# Successive Approximation of SFDEs with Finite Delay Driven byG-Brownian Motion

Authors: Litan Yan; Qinghua Zhang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider the stochastic functional differential equations with finite delay driven byG-Brownian motion. Under the global Carathéodory conditions we prove the existence and uniqueness, and as an application, we price the European call option when the underlying asset's price follows such an equation.


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## Body

## 1. Introduction

Motivated by various types of uncertainty and financial problems, Peng [1] has introduced a new notion of nonlinear expectation, the so-called G-expectation (see also Peng et al. [2–4]), which is associated with the following nonlinear heat equation:
(1)∂∂tu(t,x)=G(Δu),(t,x)∈[0,+∞)×ℝ,u(0,x)=φ(x),
where Δ is Laplacian and the sublinear function G is defined as
(2)G(α)=12(σ¯2α+-σ_2α-),α∈ℝ
with two given constants 0<σ_<σ¯. Together with the notion of G-expectations Peng also introduced the related G-normal distribution, the G-Brownian motion, and related stochastic calculus underG-expectation, and moreover an Itô’s formula for the G-Brownian motion was established. G-Brownian motion has a very rich and interesting new structure which nontrivially generalizes the classical one. Briefly speaking, a G-Brownian motion B is a continuous process with independent stationary increments Bt+s-Bt being G-normally distributed under a given sublinear expectation 𝔼^. A very interesting new phenomenon of G-Brownian motion B is that its quadratic process 〈B〉 is a continuous process with independent and stationary increments, but not a deterministic process.On the other hand, over the past decades, thanks for the contribution of Black and Scholes [5] and Merton [6] in the formulation of Black-Scholes model, the trading of derivatives has become an important area in the world of finance, and the Black-Scholes formula has been one of the most important consequences of the study of continuous time models in finance. Although Black-Scholes model is a benchmark of mathematical finance, the significant weakness of it cannot be ignored since empirical evidence shows that volatility actually depends on time in a way that is not predictable and many empirical studies have outlined that the volatility of underlying asset is highly unlikely to be constant. This is sometimes pointed out as the reason for inaccurate predictions made by the Black and Scholes formula. Therefore, a number of alternative methods have been studied for the underlying asset model. In this paper, we consider the effect of the past in the determination of the fair price of a call option under a sublinear framework. In particular, we assume that the stock price satisfies a stochastic functional differential equation (SFDE) with fixed or variable delay driven by a G-Brownian motion. We consider call options that can be exercised only at the maturity date, namely, European call options. Let {B(t),t≥0} be the G-Brownian motion with the quadratic variation process {〈B〉(t),t≥0}. We consider SFDE with finite delay of the form
(3)dX(t)=b(t,Xt)dt+h(t,Xt)d〈B〉(t)+σ(t,Xt)dB(t),dddddddddddddddddddddd.ddddt∈[t0,T],
where (i) Xt={X(t+θ);-τ≤θ≤0} and t∈[t0,T] is a C([-τ,0];ℝ)-valued stochastic process; (ii) b, h, and σ are three given functions satisfying some satiable conditions; (iii) τ>0 and 0≤t0<∞ are two given constants.As an application we price the European call option when the underlying asset’s price follows a special SFDE. In the case of classical Brownian motion, this is first studied by Arriojas et al. [7].The note is organized as follows. In Section2, we present some preliminaries for sublinear expectation and G-Brownian motion B. In Section 3, we express and prove our main result. In Section 4, we give an application to price the European call option when the underlying asset price follows a special SFDE driven by a G-Brownian motion.


## 2. Preliminaries

In this section, we briefly recall some basic notations and results forG-Brownian motion under G-framework. For more aspects on these material we refer to Peng [4], Denis et al. [8], and Hu and Peng [9]. More works for G-Brownian motion can be found in Hu and Li [10], Lin [11], Peng et al. [12], Song [13], Xu et al. [14], Yan et al. [15, 16], and the references therein.


### 2.1. Sublinear Expectation Space

LetΩ≠∅ be a given set and let ℋ be a linear space of real-valued functions defined on Ω such that 1∈ℋ and |X|∈ℋ for all X∈ℋ. It is important to note that we can suppose that
(4)φ(X1,…,Xd)∈ℋ
if Xi∈ℋ, i=1,…,d, for all φ∈Cb,Lip(ℝd), where Cb,Lip(ℝd) denotes the space of all bounded and Lipschitz functions on ℝd.Definition 1.
A sublinear expectation𝔼^ on ℋ is a functional with the following properties: for all X, Y∈ℋ, we have (i)
monotonicity: ifX≥Y, then 𝔼^[X]≥𝔼^[Y];(ii)
constant preserving:𝔼^[c]=c, for all c∈ℝ;(iii)
subadditivity:𝔼^[X]-𝔼^[Y]≤𝔼^[X-Y];(iv)
positive homogeneity:𝔼^[λX]=λ𝔼^[X], for all λ≥0. The triple (Ω,ℋ,𝔼^) is called a sublinear expectation space, and ℋ is considered as the space of random variables on Ω.In this paper we throughout letΩ=C0(ℝ+) be the space of all real-valued continuous functions on [0,∞) with initial value 0, equipped with the distance
(5)ρ(ω1,ω2)=∑i=1∞‍2-i[(maxt∈[0,i]|ωt1-ωt2|)∧1],ddddddddddddddω1,ω2∈Ω.
We denote by ℬ(Ω) the Borel-algebra on Ω. We also denote, for each t∈[0,∞),
(6)Ωt={ω·∧t,ω∈Ω}
and ℱt=ℬ(Ωt), where x∧y=min{x,y}. Let 𝕃Gp(Ω) be the closure of ℋ with respect to the norm
(7)∥X∥p=𝔼^[|X|p]1/p
with p∈[1,∞). Clearly, the space 𝕃Gp(Ω) is a Banach space and the space Cb(Ω) of bounded continuous functions on Ω is a subset of 𝕃Gp(Ω), and moreover, there exists a weakly compact family 𝒫 of probability measures on (Ω,ℬ(Ω)) such that
(8)𝔼^=supP∈𝒫EP.
So we can introduce the Choquet capacity C^ by
(9)C^(A)=supP∈𝒫P(A),A∈ℬ(Ω).Definition 2.
A setA⊂Ω is called polar if C^(A)=0. A property is said to hold “quasi sure” (q.s.) if it holds outside a polar set.The above family of probability measuresP allows characterizing the space 𝕃Gp(Ω) as follows:
(10)𝕃Gp(Ω)={C^-quasiX:ℬ(Ω)=measurablereal-valuedfunctionscdonΩsuchthatsupP∈𝒫EP[|X|p]<∞,cdandisC^-quasisurelycontinuous}.Lemma 3 (Denis et al. [8]).
Let1≤p<∞. Consider the sets 𝕃Gp(Ω) and 𝕃p=ℒp/𝒩, where
(11)ℒp={supP∈𝒫X:ℬ(Ω)=measurablereal-valuedfunctionsssonΩsuchthat𝔼^(|X|p)=supP∈𝒫EP[|X|p]<∞},𝒩={X:ℬ(Ω)=measurablereal-valuedfunctionsssonΩsuchthatX=0q.s}.
Then (i)
𝕃p is a Banach space with respect to the norm ∥·∥p;(ii)
𝕃Gp is the completion of Cb(Ω) with respect to the norm ∥·∥p.Lemma 4 (Li and Peng [17]).
For a givenp∈(0,+∞], If the sequence {Xn}⊂𝕃p converges to X in 𝕃p, then there exists a subsequence {Xnk} such that Xnk converges to X quasi surely.Lemma 5 (see [18]).
Letρ:ℝ→ℝ be an increasing and concave function. Then the inequality
(12)𝔼^[ρ(X)]≤ρ(𝔼^[X])
holds for all ℬ(Ω)-measurable real-valued functions on Ω.ForX=(X1,…,Xn)∈ℋn, the functional FX defined by
(13)FX[φ]:=𝔼^[φ(X)],φ∈Cb,Lip(ℝn)
is called the distribution of X under 𝔼^. In a sublinear expectation space (Ω,ℋ,𝔼^), a random vector Y=(Y1,…,Yn), Yi∈ℋ, is said to be independent under 𝔼^ from another random vector X=(X1,…,Xm), Xi∈ℋ, if for each test function φ∈Cb,Lip(ℝm+n) we have
(14)𝔼^[φ(X,Y)]=𝔼^[𝔼^[φ(x,Y)]x=X].


### 2.2.G-Brownian Motion

Letσ_, σ¯ be two real numbers with 0≤σ_≤σ¯.A random variableξ in a sublinear expectation space (Ω,ℋ,𝔼^) is called G-normal distributed, denoted by ξ~N(0,[σ_2,σ¯2]), if, for each φ∈Cb,Lip(ℝ), the function defined by
(15)u(t,x):=𝔼^[φ(x+tξ)],(t,x)∈[0,∞)×ℝ
is the unique viscosity solution of the following nonlinear heat equation:
(16)∂∂tu(t,x)=G(Δu),(t,x)∈[0,+∞)×ℝ,u(0,x)=φ(x),
where Δ is Laplacian and the sublinear function G is defined as
(17)G(α)=12(σ¯2α+-σ_2α-),α∈ℝ.In particular,E[φ(ξ)]=u(1,0) is the distribution of ξ.Example 6 (Peng [1]).
Letξ~N(0,[σ_2,σ¯2]). We then have
(18)𝔼^[φ(ξ)]=12πσ¯∫ℝ‍φ(x)e-(1/2σ¯2)x2dx
for all convex functions φ and
(19)𝔼^[ψ(ξ)]=12πσ_∫ℝ‍ψ(x)e-(1/2σ_2)x2dx
for all concave functions ψ.Definition 7.
A processB={B(t),t≥0}⊂ℋ in a sublinear expectation space (Ω,ℋ,𝔼^) is called a G-Brownian motion if the following properties are satisfied: (i)
B(0)=0;(ii)
for eacht,s≥0, the increment B(t+s)-B(t) is N(0,[σ_2s,σ¯2s])-distributed and is independent of (B(t1),…,B(tn)), for all n=0,1,2,… and 0≤t1≤t2≤⋯≤tn≤t.Recall that a processX={X(t):0≤t≤T} is called a G-martingale if, for each 0≤t≤T, we have X(t)∈𝕃G1(Ωt) and
(20)𝔼^[X(t)∣Ωs]=X(s)
for all s≤t, and moreover, X is called a symmetric G-martingale if both X and -X are G-martingales.Remark 8.
For simplicity throughout this paper we letσ_=σ0>0, and σ¯=1.


### 2.3. Itô’s Integral

We now recall the definition of Itô’s integral and quadratic variation process of theG-Brownian motion. In Li and Peng [17], a generalized Itô integral and a generalized Itô formula with respect to the G-Brownian motion are discussed as follows. For arbitrarily fixed p≥1 and T∈ℝ+, we first denote by MGp,0([0,T]) the set of step processes
(21)ηt(ω)=∑j=1N‍ξj(ω)I[tj-1,tj)(t),ξj(ω)∈𝕃Gp(Ωtj)
with 0=t0<⋯<tN=T. Moreover, we denote by MGp([0,T]) the completion of MGp,0([0,T]) under the norm
(22)∥η∥MGp([0,T])=(𝔼^[∫0T‍|η(s)|pds])1/p.
Now we can define the stochastic integral ∫0T‍η(s)dB(s) for η∈MGp([0,T]).Definition 9.
For everyη∈MGp,0([0,T]) with the form (21), we define Itô’s integral
(23)I(η)=∫0t‍η(s)dB(s):=∑j=1N‍ξj(B(sj)-B(sj-1)).One can show thatI(η)∈𝕃G2 and the linear mapping (see Li and Peng [17])
(24)I:MG2,0([0,T])⟶𝕃G2
is continuous. Moreover, it can be continuously extended to
(25)I:MG2([0,T])⟶𝕃G2.
And, for all η∈MG2([0,T]), we have
(26)𝔼^(∫0T‍η(s)dB(s))=0,𝔼^[(∫0T‍η(s)dB(s))2]≤σ¯2𝔼^[∫0T‍η2(s)ds].
We now recall the definition of quadratic variation process of the G-Brownian motion B.Definition 10 (quadratic variation).
LetπtN={0=t0N<t1N<⋯<tN-1N=t} be a partition of [0,t] for t>0, such that μ(πtN):=maxj{tj-tj-1}→0 as N→∞. The quadratic variation of G-Brownian motion B can be defined as
(27)〈B〉(t)=limμ(πtN)→0∑k=0N-1‍(B(tk+1N)-B(tkN))2=B2(t)-2∫0t‍B(s)dB(s)
in 𝕃G2.The functiont↦〈B〉(t) is continuous and increasing outside a polar set. We can define the integral
(28)∫0T‍η(t)d〈B〉(t):=∑j=1N‍ξj(〈B〉(tj)-〈B〉(tj-1))
as a map from MG1,0([0,T]) into 𝕃G1, and the map is linear and continuous, so it can be extended continuously to MG1([0,T]).Finally, we recall some important results in order to get our desired result. For detailed description of them, please read the related papers.Theorem 11 (Itô’s formula).
LetF∈C2,1(ℝ×[0,∞)). One then has
(29)F(B(t),t)=F(B(0),0)+∫0t‍∂∂xF(B(s),s)dB(s)+∫0t‍∂∂tF(B(s),s)ds+12∫0t‍∂2∂x2F(B(s),s)d〈B〉(s)
for all t≥0.Lemma 12 (see [14]).
If there exists anε0>0 such that
(30)𝔼^[exp{(12+ε0)∫0T‍H2(s,ω)d〈B〉(s)}]<∞
then the process
(31)B(t)-∫0t‍H(s,ω)d〈B〉(s),0≤t≤T
is a G-Brownian motion under some G-expectation.Lemma 13 (see [19]).
Letp≥2 and η={η(t),t∈[0,T]}∈MGp([0,T]). Then
(32)𝔼^[sups≤u≤t|∫su‍η(r)dB(r)|p]≤Cp𝔼^(∫st‍|η(r)|2dr)p/2
for all 0≤s≤t≤T, where Cp is a positive constant depending only on p.Lemma 14 (see [19]).
Letp≥1 and η∈MGp([0,T]). Then we have
(33)𝔼^[sups≤u≤t|∫su‍η(r)d〈B〉(r)|p]≤(t-s)p-1𝔼^∫st‍[|η(r)|p]dr
for all 0≤s≤t≤T.


## 2.1. Sublinear Expectation Space

LetΩ≠∅ be a given set and let ℋ be a linear space of real-valued functions defined on Ω such that 1∈ℋ and |X|∈ℋ for all X∈ℋ. It is important to note that we can suppose that
(4)φ(X1,…,Xd)∈ℋ
if Xi∈ℋ, i=1,…,d, for all φ∈Cb,Lip(ℝd), where Cb,Lip(ℝd) denotes the space of all bounded and Lipschitz functions on ℝd.Definition 1.
A sublinear expectation𝔼^ on ℋ is a functional with the following properties: for all X, Y∈ℋ, we have (i)
monotonicity: ifX≥Y, then 𝔼^[X]≥𝔼^[Y];(ii)
constant preserving:𝔼^[c]=c, for all c∈ℝ;(iii)
subadditivity:𝔼^[X]-𝔼^[Y]≤𝔼^[X-Y];(iv)
positive homogeneity:𝔼^[λX]=λ𝔼^[X], for all λ≥0. The triple (Ω,ℋ,𝔼^) is called a sublinear expectation space, and ℋ is considered as the space of random variables on Ω.In this paper we throughout letΩ=C0(ℝ+) be the space of all real-valued continuous functions on [0,∞) with initial value 0, equipped with the distance
(5)ρ(ω1,ω2)=∑i=1∞‍2-i[(maxt∈[0,i]|ωt1-ωt2|)∧1],ddddddddddddddω1,ω2∈Ω.
We denote by ℬ(Ω) the Borel-algebra on Ω. We also denote, for each t∈[0,∞),
(6)Ωt={ω·∧t,ω∈Ω}
and ℱt=ℬ(Ωt), where x∧y=min{x,y}. Let 𝕃Gp(Ω) be the closure of ℋ with respect to the norm
(7)∥X∥p=𝔼^[|X|p]1/p
with p∈[1,∞). Clearly, the space 𝕃Gp(Ω) is a Banach space and the space Cb(Ω) of bounded continuous functions on Ω is a subset of 𝕃Gp(Ω), and moreover, there exists a weakly compact family 𝒫 of probability measures on (Ω,ℬ(Ω)) such that
(8)𝔼^=supP∈𝒫EP.
So we can introduce the Choquet capacity C^ by
(9)C^(A)=supP∈𝒫P(A),A∈ℬ(Ω).Definition 2.
A setA⊂Ω is called polar if C^(A)=0. A property is said to hold “quasi sure” (q.s.) if it holds outside a polar set.The above family of probability measuresP allows characterizing the space 𝕃Gp(Ω) as follows:
(10)𝕃Gp(Ω)={C^-quasiX:ℬ(Ω)=measurablereal-valuedfunctionscdonΩsuchthatsupP∈𝒫EP[|X|p]<∞,cdandisC^-quasisurelycontinuous}.Lemma 3 (Denis et al. [8]).
Let1≤p<∞. Consider the sets 𝕃Gp(Ω) and 𝕃p=ℒp/𝒩, where
(11)ℒp={supP∈𝒫X:ℬ(Ω)=measurablereal-valuedfunctionsssonΩsuchthat𝔼^(|X|p)=supP∈𝒫EP[|X|p]<∞},𝒩={X:ℬ(Ω)=measurablereal-valuedfunctionsssonΩsuchthatX=0q.s}.
Then (i)
𝕃p is a Banach space with respect to the norm ∥·∥p;(ii)
𝕃Gp is the completion of Cb(Ω) with respect to the norm ∥·∥p.Lemma 4 (Li and Peng [17]).
For a givenp∈(0,+∞], If the sequence {Xn}⊂𝕃p converges to X in 𝕃p, then there exists a subsequence {Xnk} such that Xnk converges to X quasi surely.Lemma 5 (see [18]).
Letρ:ℝ→ℝ be an increasing and concave function. Then the inequality
(12)𝔼^[ρ(X)]≤ρ(𝔼^[X])
holds for all ℬ(Ω)-measurable real-valued functions on Ω.ForX=(X1,…,Xn)∈ℋn, the functional FX defined by
(13)FX[φ]:=𝔼^[φ(X)],φ∈Cb,Lip(ℝn)
is called the distribution of X under 𝔼^. In a sublinear expectation space (Ω,ℋ,𝔼^), a random vector Y=(Y1,…,Yn), Yi∈ℋ, is said to be independent under 𝔼^ from another random vector X=(X1,…,Xm), Xi∈ℋ, if for each test function φ∈Cb,Lip(ℝm+n) we have
(14)𝔼^[φ(X,Y)]=𝔼^[𝔼^[φ(x,Y)]x=X].


## 2.2.G-Brownian Motion

Letσ_, σ¯ be two real numbers with 0≤σ_≤σ¯.A random variableξ in a sublinear expectation space (Ω,ℋ,𝔼^) is called G-normal distributed, denoted by ξ~N(0,[σ_2,σ¯2]), if, for each φ∈Cb,Lip(ℝ), the function defined by
(15)u(t,x):=𝔼^[φ(x+tξ)],(t,x)∈[0,∞)×ℝ
is the unique viscosity solution of the following nonlinear heat equation:
(16)∂∂tu(t,x)=G(Δu),(t,x)∈[0,+∞)×ℝ,u(0,x)=φ(x),
where Δ is Laplacian and the sublinear function G is defined as
(17)G(α)=12(σ¯2α+-σ_2α-),α∈ℝ.In particular,E[φ(ξ)]=u(1,0) is the distribution of ξ.Example 6 (Peng [1]).
Letξ~N(0,[σ_2,σ¯2]). We then have
(18)𝔼^[φ(ξ)]=12πσ¯∫ℝ‍φ(x)e-(1/2σ¯2)x2dx
for all convex functions φ and
(19)𝔼^[ψ(ξ)]=12πσ_∫ℝ‍ψ(x)e-(1/2σ_2)x2dx
for all concave functions ψ.Definition 7.
A processB={B(t),t≥0}⊂ℋ in a sublinear expectation space (Ω,ℋ,𝔼^) is called a G-Brownian motion if the following properties are satisfied: (i)
B(0)=0;(ii)
for eacht,s≥0, the increment B(t+s)-B(t) is N(0,[σ_2s,σ¯2s])-distributed and is independent of (B(t1),…,B(tn)), for all n=0,1,2,… and 0≤t1≤t2≤⋯≤tn≤t.Recall that a processX={X(t):0≤t≤T} is called a G-martingale if, for each 0≤t≤T, we have X(t)∈𝕃G1(Ωt) and
(20)𝔼^[X(t)∣Ωs]=X(s)
for all s≤t, and moreover, X is called a symmetric G-martingale if both X and -X are G-martingales.Remark 8.
For simplicity throughout this paper we letσ_=σ0>0, and σ¯=1.


## 2.3. Itô’s Integral

We now recall the definition of Itô’s integral and quadratic variation process of theG-Brownian motion. In Li and Peng [17], a generalized Itô integral and a generalized Itô formula with respect to the G-Brownian motion are discussed as follows. For arbitrarily fixed p≥1 and T∈ℝ+, we first denote by MGp,0([0,T]) the set of step processes
(21)ηt(ω)=∑j=1N‍ξj(ω)I[tj-1,tj)(t),ξj(ω)∈𝕃Gp(Ωtj)
with 0=t0<⋯<tN=T. Moreover, we denote by MGp([0,T]) the completion of MGp,0([0,T]) under the norm
(22)∥η∥MGp([0,T])=(𝔼^[∫0T‍|η(s)|pds])1/p.
Now we can define the stochastic integral ∫0T‍η(s)dB(s) for η∈MGp([0,T]).Definition 9.
For everyη∈MGp,0([0,T]) with the form (21), we define Itô’s integral
(23)I(η)=∫0t‍η(s)dB(s):=∑j=1N‍ξj(B(sj)-B(sj-1)).One can show thatI(η)∈𝕃G2 and the linear mapping (see Li and Peng [17])
(24)I:MG2,0([0,T])⟶𝕃G2
is continuous. Moreover, it can be continuously extended to
(25)I:MG2([0,T])⟶𝕃G2.
And, for all η∈MG2([0,T]), we have
(26)𝔼^(∫0T‍η(s)dB(s))=0,𝔼^[(∫0T‍η(s)dB(s))2]≤σ¯2𝔼^[∫0T‍η2(s)ds].
We now recall the definition of quadratic variation process of the G-Brownian motion B.Definition 10 (quadratic variation).
LetπtN={0=t0N<t1N<⋯<tN-1N=t} be a partition of [0,t] for t>0, such that μ(πtN):=maxj{tj-tj-1}→0 as N→∞. The quadratic variation of G-Brownian motion B can be defined as
(27)〈B〉(t)=limμ(πtN)→0∑k=0N-1‍(B(tk+1N)-B(tkN))2=B2(t)-2∫0t‍B(s)dB(s)
in 𝕃G2.The functiont↦〈B〉(t) is continuous and increasing outside a polar set. We can define the integral
(28)∫0T‍η(t)d〈B〉(t):=∑j=1N‍ξj(〈B〉(tj)-〈B〉(tj-1))
as a map from MG1,0([0,T]) into 𝕃G1, and the map is linear and continuous, so it can be extended continuously to MG1([0,T]).Finally, we recall some important results in order to get our desired result. For detailed description of them, please read the related papers.Theorem 11 (Itô’s formula).
LetF∈C2,1(ℝ×[0,∞)). One then has
(29)F(B(t),t)=F(B(0),0)+∫0t‍∂∂xF(B(s),s)dB(s)+∫0t‍∂∂tF(B(s),s)ds+12∫0t‍∂2∂x2F(B(s),s)d〈B〉(s)
for all t≥0.Lemma 12 (see [14]).
If there exists anε0>0 such that
(30)𝔼^[exp{(12+ε0)∫0T‍H2(s,ω)d〈B〉(s)}]<∞
then the process
(31)B(t)-∫0t‍H(s,ω)d〈B〉(s),0≤t≤T
is a G-Brownian motion under some G-expectation.Lemma 13 (see [19]).
Letp≥2 and η={η(t),t∈[0,T]}∈MGp([0,T]). Then
(32)𝔼^[sups≤u≤t|∫su‍η(r)dB(r)|p]≤Cp𝔼^(∫st‍|η(r)|2dr)p/2
for all 0≤s≤t≤T, where Cp is a positive constant depending only on p.Lemma 14 (see [19]).
Letp≥1 and η∈MGp([0,T]). Then we have
(33)𝔼^[sups≤u≤t|∫su‍η(r)d〈B〉(r)|p]≤(t-s)p-1𝔼^∫st‍[|η(r)|p]dr
for all 0≤s≤t≤T.


## 3. Existence and Uniqueness Theorem

In this section, we prove the existence and uniqueness of solutions to (3) under global Carathéodory conditions. More works for stochastic differential equations driven by G-Brownian motion can be found in Bai and Lin [18], Chen and Zhang [20], Gao [19], Lin [21], Lin [22], Ren et al. [23, 24], and the references therein.Letτ>0 and denote by C([-τ,0];ℝ) the family of continuous functions φ from [-τ,0] to ℝ with the norm
(34)∥φ∥=sup-τ≤θ≤0|φ(θ)|.
Given 0≤t0<T<∞ let
(35)b,h,σ:[t0,T]×C([-τ,0];ℝ)⟶ℝ
be three Borel measurable functions satisfying
(36)b(·,x),h(·,x),σ(·,x)∈MG2([t0,T])
for any x∈C([-τ,0];ℝ). Consider the following SFDE with finite delay of the form
(37)dX(t)=b(t,Xt)dt+h(t,Xt)d〈B〉(t)+σ(t,Xt)dB(t),
where Xt={X(t+θ);-τ≤θ≤0} and t∈[t0,T] is regarded as a C([-τ,0];ℝ)-valued stochastic process. Then we can impose the initial data: (i)
Xt0=ξt0={ξt0(θ);-τ≤θ≤0} is an ℱt0-measurable, C([-τ,0];ℝ)-valued random variable such that 𝔼^∥ξ∥2<∞. Thus we can get the following equivalent form:
(38)X(t)=ξt0(0)+∫t0t‍b(s,Xs)ds+∫t0t‍h(s,Xs)d〈B〉(s)+∫t0t‍σ(s,Xs)dB(s)
with t∈[t0,T]. In order to give the existence and uniqueness we now propose the following assumptions:(H1)
There exists a functionH(t,u):[t0,T]×ℝ+→ℝ+, such that
(1a)
|b(t,x)|2∨|h(t,x)|2∨|σ(t,x)|2≤H(t,|x|2)(1b)
H(t,u) is integrable in t for each fixed u∈ℝ+, and is continuous, concave, nondecreasing in u for each fixed t∈[t0,T],(1c)
for any constantK>0 the deterministic ODE
(39)dudt=KH(t,u),t∈[t0,T],
has a global solution for any initial value.(H2)
There exists a functionΓ(t,u):[t0,T]×ℝ+→ℝ+ such that
(2a)
(40)|b(t,x)-b(t,y)|2∨|h(t,x)-h(t,y)|2∨|σ(t,x)-σ(t,y)|2≤Γ(t,|x-y|2)
for all x,y∈ℝ and t∈[t0,T];(2b)
Γ(t,u) is integrable in t for each fixed u∈ℝ+ and is continuous, concave, nondecreasing in u for each fixed t≥t0. Moreover, Γ(t,0)≡0, and if there exists a nonnegative continuous function Z(t), t∈[t0,T] satisfies
(41)Z(t)≤D∫t0t‍Γ(s,Z(s))ds,t∈[t0,T],
where D>0 is a positive constant, then Z(t)≡0 for all t∈[t0,T].Example 15 (see [25]).
LetΓ(t,u)=λ(t)α(u), t≥t0, u≥0, where λ(t)≥0 is integrable and α:ℝ+→ℝ+ is a continuous, concave, monotone nondecreasing function with α(0)=0 such that ∫0+‍(1/α(u))du=∞. Let u(t) be a solution of du/dt=λ(t)α(u). If u(t0)=0, then we get u(t)≡0 for all t≥t0. If not, we can suppose without loss of generality that there exists t1>t0 such that u(t) is positive for all t∈(t0,t1]. Let t0<t<t1. Thus,
(42)∫u(t)u(t1)‍(1α(u))du=∫tt1‍λ(s)ds,
which implies a contradiction letting t→t0. Therefore, u(t)≡0 for all t≥t0.Our main object of this section is to explain and prove the following theorem.Theorem 16.
Let the assumptions (H1) and (H2) hold. Then, SFDE (37) brings a unique solution X with the above initial data and X∈𝕃G2.Proof.
DefineXt00=ξt0, X0(t)=ξt0(0) for t∈[t0,T], and let Xt0n=ξt0 for each n=1,2,…. Consider the following Picard iterations:
(43)Xn(t)=ξt0(0)+∫t0t‍b(s,Xsn-1)ds+∫t0t‍h(s,Xsn-1)d〈B〉(s)+∫t0t‍σ(s,Xsn-1)dB(s)
for n=0,1,2,…. Then we get
(44)X1(t)=ξt0(0)+∫t0t‍b(s,Xs0)ds+∫t0t‍h(s,Xs0)d〈B〉(s)+∫t0t‍σ(s,Xs0)dB(s).
Step  1. We claim that
(45)𝔼^|Xn(t)|2≤u(t),𝔼^|Xn(t)-X0(t)|2≤C
for all n=1,2,… and all t∈[t0,T], where C is a positive constant, which points out that there exists a real number R>0 depending only on t0 and T such that
(46)𝔼^|Xn(t)|2≤R
for all t∈[t0,T] and all n=0,1,2,…, since u(t) is continuous on [t0,T].
We will use the induction to prove this. It follows from Lemmas5, 13, and 14 and assumptions (1a), (1b) that there exist three positive constants k1, k2, and k3 independent of t such that
(47)𝔼^|X1(t)|2≤4𝔼^|ξt0(0)|2+4k1t𝔼^∫t0t‍|b(s,Xs0)|2ds+4k2t𝔼^∫t0t‍|h(s,Xs0)|2ds+4k3𝔼^∫t0t‍|σ(s,Xs0)|2ds≤4𝔼^∥ξ∥2+4(k1T+k2T+k3)𝔼^∫t0t‍H(s,|Xs0|2)ds≤4𝔼^∥ξ∥2+4(k1T+k2T+k3)∫t0t‍H(s,𝔼^|Xs0|2)ds
for all t∈[t0,T]. Notice that du/dt=KH(t,u) has a global solution with any initial value (t0,u0). We can take u0∈ℝ+, such that u0>4𝔼^∥ξ∥2, and may assume that u(t)=u(t;t0,u0) is the solution of the above ODE with the initial value (t0,u0), which implies that
(48)u(t)-𝔼^|X1(t)|2>4(k1T+k2T+k3)×∫t0t(H(s,u(s))-H(s,𝔼^|Xs0|2))ds≥0.
Denote thatp0=max{u(t)∣t∈[t0,T]}. Then continuity of u(t) on [t0,T] deduces p0<∞ and
(49)H(s,u(s))≤H(s,p0)
for each s∈[t0,T]. Therefore, there exists a positive constant C such that
(50)𝔼^|X1(t)-X0(t)|2≤3(k1T+k2T+k3)∫t0t‍H(s,p0)ds≤C.
We now suppose that the inequalities
(51)𝔼^|Xk(t)|2≤u(t),𝔼^|Xk(t)-X0(t)|2≤C
hold for all t∈[t0,T] and k≥1. Then we have
(52)Xk+1(t)=ξt0(0)+∫t0t‍b(s,Xsk)ds+∫t0t‍h(s,Xsk)d〈B〉(s)+∫t0t‍σ(s,Xsk)dB(s),𝔼^|Xk+1(t)|2≤4𝔼^∥ξ∥2+4(k1T+k2T+k3)×∫t0t‍H(s,𝔼^|Xsk|2)ds.
Since u0>4𝔼^∥ξ∥2, we obtain that
(53)u(t)-𝔼^|Xk+1(t)|2>4(k1T+k2T+k3)×∫t0t(H(s,u(s))-H(s,𝔼^|Xsk|2))ds≥0;
then 𝔼^|Xk+1(t)|2≤u(t) for all t∈[t0,T], and
(54)𝔼^|Xk+1(t)-X0(t)|2≤3(k1T+k2T+k3)𝔼^|Xk+1(t)-X0(t)|2≤×∫t0t‍H(s,p0)ds≤C
for all t∈[t0,T]. Thus, we obtain our assertion by induction.
Step  2. We claim that {Xn(t)} is a Cauchy sequence in 𝕃G2. Thus, Xn(t)→X(t) in 𝕃G2 as n→∞. From Lemmas 5, 13, and 14 and assumptions (2a) and (2b), we get, for all m≥n≥0,
(55)𝔼^(supt0≤v≤t|Xm(v)-Xn(v)|2)=𝔼^(supt0≤v≤t|∫t0v‍(b(s,Xsm-1)-b(s,Xsn-1))dsddddddddc+∫t0v‍(h(s,Xsm-1)-h(s,Xsn-1))d〈B〉(s)ddddddddc+∫t0v‍(σ(s,Xsm-1)-σ(s,Xsn-1))dB(s)|2)≤3k1T𝔼^∫t0t‍|b(v,Xvm-1)-b(v,Xvn-1)|2dv+3k2T𝔼^∫t0T‍|h(v,Xvm-1)-h(v,Xvn-1)|2dv+3k3𝔼^∫t0t‍|σ(v,Xvm-1)-σ(v,Xvn-1)|2dv≤(3k1T+3k2T+3k3)×∫t0t‍Γ(v,𝔼^|Xvm-1-Xvn-1|2)dv≤(3k1T+3k2T+3k3)×∫t0t‍Γ(v,𝔼^(supt0≤s≤v|Xm-1(s)-Xn-1(s)|2))dv.
Let a(t)=lim¯m,n→∞𝔼^(supt0≤v≤t|Xm(v)-Xn(v)|2); then we get
(56)a(t)≤(3k1T+3k2T+3k3)·∫t0t(lim¯m,n→∞Γ(v,𝔼^(supt0≤s≤v|Xm-1(s)-Xn-1(s)|2)))dv=(3k1T+3k2T+3k3)×∫t0t‍Γ(v,lim¯m,n→∞𝔼^(supt0≤s≤v|Xm-1(s)-Xn-1(s)|2))dv,
which implies that
(57)a(t)≤(3k1T+3k2T+3k3)∫t0t‍Γ(s,a(s))ds.
It follows that a(t)≡0 for all t∈[t0,T]. That is,
(58)lim¯m,n→∞𝔼^(supt0≤t≤T|Xm(t)-Xn(t)|2)=0.
This shows that the sequence {Xn(·)} is a Cauchy sequence in 𝕃G2, which deduces Xn(·)→X(·) in 𝕃G2 as n→∞. Moreover, it is uniformly convergent on [t0,T], and therefore, X(t) is continuous.
Step  3. We show that the stochastic process X(t) given above is a solution of (37). We have
(59)𝔼^|∫t0t‍(b(s,Xsn)-b(s,Xs))ds|2+𝔼^|∫t0t‍(h(s,Xsn)-h(s,Xs))d〈B〉(s)|2+𝔼^|∫t0t‍(σ(s,Xsn)-σ(s,Xs))dB(s)|2≤k1T𝔼^∫t0t‍|b(s,Xsn)-b(s,Xs)|2ds+k2T𝔼^∫t0t‍|h(s,Xsn)-h(s,Xs)|2ds+k3𝔼^∫t0t‍|σ(s,Xsn)-σ(s,Xs)|2ds≤(k1T+k2T+k3)∫t0t‍Γ(s,𝔼^|Xsn-Xs|2)ds
for all t∈[t0,T] and n≥0. Noting that the sequence {Xn(t)} uniformly converges on [t0,T], we get
(60)∫t0t‍b(s,Xsn)ds⟶∫t0t‍b(s,Xs)ds,∫t0t‍h(s,Xsn)d〈B〉(s)⟶∫t0t‍h(s,Xs)d〈B〉(s),∫t0t‍σ(s,Xsn)dB(s)⟶∫t0t‍σ(s,Xs)dB(s)
in 𝕃G2, as n→∞, which deduces
(61)X(t)=ξt0(0)+∫t0t‍b(s,Xs)ds+∫t0t‍h(s,Xs)d〈B〉(s)+∫t0t‍σ(s,Xs)dB(s).
Thanks to Lemmas 13, 14, and 5 and assumptions (1a), (1b), and (1c), we can also get X(t)∈𝕃G2 for all t∈[t0,T].
Step  4. Finally, we will show that the uniqueness of the solution. Let X and Y be two solutions existing on [t0,T] and X(t0)=Y(t0)=ξt0(0). Then, we have
(62)𝔼^(supt0≤v≤t|X(v)-Y(v)|2)=𝔼^[supt0≤v≤t|∫t0v‍(b(s,Xs)-b(s,Ys))dsddddddddddd+∫t0v‍(h(s,Xs)-h(s,Ys))d〈B〉(s)ddddddddddd+∫t0v‍(σ(s,Xs)-σ(s,Ys))dB(s)|2]≤(3k1T+3k2T+3k3)∫t0t‍Γ(s,𝔼^|Xs-Ys|2)ds≤(3k1T+3k2T+3k3)×∫t0t‍Γ(s,𝔼^(supt0≤v≤s|X(v)-Y(v)|2))ds,
which deduces X(t)=Y(t) for all t∈[t0,T] q.s. This shows that the uniqueness of the solution and the theorem follows.


## 4. An Application

In this section, we consider an application of the above theorem. Suppose that the stock price satisfies the following equation:(63)dS(t)=b(St)S(t)dt+σ(St)S(t)dB(t)+h(St)S(t)d〈B〉(t),S0=ξ={ξ(θ);-τ≤θ≤0},
where St={S(t+θ);-τ≤θ≤0,t∈[0,T]} is regarded as a C([-τ,0];ℝ)-valued stochastic process. {〈B〉(t),t≥0} is the quadratic variation process of G-Brownian motion {B(t),t≥0}.In the rest of this section, we want to consider a special case of the above equation. We let the functions(64)b,σ,h:C([-τ,0];ℝ)⟶ℝ
satisfy the following conditions: (1)
there exists aε1>0 such that |σ(x)|>ε1;(2)
|b(x)|>r, where r>0 denotes the rate of return on a riskless asset, which is compounded continuously;(3)
there exists aM>0, such that |b(x)|∨|σ(x)|∨|h(x)|≤M.Corollary 17.
Under the above assumption, the above SDDE brings a unique solution and(65)S(t)=ξ(0)exp{∫0t‍[12]∫0t‍b(Sv)dv+∫0t‍σ(Sv)dB(v)dddddddd+∫0t‍[(h(Sv))2-12(σ(Sv))2]d〈B〉(v)}.
Furthermore, if ξ(0)≥0(ξ(0)>0) q.s., then S(t)≥0(S(t)>0) for all t≥0 q.s.Proof.
Letθ0∈[-τ,0]. Consider the case t∈[0,-θ0], and in this situation, we have θ0≤t+θ0≤0 and
(66)St=S(t+θ0)=ξ(t+θ0),
which deduces
(67)dS(t)=b(ξ(t+θ0))S(t)dt+σ(ξ(t+θ0))S(t)dB(t)+h(ξ(t+θ0))S(t)d〈B〉(t),S(0)=ξ(0).
It follows from Theorem 16 and Itô’s formula that the unique solution of (67)
(68)S(t)=ξ(0)exp{∫0tb(ξ(v+θ0))dv+∫0tσ(ξ(v+θ0))dB(v)ddddddddddd+∫0t[(h(ξ(v+θ0)))2-12(σ(ξ(v+θ0)))2]ddddddddddd×d〈B〉(v)∫0tb(ξ(v+θ0))dv}
for all t∈[0,-θ0]. This implies that S(t)>0 for all t∈[0,-θ0], when ξ(0)>0 q.s. By a similar argument, it follows that S(t)>0 for all t∈[-θ0,-2θ0] q.s. Therefore S(t)>0 for all t≥0 q.s. By induction, we can also get the solution to (67) as follows:
(69)S(t)=ξ(0)exp{∫0t‍b(S(v+θ0))dv+∫0t‍σ(S(v+θ0))dB(v)ddddddddddd+∫0t‍[(h(S(v+θ0)))2-12(σ(S(v+θ0)))2]ddddddddddd×d〈B〉(v)∫0t‍}
for t∈[0,T]. Since that equality is true for all -τ≤θ0≤0, we get our result.We denote the wealth process byV(t) and units of the stock that we hold by α(t). Thanks to the assumption of self-financing, we can get
(70)dV(t)=r(V(t)-α(t)S(t))dt+α(t)dS(t)=r(V(t)-α(t)S(t))dt+α(t)[b(St)S(t)dt+σ(St)S(t)dB(t).α(t)d+h(St)S(t)d〈B〉(t)]
for all t∈[0,T]. Let
(71)H(t,ω)=((r-b(St))/σ02)-h(St)σ(St).
Then there exists a ε0>0 such that
(72)𝔼^exp[(12+ε0)∫0TH2(t,ω)d〈B〉(t)]<∞,
which shows that
(73)B~(t)=B(t)-∫0tH(s,ω)d〈B〉(s)
is G-Brownian motion under some G-expectation 𝔼~ by G-Girsanov theorem. Let now Γ(t)=e-rt; then
(74)dΓ(t)=-rΓ(t)dt,dΓ(t)V(t)=Γ(t)dV(t)+V(t)dΓ(t)+dΓ(t)dV(t)=-rΓ(t)V(t)dt+Γ(t)dV(t)=α(t)Γ(t)S(t)[(b(St)-r)dt+σ(St)dB(t)+h(St)d〈B〉(t)]=α(t)Γ(t)S(t)[r-b(St)σ02(b(St)-r)dt+σ(St)dB~(t)+r-b(St)σ02d〈B〉(t)]=α(t)Γ(t)D(t)S(t)[d(σ02t)-d〈B〉(t)]+α(t)Γ(t)S(t)σ(St)dB~(t),
where D(t)=(b(St)-r)/σ02; we can get that the discounted wealth process is G-martingale under some G-expectation. Thus, we can obtain that the price of European call option C(0) is
(75)C(0)≤𝔼~[V(T)Γ(T)].



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*Source: 637106-2013-12-28.xml*