˆ (Eˆ t )t≥0 ), where L p (Ω ) is a Banach space under the natural norm ∥X ∥ p = (Ω , L1G , E, G ˆ E[|X | p ]1/ p . In this space the corresponding canonical process Bt (ω) = ωt is a G-Brownian p p motion. Denote by L b (Ω ) the completion of Bb (Ω ). Denis et al. [2] proved that L b (Ω ) ⊃ p LG (Ω ) ⊃ Cb (Ω ), and there exists a weakly compact family P of probability measures defined
3872
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
on (Ω , B(Ω )) such that ˆ ] = sup E P [X ], E[X P∈P
X ∈ L1G (Ω ).
ˆ Consider a 1-dimensional Remark 2.6. Denis et al. [2] gave a concrete set P M that represents E. Brownian motion Bt on (Ω , F, P), then t −1 2 2 2 P M := Pθ : Pθ = P ◦ X , X t = θs d Bs , θ ∈ L F ([0, T ]; [σ , σ ]) 0
ˆ where L 2 ([0, T ]; [σ 2 , σ 2 ]) is the collection of all F-adapted measuris a set that represents E, F 2 able processes with σ ≤ |θ (s)|2 ≤ σ 2 . Now we introduce the natural Choquet capacity c(A) := sup P(A), P∈P
A ∈ B(Ω ).
Definition 2.7. A set A ⊂ B(Ω ) is polar if c(A) = 0. A property holds “quasi-surely” (q.s.) if it holds outside a polar set. Definition 2.8. A real function X on Ω is said to be quasi-continuous if for each ε > 0, there exists an open set O with c(O) < ε such that X | O c is continuous. Definition 2.9. We say that X : Ω → R has a quasi-continuous version if there exists a quasicontinuous function Y : Ω → R such that X = Y , q.s. p
p
Then L b (Ω ) and LG (Ω ) can be characterized as follows: p ˆ | p I|X |≥N ] = 0} L b (Ω ) = {X ∈ L 0 (Ω )| lim E[|X N →∞
and p
p
LG (Ω ) = {X ∈ L b (Ω ) | X has a quasi-continuous version}. 2.3. G-stochastic calculus Peng [17] also introduced the related stochastic calculus of Itˆo’s type with respect to GBrownian motion (see Li and Peng [7], Lin [12] for more general and systematic research). Let T ∈ R+ be fixed. Definition 2.10. For each p ≥ 1, consider the following simple type of processes: N −1 0, p MG (0, T ) = η := ηt (ω) = ξ j (ω)I[t j ,t j+1 ) (t) ∀ N > 0, j=0
0 = t0 < · · · < t N = T, ξ j ∈ 0, p
p
p LG (Ωt j ),
j = 0, 1, 2, . . . , N − 1 .
Denote by MG (0, T ) the completion of MG (0, T ) under the norm T 1/ p p ˆ ∥η∥ M p (0,T ) = E[|η(t)| ]dt . G
0
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3873
Definition 2.11. For each η ∈ MG0,2 (0, T ) with the form N −1
ηt (ω) =
ξk (ω)I[tk ,tk+1 ) (t),
k=0
define I (η) =
T
ηs d Bs :=
0
N −1
ξk (Bt N − Bt N ). k
k+1
k=0
The mapping I : MG0,2 (0, T ) → L2G (ΩT ) can be continuously extended to I : MG2 (0, T ) → L2G (ΩT ). For each η ∈ MG2 (0, T ), the stochastic integral is defined by T I (η) := ηs d Bs , η ∈ MG2 (0, T ). 0
Unlike the classical theory, the quadratic variation process of G-Brownian motion B is not always a deterministic process and it can be formulated in L2G (Ωt ) by ⟨B⟩t := lim
N −1
N →∞
where tiN =
iT N
i=0
(Bt N − Bt N ) = 2
i
i+1
Bt2
t
−2
Bs d Bs ,
0
for each integer N ≥ 1.
Definition 2.12. Define a mapping MG0,1 (0, T ) → L1G (ΩT ): T
Q(η) = 0
ηs d⟨B⟩s :=
N −1
ξk [⟨B⟩t N − ⟨B⟩t N ]. k
k+1
k=0
Then Q can be uniquely extended to MG1 (0, T ) → L1G (ΩT ). We also denote this mapping by T Q(η) := ηs d⟨B⟩s , η ∈ MG1 (0, T ). 0
In view of the dual formulation of G-expectation as well as the properties of the quadratic variation process ⟨B⟩ in G-framework, Gao [4] obtained the following BDG type inequalities. p
Lemma 2.13. For each p ≥ 1 and η ∈ MG (0, T ), t p T ˆE sup ηs d⟨B⟩s ≤ σ¯ 2 p T p−1 ˆ s | p ]ds. E[|η 0≤t≤T
0
0
p
Lemma 2.14. Let p ≥ 2 and η ∈ MG (0, T ). Then there exists some constant C p depending only on p and T such that t p T 2p Eˆ sup ηs d Bs ≤ C p Eˆ |ηs |2 ds . 0≤t≤T
0
0
3874
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3. G-Stochastic differential equation Let us first recall some notations, • C n (Rd ): the space of all functions of class C n from Rd into R, n (Rd ): the space of all bounded functions of class C n (Rd ) whose partial derivatives of • Cb,lip order less than or equal to n are bounded Lipschitz continuous functions, • C n ([0, T ] × Rd ): the space of all functions of class C n from [0, T ] × Rd into R, n ([0, T ] × Rd ): the space of all bounded functions of class C n ([0, T ] × Rd ) whose partial • Cb,lip derivatives of order less than or equal to n are bounded Lipschitz continuous functions. Consider the following SDE driven by a 1-dimensional G-Brownian motion: t t t Xt = X0 + b(s, X s )ds + h(s, X s )d⟨B⟩s + σ (s, X s )d Bs , t ∈ [0, T ], 0
0
(1)
0
where the initial condition X 0 ∈ R is a given constant. We recall the following assumption. (H) b, h, σ : Ω × [0, T ] × R → R are given functions satisfying b(·, x), h(·, x), σ (·, x) ∈ MG2 (0, T ) for each x ∈ R. Moreover, there exists some constant K such that |ϕ(t, x) − ϕ(t, y)| ≤ K |x − y| for each t ∈ [0, T ], x, y ∈ R, ϕ = b, h and σ , respectively. From Peng [17], we have the following. Theorem 3.1. Under the assumption (H), there exists a unique solution X ∈ MG2 (0, T ) to the stochastic differential equation (1). Remark 3.2. We remark that there is a potential to extend our results to a much more general setting. However, in order to focus on the main ideas, in this paper we content ourselves with the case that the coefficients are 1-dimensional satisfying bounded conditions. In particular, by slightly more involved estimates, we can extend our results to the multi-dimensional case without bounded conditions. 3.1. A simple case In order to explain the main ideas, we first consider a simple G-SDE, t t 1 t σ (X s )∂x σ (X s )d⟨B⟩s + σ (X s )d Bs , Xt = X0 + b(X s )ds + 2 0 0 0
t ∈ [0, T ], (2)
1 (R) and b(x) ∈ C where σ (x) ∈ Cb,lip b,lip (R). By Theorem 3.1, G-SDE (2) admits a unique 2 solution X ∈ MG (0, T ). Now consider the following ODE
dy = σ (y), dx
y(0) = v ∈ R.
The above ODE has a unique solution y = ϕ(x, v) ∈ C(R2 ). Then, ∂x ϕ = σ (ϕ),
ϕ(0, v) = v.
(3)
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3875
Consequently, ∂v ϕ(x, v) = exp
x
∂x σ (ϕ(y, v))dy ,
0
∂x2x ϕ(x, v) = (∂x σ σ )(ϕ(x, v)).
Next we introduce the following ODE with parameter ω: Bt (ω) d V = exp − ∂x σ (ϕ(y, Vt ))dy b(ϕ(Bt (ω), Vt ))dt, t 0 V0 = X 0 .
(4)
For every fixed ω, recalling the Cauchy–Lipschitz theorem, Eq. (4) has a unique solution Vt = Vt (ω) and Vt is a continuous finite variation process. Moreover, Vt (ω) is a continuous function on (Ω , ρ). The following result is important in our future discussion. Lemma 3.3. For any p ≥ 0, there exists a constant C p depending only on p such that, ˆ sup e p|Bt | ] ≤ C p . E[ 0≤t≤T
Proof. For any p ≥ 0, we have n | p B | t p|B | t ˆ sup e Eˆ sup E[ ]≤ . n! 0≤t≤T 0≤t≤T n=0 Applying Doob’s maximal inequality yields that n ˆE[ sup | p Bt |n ] ≤ 1 + 1 ˆ p BT |n ]. E[| n−1 0≤t≤T By Exercise 1.7 in Chapter 3 of Peng [17], one can show that for some constant C ′p depending only on p, | p BT |n Eˆ ≤ C ′p . n! n=0 Since limn 1 +
1 n−1
n
= e, we can find some constant C p depending only on p such that,
ˆ sup e p|Bt | ] ≤ C p , E[ 0≤t≤T
which is the desired result.
p
Lemma 3.4. For each p ≥ 1, Vt ∈ LG (Ωt ). Moreover, there exists some constant C p depending only on p such that for each s ≤ t ∈ [0, T ], ˆ tV − TsV | p ] ≤ C p |t − s| p , E[|T where T V is the total variation process of V .
3876
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
Proof. By Eq. (4), t Vt = V0 + exp − 0
Bu (ω)
∂x σ (ϕ(y, Vu ))dy b(ϕ(Bu (ω), Vu ))du.
0
Denote by C p a constant depending only on p, which is allowed to change from line to line. Then applying Lemma 3.3, we conclude T Bu (ω) ˆE[ sup |Vt | p ] ≤ C p Eˆ |V0 | p + exp − p ∂x σ (ϕ(y, Vu ))dy du 0
0≤t≤T
0
ˆ sup eC p|Bt | ]) ≤ C p . ≤ C p (|V0 | p + C E[ 0≤t≤T p
Since Vt (ω) is a continuous function on (Ω , ρ), recalling the pathwise description of LG (Ωt ), p Vt ∈ LG (Ωt ) for each p ≥ 1. Note that Bu t TtV = exp − ∂x σ (ϕ(y, Vu ))dy |b(ϕ(Bu , Vu ))|du, 0
0
applying Lemma 3.3 again, we obtain for each s ≤ t ∈ [0, T ], ˆ tV − TsV | p ] ≤ C p |t − s| p , E[|T which completes the proof.
By Lemma 3.4, we deduce that ϕ(Bt , Vt ) ∈ MG2 (0, T ). Since ϕ satisfies the conditions of Theorem A.1, applying the G-Itˆo formula, we get 1 dϕ(Bt , Vt ) = ∂x ϕ(Bt , Vt )d Bt + ∂v ϕ(Bt , Vt )d Vt + ∂x2x ϕ(Bt , Vt )d⟨B⟩t 2 1 = b(ϕ(Bt , Vt ))dt + ∂x σ (ϕ(Bt , Vt ))σ (ϕ(Bt , Vt ))d⟨B⟩t + σ (ϕ(Bt , Vt ))d Bt . 2 Consequently, X t = ϕ(Bt , Vt ) is the unique MG2 (0, T )-solution of G-SDE (2). 3.2. The general case In this section, we will extend the above result to a more general case, where all the coefficients are functions in t, Bt and x. Assume b(t, x, y), h(t, x, y) ∈ Cb,lip ([0, T ] × R2 ) 1 ([0, T ] × R2 ). It is obvious G-SDE and σ (t, x, y) ∈ Cb,lip t t t Xt = X0 + b(s, Bs , X s )ds + h(s, Bs , X s )d⟨B⟩s + σ (s, Bs , X s )d Bs , 0
0
0
(5)
t ∈ [0, T ] has a unique solution X ∈ MG2 (0, T ). Then the following ODE dy = σ (t, x, y), dx
y(t, 0) = v ∈ R
(6)
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3877
admits a unique solution y = ϕ(t, x, v) ∈ C([0, T ] × R2 ). Moreover, we can get x ∂ y σ (t, u, ϕ(t, u, v))du ∂v ϕ(t, x, v) = exp 0
and
∂t ϕ(t, x, v) = exp ×
0 x
x
∂ y σ (t, z, ϕ(t, z, v))dz
∂t σ (t, u, ϕ(t, u, v))e
−
u 0
∂ y σ (t,z,ϕ(t,z,v))dz
du .
0
Set g(t, x, v) := ∂v ϕ −1 (t, x, v)(b(t, x, ϕ(t, x, v)) − ∂t ϕ(t, x, v)), 1 f (t, x, v) := ∂v ϕ −1 (t, x, v) h(t, x, ϕ(t, x, v)) − (∂x σ + ∂ y σ σ )(t, x, ϕ(t, x, v)) . 2 Then consider the following initial value problem with parameter ω: d Vt = g(t, Bt (ω), Vt )dt + f (t, Bt (ω), Vt )d⟨B⟩t (ω), V0 = X 0 .
(7)
Note that ⟨B⟩t is a continuous finite variation process, then the ODE (7) has a unique solution V = Vt (ω) and Vt is a continuous finite variation process. Since ⟨B⟩t (ω) is not always a deterministic process, in general we cannot get Vt (ω) is a continuous function on (Ω , ρ) as the above section. However, we also have the following result. Lemma 3.5. For each p ≥ 1, there exists some constant C p depending only on p such that, for each s ≤ t ∈ [0, T ], ˆ tV − TsV | p ] ≤ C p |t − s| p , E[|T where T V is the total variation process of V . Proof. The proof is immediate in light of Lemma 3.4.
Now we shall give the main result of this section. Theorem 3.6. Assume b(t, x, y), h(t, x, y) ∈ Cb,lip ([0, T ] × R2 ) and σ (t, x, y) ∈ 1 ([0, T ] × R2 ), then for each p ≥ 1, V ∈ L p (Ω ) and ϕ(t, B , V ) is the unique M 2 (0, T )Cb,lip t t t t G G solution of G-SDE (5). p
Proof. It is obvious Vt ∈ L b (Ωt ). Then applying Theorem A.1, we obtain q.s. dϕ(t, Bt , Vt ) = ∂t ϕ(t, Bt , Vt )dt + ∂x ϕ(t, Bt , Vt )d Bt + ∂v ϕ(t, Bt , Vt )d Vt 1 + ∂x2x ϕ(t, Bt , Vt )d⟨B⟩t 2 = b(t, Bt , ϕ(t, Bt , Vt ))dt + h(t, Bt , ϕ(t, Bt , Vt ))d⟨B⟩t + σ (t, Bt , ϕ(t, Bt , Vt ))d Bt . By a standard argument, there exists some constant C such that, t ˆE[|ϕ(t, Bt , Vt ) − X t |2 ] ≤ C ˆ E[|ϕ(s, Bs , Vs ) − X s |2 ]ds. 0
Applying Gronwall’s lemma, we obtain ϕ(t, Bt , Vt ) = X t , q.s.
3878
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
By the uniqueness of solution of ODE (6), v = ϕ(t, −x, ϕ(t, x, v)), p
thus, Vt = ϕ(t, −Bt , X t ) q.s. In particular, Vt has a quasi-continuous version and Vt ∈ LG (Ωt ). The proof is completed. 4. G-diffusion process The objective of this section is to remove the condition that σ is continuously differentiable and to obtain a more general result on this topic. By an approximation approach, we can also represent the solution of G-SDE as a function of Bt and a continuous finite variation process Vt as the above section. Theorem 4.1. If b, σ, h ∈ Cb,lip (R), then there exists a unique continuous finite variation p process Vt ∈ LG (Ωt ) for each p ≥ 1 such that, X t = ϕ(Bt , Vt ), where ϕ is the solution of the ODE: ∂x ϕ(x, v) = σ (ϕ(x, v)),
ϕ(0, v) = v.
1 (R), then for q.s. ω, V (ω) is the solution of the following ODE: Moreover if σ ∈ Cb,lip t
Bt d Vt = exp − ∂x σ (ϕ(y, Vt ))dy b(ϕ(Bt , Vt ))dt + (h(ϕ(Bt ), Vt )) 0 1 − ∂x σ σ (ϕ(Bt , Vt ))d⟨B⟩t , 2 V0 = X 0 .
(8)
1 (R), then the theorem holds true. If σ ∈ C Proof. If σ ∈ Cb,lip b,lip (R), one can define σ n (x) := σ (y)ρn (y − x)dy = σ (y + x)ρn (y)dy, R
R
where ρn is a nonnegative function defined on {x : |x| ≤ n1 } with R ρn (y)dy = 1. From this definition, we conclude that K n |σ (x) − σ (x)| ≤ |σ (y + x) − σ (x)|ρn (y)dy ≤ K |y|ρn (y)dy ≤ , n R R C∞
where K is the Lipschitz coefficient of b, h and σ . 1 (R). Thus, X n := ϕ n (B , V n ) is the solution of G-SDE: For each n, it is obvious σ n ∈ Cb,lip t t t t t t X tn = X 0 + b(X sn )ds + h(X sn )d⟨B⟩s + σ n (X sn )d Bs , t ∈ [0, T ], 0
0
0
where ϕ n satisfies ∂x ϕ n (x, v) = σ n (ϕ n (x, v)),
ϕ n (0, v) = v
3879
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
and Bt n n n n n n d Vt = exp − ∂x σ (ϕ (y, Vt ))dy b(ϕ (Bt , Vt ))dt + h(ϕ n (Bt , Vtn )) 0 1 n n n n − ∂x σ σ (ϕ (Bt , Vt ))d⟨B⟩t , 2 n V0 = X 0 . For each n, there exists some constant C depending only on T and K such that, ˆ sup |X tn − X t |2 ] ≤ C . E[ n2 t∈[0,T ] Indeed, applying BDG inequalities, we obtain for some constant C, which is allowed to change from line to line, t ˆE[ sup |X tn − X t |2 ] ≤ Eˆ sup b(X sn ) − b(X s )ds t∈[0,T ]
0
t∈[0,T ] t
+ 0
≤ C Eˆ
K |X tn
− X t |ds
K |X tn
K − Xt | + n
0 T
+ 0
1 + n2
1 + n2
≤C ≤C
T
0
n
0
2
(X sn ) − σ (X s )d Bs
2
T
+ 0
K |X tn
− X t |d⟨B⟩s
2
d Bs
n 2 ˆ E[|X t − X t | ]dt
T
ˆ sup E[ 0
σ
+
2
T
t
h(X sn ) − h(X s )d⟨B⟩s
s∈[0,t]
|X sn
− X s | ]dt . 2
By Gronwall’s lemma, we can get the desired result. Moreover, choosing a subsequence if necessary, X n → X uniformly in [0, T ] q.s. For each v1 , v2 , x ∈ R, |ϕ n (x, v1 ) − ϕ(x, v2 )| ≤ |ϕ n (x, v1 ) − ϕ n (x, v2 )| + |ϕ n (x, v2 ) − ϕ(x, v2 )|. Applying Taylor’s formula yields that |ϕ n (x, v1 ) − ϕ n (x, v2 )| ≤ |∂v ϕ n (x, v ∗ )| |v1 − v2 | ≤ |v1 − v2 |eC|x| . By the definitions of ϕ n and ϕ, we obtain x n n n |ϕ (x, v2 ) − ϕ(x, v2 )| ≤ σ (ϕ (s, v2 )) − σ (ϕ(s, v2 ))ds
0 |x|
≤ 0
1 ds. K |ϕ n (s, v2 ) − ϕ(s, v2 )| + n
3880
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
From Gronwall’s lemma, we conclude for some constant C |x| C|x| e . |ϕ n (x, v1 ) − ϕ(x, v2 )| ≤ C |v1 − v2 | + n Define Vt := ϕ(−Bt , X t ), thus X t = ϕ(Bt , Vt ) and Vt has a quasi-continuous version. Moreover, lim
sup |ϕ n (−Bt , X tn ) − ϕ(−Bt , X t )|
sup |Vtn − Vt | = lim
n→∞ t∈[0,T ]
n→∞ t∈[0,T ]
≤ C lim
n→∞
sup t∈[0,T ]
|X tn
sup |Bt | |Bt | C t∈[0,T ] − X t | + sup e = 0. t∈[0,T ] n
Since for each n and t, s ∈ [0, T ], there exists some constant C such that |Vtn − Vsn | ≤ C sup eC|Bt | |t − s|. t∈[0,T ]
Thus |Vt − Vs | ≤ C sup eC|Bt | |t − s|. t∈[0,T ] p
p
By the pathwise description of LG (Ωt ), Vt ∈ LG (Ωt ) for each p ≥ 1 and the proof is completed. In general, we can also get Theorem 4.2. If b, σ, h ∈ Cb,lip ([0, T ] × R2 ), then there exists a unique continuous finite p variation process Vt ∈ LG (Ωt ) for each p ≥ 1 such that X t = ϕ(t, Bt , Vt ), 1 ([0, T ]×R2 ), then for q.s. ω, V is the solution where ϕ is given by Eq. (6). Moreover if σ ∈ Cb,lip t of ODE (7).
5. Comparison theorem for G-SDEs In the above sections, we establish the relations between G-SDEs and ODEs. From these results, we shall study the comparison theorem for G-SDEs. We refer to Lin [8] for some sufficient condition under which a comparison theorem for G-SDEs is also obtained by virtue of a stochastic calculus approach. We begin with a lemma, which is from [5]. Lemma 5.1. Assume that two functions f (t, x) and f˜(t, x) are defined on R2 , satisfying the Carath´eodory conditions, that is, they are measurable in t, continuous in x and dominated by a locally integrable function m t in R2 . Let (t0 , x0 ), (t0 , x˜0 ) be two points in R2 such that x0 ≤ x˜0 . Moreover, xt is a solution to the initial value problem d xt = f (t, xt )dt,
x t0 = x 0 ,
and x˜t is the maximal solution to the problem d x˜t = f˜(t, x˜t )dt,
x˜t0 = x˜0 .
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3881
If the inequality (t − t0 ) f (t, x) ≤ (t − t0 ) f˜(t, x) holds a.e. in R2 , then x(t) ≤ x(t) ˜ for every t in the common interval of existence of the solutions xt and x˜t . Then we have the following comparison theorem. 1 ([0, T ]×R2 ) Theorem 5.2. Let b(t, x, y), h(t, x, y) ∈ Cb,lip ([0, T ]×R2 ) and σ (t, x, y) ∈ Cb,lip be given. If there exists three functions σ˜ , f˜ and g˜ satisfying the Carath´eodory conditions and the inequalities
xσ (t, x, y) ≤ x σ˜ (t, x, y), 2G( f (t, x, y) − f˜(t, x, y)) + g(t, x, y) − g(t, ˜ x, y) ≤ 0
in [0, T ] × R2 .
Then for the unique solution X t of SDE (5) X t ≤ ϕ(t, ˜ Bt , V˜t ) holds for q.s. ω and every t in the common interval where both sides are defined. Here ϕ˜ and V˜ are the maximal solutions to the problems d ϕ˜ = σ˜ (t, x, ϕ), ˜ dx
ϕ(t, ˜ 0, v) = v ∈ R
and d V˜t = g(t, ˜ Bt , V˜t )dt + f˜(t, Bt (ω), V˜t )d⟨B⟩t ,
V˜0 = X˜ 0
with X 0 ≤ X˜ 0 , respectively. Proof. According to Lemma 5.1, we get ϕ(t, x, v) ≤ ϕ(t, ˜ x, v) ˜ provided v ≤ v. ˜ From [2] or [18], d⟨B⟩t = αˆ t (ω)dt, where αˆ is well defined for each ω and q.s. takes value in [σ 2 , σ 2 ]. Since 2G( f (t, x, v) − f˜(t, x, v)) + g(t, x, v) − g(t, ˜ x, v) ≤ 0, we obtain ( f (t, x, v) − f˜(t, x, v))αˆ t + g(t, x, v) − g(t, ˜ x, v) ≤ 0. Then applying Lemma 5.1 again, we also have Vt ≤ V˜t in the common interval where both sides are defined, which is the desired result. Now we consider some examples of its applications. Example 5.3. Consider two G-SDEs with the same diffusion coefficient σ : i d X t = bi (t, Bt , X ti )dt + h i (t, Bt , X ti )d⟨B⟩t + σ (t, Bt , X ti )d Bt , X 0i = X 0i (i = 1, 2), where σ, b1 , b2 , h 1 , h 2 satisfy the conditions in Theorem 5.2 and X 01 ≤ X 02 , b1 − b2 + 2G(h 1 − h 2 ) ≤ 0. Denote: g i (t, x, v) = ∂v ϕ −1 (t, x, v)(bi (t, x, ϕ(t, x, v)) − ∂t ϕ(t, x, v)), 1 f i (t, x, v) = ∂v ϕ −1 (t, x, v) h i (t, x, ϕ(t, x, v)) − (∂x σ + ∂ y σ σ )(t, x, ϕ(t, x, v)) , 2 (i = 1, 2).
3882
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
One can easily show that g 1 − g 2 + 2G( f 1 − f 2 ) ≤ 0. Applying Theorem 5.2, we obtain X t1 ≤ X t2 q.s. Remark 5.4. In Example 5.3, we can also assume that σ ∈ Cb,lip ([0, T ] × R2 ). Indeed, applying Theorem 4.1, there exists a sequence X i,n → X i uniformly in [0, T ]. Then we conclude X t1 ≤ X t2 from X t1,n ≤ X t2,n for each t ∈ [0, T ]. In particular, we obtain a necessary and sufficient condition for comparison theorem of 1dimensional G-SDEs. Theorem 5.5. Consider two G-SDEs: d X ti,xi = bi (X ti,xi )dt + h i (X ti,xi )d⟨B⟩t + σ i (X ti,xi )d Bt , X 0i,xi = xi ,
(9)
where σ i , bi , h i ∈ Cb,lip (R) and i ∈ {1, 2}, then for each x1 ≤ x2 , X t1,x1 ≤ X t2,x2 if and only if b1 (x) − b2 (x) + 2G(h 1 (x) − h 2 (x)) ≤ 0,
σ 1 (x) = σ 2 (x),
∀x ∈ R.
Proof. We shall only have to prove that from X t1,x ≤ X t2,x for each x ∈ R, we infer that b1 (x) − b2 (x) + 2G(h 1 (x) − h 2 (x)) ≤ 0,
σ 1 (x) = σ 2 (x).
By X t1,x ≤ X t2,x , we get t t t b1 (X s1,x )ds + h 1 (X s1,x )d⟨B⟩s + σ 1 (X s1,x )d Bs 0 0 0 t t t 2 2,x 2 2,x ≤ b (X s )ds + h (X s )d⟨B⟩s + σ 2 (X s2,x )d Bs . 0
0
bi (X si,x ) − bi (x),
αsi
0
h i (X si,x ) − h i (x)
βsi
Set = = there exists some constant C such that,
and γsi = σ i (X si,x ) − σ i (x). From Peng [17],
ˆ sup (|αsi |2 + |βsi |2 + |γsi |2 )] ≤ Ct. E[ s∈[0,t]
For each t ∈ [0, T ], we have (b1 (x) − b2 (x))t + (h 1 (x) − h 2 (x))⟨B⟩t + (σ 1 (x) − σ 2 (x))Bt t t t 1 2 1 2 (γs2 − γs1 )d Bs . ≤ (αs − αs )ds + (βs − βs )d⟨B⟩s + 0
0
0
Applying BDG inequalities, we can find some constant C so that 2 t t i ˆ si |2 ]ds ≤ Ct 2 . Eˆ γs d Bs ≤ C E[|γ 0
0
Thus q.s. lim t↓0
0
t
γi √s d Bs = 0. t
(10)
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3883
In a similar way we can also obtain q.s. t t 1 1 i lim √ αs ds = 0, lim √ βsi d⟨B⟩s = 0. t↓0 t↓0 t 0 t 0 Recalling that Bt Bt c lim sup √ = +∞ = 1, c lim inf √ = −∞ = 1, t↓0 t t t↓0 then there exists a subset Ω0 ⊂ Ω with c(Ω0 ) = 1, such that for each ω ∈ Ω0 , we can find a B sequence (rn := rn (ω)) so that limrn ↓0 √rrnn = +∞. By Eq. (10), we derive that Br (σ 1 (x) − σ 2 (x)) lim √ n ≤ 0. rn ↓0 rn Consequently, σ 1 (x) ≤ σ 2 (x). Similarly we can prove σ 1 (x) ≥ σ 2 (x), then, σ 1 (x) = σ 2 (x). Finally, taking expectation on both sides of Eq. (10) yields b1 (x) − b2 (x) + 2G(h 1 (x) − h 2 (x)) t t 1 ≤ lim Eˆ (αs2 − αs1 )ds + (βs2 − βs1 )d⟨B⟩s = 0, t→0 t 0 0 which completes the proof.
Remark 5.6. Let σ 1 = σ 2 = b1 = h 2 = 0, b2 = σ 2 , h 1 = 1, one can show that b1 (x) − b2 (x) + 2G(h 1 (x) − h 2 (x)) ≤ 0 and X t1.x = x + ⟨B⟩t ≤ x + σ 2 t = X t2,x q.s. Thus b1 (x) − b2 (x) + 2G(h 1 (x) − h 2 (x)) ≤ 0 does not imply b1 (x) ≤ b2 (x) and h 1 (x) ≤ h 2 (x). Example 5.7. Consider two G-SDEs with different diffusion coefficients: d X i = σ (X i )d B + 1 σ (X i )σ ′ (X i )d⟨B⟩ , i t i t t t t i t 2 i i X 0 = X 0 (i = 1, 2), 1 (R). Consider the following initial value problems: where σi > 0 and σi ∈ Cb,lip
dϕi = σi (ϕi ), ϕi (0) = vi (i = 1, 2). dx Clearly, the solutions ϕi (x, vi ) satisfy the equalities ϕ1 (x,v1 ) ϕ2 (x,v2 ) ds ds =x= . σ (s) σ (s) 1 2 v1 v2 Note that bi ≡ 0, we can obtain Vi ≡ X 0i by Eq. (4). Hence, if for every x ∈ R the inequality x x dy dy ≥ 1 2 σ (y) σ X0 1 X 0 2 (y) holds, then ϕ1 (x, X 01 ) ≤ ϕ2 (x, X 02 ) and therefore q.s. X t1 = ϕ1 (Bt , X 01 ) ≤ ϕ2 (Bt , X 02 ) = X t2 .
3884
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
Example 5.8. Consider the following G-SDE: d X t = b(X t )dt + h(X t )d⟨B⟩t + σ (X t )d Bt , X 0 = X 0, 1 (R). Then we have for some constant C where b, h ∈ Cb,lip (R) and σ ∈ Cb,lip
|σ (x)| ≤ C,
|g(x, v)| ≤ CeC|x| ,
| f (x, v)| ≤ CeC|x| .
Let σ˜ (x) = Csgn(x), g(x, ˜ v) = CeC|x| and f˜(x, v) = CeC|x| , combining these three inequalities and using Theorem 5.2, we obtain an asymptotic estimation for the paths of G-Itˆo diffusion process X t , for q.s. ω, t X t ≤ C|Bt | + C eC|Bs | ds + X 0 . 0
A symmetric argument shows that, for q.s. ω, t X t ≥ −C|Bt | − C eC|Bs | ds + X 0 . 0
Acknowledgments The authors would like to thank Prof. Peng, S. for his helpful discussions and suggestions. The authors also thank the editor and two anonymous referees for their careful reading, helpful suggestions. The second author was partially supported by Graduate Independent Innovation Foundation of Shandong University (No. YZC12062). Appendix The G-Itˆo formula for a G-Itˆo process was obtained by Peng [17] and improved by Gao [4], p Zhang et al. [19] in LG (Ω ). Li and Peng [7] significantly improved the previous ones for a general p p C 1,2 -function in a larger space L b (Ω ) instead of LG (Ω ) (see also Lin [9,12]). For reader’s convenience, we give the following G-Itˆo formula. Indeed, it can be viewed as a special case of Theorem 2.33 of Lin [9]. For each 0 ≤ t ≤ T , consider a G-Itˆo process: t t t Xt = X0 + f u du + h u d⟨B⟩u + gu d Bu . 0
0
0
Theorem A.1. Suppose ϕ ∈ C([0, T ] × R2 ) satisfies for each t1 , t2 ∈ [0, T ], x1 , x2 , v1 , v2 ∈ R, |ψ(t1 , x1 , v1 ) − ψ(t2 , x2 , v2 )| ≤ C(1 + |x1 | + |x2 |)eC(|x1 |+|x2 |) × (|t1 − t2 | + |x1 − x2 | + |v1 − v2 |), where ψ = ∂t ϕ, ∂x ϕ, ∂x2x ϕ and ∂v ϕ. Let f , h and g be bounded processes in MG2 (0, T ). If for p each p ≥ 1, the continuous finite variation process Vt ∈ L b (Ωt ) and there exists some constant
P. Luo, F. Wang / Stochastic Processes and their Applications 124 (2014) 3869–3885
3885
C p such that for each s ≤ t ∈ [0, T ]: ˆ tV − TsV | p ] ≤ C p |t − s| p , E[|T where T V is the total variation process of V . Then in L 2b (Ωt ), t t ϕ(t, X t , Vt ) = ϕ(0, X 0 , V0 ) + ∂u ϕ(u, X u , Vu )du + ∂x ϕ(u, X u , Vu ) f u du 0 0 t t + ∂x ϕ(u, X u , Vu )h u d⟨B⟩u + ∂x ϕ(u, X u , Vu )gu d Bu 0 0 t 1 t 2 ∂v ϕ(u, X u , Vu )d Vu + + ∂x x ϕ(u, X u , Vu )gu2 d⟨B⟩u . 2 0 0 Proof. The proof is immediate in light of Lemma 3.3, Theorem 5.4 of Li and Peng [7] and Theorem 2.33 of Lin [9]. References [1] X. Bai, Y. Lin, On the existence and uniqueness of solutions to stochastic differential equations driven by GBrownian motion with integral-Lipschitz coefficients, 2010. arXiv:1002.1046. [2] L. Denis, M. Hu, S. Peng, Function spaces and capacity related to a sublinear expectation: application to GBrownian motion paths, Potential Anal. 34 (2) (2011) 139–161. [3] H. Doss, Liens entre e´ quations diff´erentielles stochastiques et ordinaires, Ann. Inst. H. Poincar´e Sect. A (N.S.) 13 (1977) 99–125. [4] F. Gao, Pathwise properties and homomorphic flows for stochastic differential equations driven by G-Brownian motion, Stochastic Process. Appl. 119 (2009) 3356–3382. [5] Z. Huang, A comparison theorem for solutions of stochastic differential equations and applications, Proc. Amer. Math. Soc. 91 (1984) 611–617. [6] Z. Huang, M. Xu, Z. Hu, On the generalized sample solutions of stochastic differential equations., Wuhan Univ. J. (Natuaral Sci. Ed.) 2 (1981) 11–21 (Chinese). [7] X. Li, S. Peng, Stopping times and related Itˆo calculus with G-Brownian motion, Stochastic Process. Appl. 121 (2009) 1492–1508. [8] Q. Lin, Differentialbility of stochastic differential equations driven by G-Brownian motion, Sci. China Math. 56 (2013) 1087–1107. ´ [9] Y. Lin, Equations diff´erentielles stochastiques sous les esp´erances math´ematiques non-lin´eaire et applications (Ph.D. thesis), Universit´e de Rennes 1, 2013, www.tel.archives-ouvertes.fr/tel-00955814. [10] Q. Lin, Local time and Tanaka formula for the G-Brownian motion, J. Math. Anal. Appl. 398 (2013) 315–334. [11] Q. Lin, Some properties of stochastic differential equations driven by G-Brownian motion, Acta Math. Sin. (Engl. Ser.) 29 (2013) 923–942. [12] Y. Lin, Stochastic differential eqations driven by G-Brownian motion with reflecting boundary, Electron. J. Probab. 18 (9) (2013) 1–23. [13] S. Peng, Filtration consistent nonlinear expectations and evaluations of contingent claims, Acta Math. Appl. Sin. Engl. Ser. 20 (2) (2004) 1–24. [14] S. Peng, Nonlinear expectations and nonlinear Markov chains, Chinese Ann. Math. 26B (2) (2005) 159–184. [15] S. Peng, Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Sci. China Ser. A 52 (7) (2009) 1391–1411. [16] S. Peng, Backward stochastic differential equation, nonlinear expectation and their applications, in: Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010. [17] S. Peng, Nolinear expectations and stochastic calculus under uncertainty, 2010. arxiv:1002.4546v1. [18] H.M. Soner, N. Touzi, J. Zhang, Wellposedness of second order Backward SDEs, Probab. Theory Related Fields 153 (2012) 149–190. [19] B. Zhang, J. Xu, D. Kannan, Extension and application of Itˆo’s formula under G-framework, Stoch. Anal. Appl. 28 (2010) 322–349.