Successive Approximation of SFDEs with Finite Delay Driven by 𝐺-Brownian Motion

Abstract

We consider the stochastic functional differential equations with finite delay driven by 𝐺-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.

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: 𝜕𝑢()(),()∈[)𝑢()(𝑥),𝜕𝑡𝑡,𝑥=𝐺Δ𝑢𝑡,𝑥0,+∞×ℝ,0,𝑥=𝜑 (1) where Δ is Laplacian and the sublinear function 𝐺 is defined as 𝐺(𝛼)=12𝜎²𝛼⁺−𝜎²𝛼⁻,𝛼∈ℝ (2) 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 under 𝐺-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 𝐵_𝑡+𝑠−𝐵_𝑡 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 𝑑𝑋(𝑡)=𝑏𝑡,𝑋_𝑡𝑑𝑡+ℎ𝑡,𝑋_𝑡𝑑⟨𝐵⟩(𝑡)+𝜎𝑡,𝑋_𝑡(𝑡),𝑡𝑑𝐵𝑡∈₀,𝑇, (3) where (i) 𝑋_𝑡={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 Section 2, 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 for 𝐺-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

Let Ω≠∅ 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 𝜑𝑋₁,…,𝑋_𝑑∈ℋ (4) if 𝑋_𝑖∈ℋ, 𝑖=1,…,𝑑, for all 𝜑∈𝐶_𝑏,Lip(ℝ^𝑑), where 𝐶_𝑏,Lip(ℝ^𝑑) denotes the space of all bounded and Lipschitz functions on ℝ^𝑑.

Definition 1. A sublinear expectation 𝔼 on ℋ is a functional with the following properties: for all 𝑋, 𝑌∈ℋ, we have (i)monotonicity: if 𝑋≥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 let Ω=𝐶₀(ℝ⁺) be the space of all real-valued continuous functions on [0,∞) with initial value 0, equipped with the distance 𝜌𝜔¹,𝜔²=^∞_𝑖=12^−𝑖max_{𝑡∈[0,𝑖]}𝜔¹_𝑡−𝜔²_𝑡,𝜔∧1₁,𝜔²∈Ω. (5) We denote by ℬ(Ω) the Borel-algebra on Ω. We also denote, for each 𝑡∈[0,∞), Ω_𝑡=𝜔_⋅∧𝑡,𝜔∈Ω (6) and ℱ_𝑡=ℬ(Ω_𝑡), where 𝑥∧𝑦=min{𝑥,𝑦}. Let 𝕃^𝑝_𝐺(Ω) be the closure of ℋ with respect to the norm ‖𝑋‖_𝑝=𝔼|𝑋|^𝑝1/𝑝 (7) with 𝑝∈[1,∞). Clearly, the space 𝕃^𝑝_𝐺(Ω) is a Banach space and the space 𝐶_𝑏(Ω) of bounded continuous functions on Ω is a subset of 𝕃^𝑝_𝐺(Ω), and moreover, there exists a weakly compact family 𝒫 of probability measures on (Ω,ℬ(Ω)) such that 𝔼=sup_𝑃∈𝒫𝐸_𝑃. (8) So we can introduce the Choquet capacity 𝐶 by 𝐶(𝐴)=sup_𝑃∈𝒫𝑃(𝐴),𝐴∈ℬ(Ω). (9)

Definition 2. A set 𝐴⊂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 measures 𝑃 allows characterizing the space 𝕃^𝑝_𝐺(Ω) as follows: 𝕃^𝑝_𝐺(Ω)=𝑋∶ℬ(Ω)=measurablereal-valuedfunctionsonΩsuchthatsup_𝑃∈𝒫𝐸_𝑃|𝑋|^𝑝<∞,andis𝐶-quasisurelycontinuous. (10)

Lemma 3 (Denis et al. [8]). Let 1≤𝑝<∞. Consider the sets 𝕃^𝑝_𝐺(Ω) and 𝕃^𝑝=ℒ^𝑝/𝒩, where ℒ^𝑝=𝑋∶ℬ(Ω)=𝑚𝑒𝑎𝑠𝑢𝑟𝑎𝑏𝑙𝑒𝑟𝑒𝑎𝑙-𝔼|𝑋|𝑣𝑎𝑙𝑢𝑒𝑑𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠𝑜𝑛Ω𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡^𝑝=sup_𝑃∈𝒫𝐸_𝑃|𝑋|^𝑝,(Ω)<∞𝒩=𝑋∶ℬ=𝑚𝑒𝑎𝑠𝑢𝑟𝑎𝑏𝑙𝑒𝑟𝑒𝑎𝑙-.𝑣𝑎𝑙𝑢𝑒𝑑𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠𝑜𝑛Ω𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑋=0𝑞.𝑠 (11) Then (i)𝕃^𝑝 is a Banach space with respect to the norm ‖⋅‖_𝑝;(ii)𝕃^𝑝_𝐺 is the completion of 𝐶_𝑏(Ω) with respect to the norm ‖⋅‖_𝑝.

Lemma 4 (Li and Peng [17]). For a given 𝑝∈(0,+∞], If the sequence {𝑋_𝑛}⊂𝕃^𝑝 converges to 𝑋 in 𝕃^𝑝, then there exists a subsequence {𝑋_𝑛𝑘} such that 𝑋_𝑛𝑘 converges to 𝑋 quasi surely.

Lemma 5 (see [18]). Let 𝜌∶ℝ→ℝ be an increasing and concave function. Then the inequality 𝔼𝜌(𝑋)≤𝜌𝔼[𝑋] (12) holds for all ℬ(Ω)-measurable real-valued functions on Ω.

For 𝑋=(𝑋₁,…,𝑋_𝑛)∈ℋ^𝑛, the functional 𝐹_𝑋 defined by 𝐹_𝑋𝜑∶=𝔼𝜑(𝑋),𝜑∈𝐶_𝑏,Lipℝ^𝑛 (13) is called the distribution of 𝑋 under 𝔼. In a sublinear expectation space (Ω,ℋ,𝔼), a random vector 𝑌=(𝑌₁,…,𝑌_𝑛), 𝑌_𝑖∈ℋ, is said to be independent under 𝔼 from another random vector 𝑋=(𝑋₁,…,𝑋_𝑚), 𝑋_𝑖∈ℋ, if for each test function 𝜑∈𝐶_𝑏,Lip(ℝ^𝑚+𝑛) we have 𝔼𝜑(𝑋,𝑌)=𝔼𝔼𝜑(𝑥,𝑌)_𝑥=𝑋. (14)

2.2. 𝐺-Brownian Motion

Let 𝜎, 𝜎 be two real numbers with 0≤𝜎≤𝜎.

A random variable 𝜉 in a sublinear expectation space (Ω,ℋ,𝔼) is called 𝐺-normal distributed, denoted by 𝜉∼𝑁(0,[𝜎²,𝜎²]), if, for each 𝜑∈𝐶_𝑏,Lip(ℝ), the function defined by 𝑢(𝑡,𝑥)∶=𝔼𝜑𝑥+√𝑡𝜉,(𝑡,𝑥)∈[0,∞)×ℝ (15) is the unique viscosity solution of the following nonlinear heat equation: 𝜕𝑢()(),()∈[)𝑢()(𝑥),𝜕𝑡𝑡,𝑥=𝐺Δ𝑢𝑡,𝑥0,+∞×ℝ,0,𝑥=𝜑 (16) where Δ is Laplacian and the sublinear function 𝐺 is defined as 𝐺(𝛼)=12𝜎²𝛼⁺−𝜎²𝛼⁻,𝛼∈ℝ. (17)

In particular, 𝐸[𝜑(𝜉)]=𝑢(1,0) is the distribution of 𝜉.

Example 6 (Peng [1]). Let 𝜉∼𝑁(0,[𝜎²,𝜎²]). We then have 𝔼𝜑(𝜉=1)√2𝜋𝜎^ℝ𝜑(𝑥)𝑒^{−(1/2𝜎2)𝑥2}𝑑𝑥 (18) for all convex functions 𝜑 and 𝔼𝜓(𝜉=1)√2𝜋𝜎^ℝ𝜓(𝑥)𝑒^{−(1/2𝜎2)𝑥2}𝑑𝑥 (19) for all concave functions 𝜓.

Definition 7. A process 𝐵={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 each 𝑡,𝑠≥0, the increment 𝐵(𝑡+𝑠)−𝐵(𝑡) is 𝑁(0,[𝜎²𝑠,𝜎²𝑠])-distributed and is independent of (𝐵(𝑡₁),…,𝐵(𝑡_𝑛)), for all 𝑛=0,1,2,… and 0≤𝑡₁≤𝑡₂≤⋯≤𝑡_𝑛≤𝑡.

Recall that a process 𝑋={𝑋(𝑡)∶0≤𝑡≤𝑇} is called a 𝐺-martingale if, for each 0≤𝑡≤𝑇, we have 𝑋(𝑡)∈𝕃¹_𝐺(Ω_𝑡) and 𝔼𝑋(𝑡)∣Ω_𝑠=𝑋(𝑠) (20) 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 let 𝜎=𝜎₀>0, and 𝜎=1.

2.3. Itô’s Integral

We now recall the definition of Itô’s integral and quadratic variation process of the 𝐺-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 𝑀_𝐺^𝑝,0([0,𝑇]) the set of step processes 𝜂_𝑡(𝜔)=^𝑁_𝑗=1𝜉_𝑗(𝜔)𝐼_{[𝑡𝑗−1,𝑡𝑗)}(𝑡),𝜉_𝑗(𝜔)∈𝕃^𝑝_𝐺Ω_𝑡𝑗 (21) with 0=𝑡₀<⋯<𝑡_𝑁=𝑇. Moreover, we denote by 𝑀^𝑝_𝐺([0,𝑇]) the completion of 𝑀_𝐺^𝑝,0([0,𝑇]) under the norm 𝜂_𝑀^𝑝_{𝐺([0,𝑇])}=𝔼^𝑇₀𝜂(𝑠)^𝑝𝑑𝑠^1/𝑝. (22) Now we can define the stochastic integral ∫^𝑇₀𝜂(𝑠)𝑑𝐵(𝑠) for 𝜂∈𝑀^𝑝_𝐺([0,𝑇]).

Definition 9. For every 𝜂∈𝑀_𝐺^𝑝,0([0,𝑇]) with the form (21), we define Itô’s integral 𝐼𝜂=^𝑡₀𝜂(𝑠)𝑑𝐵(𝑠)∶=^𝑁_𝑗=1𝜉_𝑗𝐵𝑠_𝑗−𝐵𝑠_𝑗−1. (23)

One can show that 𝐼(𝜂)∈𝕃²_𝐺 and the linear mapping (see Li and Peng [17]) 𝐼∶𝑀_𝐺^2,0([0,𝑇])⟶𝕃²_𝐺 (24) is continuous. Moreover, it can be continuously extended to 𝐼∶𝑀²_𝐺([0,𝑇])⟶𝕃²_𝐺. (25) And, for all 𝜂∈𝑀²_𝐺([0,𝑇]), we have 𝔼^𝑇₀𝜂(𝑠)(𝑠)𝔼𝑑𝐵=0,_𝑇0𝜂(𝑠)𝑑𝐵(𝑠)²≤𝜎²𝔼^𝑇₀𝜂²(𝑠)𝑑𝑠. (26) We now recall the definition of quadratic variation process of the 𝐺-Brownian motion 𝐵.

Definition 10 (quadratic variation). Let 𝜋^𝑁_𝑡={0=𝑡^𝑁₀<𝑡^𝑁₁<⋯<𝑡^𝑁_𝑁−1=𝑡} be a partition of [0,𝑡] for 𝑡>0, such that 𝜇(𝜋^𝑁_𝑡)∶=max_𝑗{𝑡_𝑗−𝑡_𝑗−1}→0 as 𝑁→∞. The quadratic variation of 𝐺-Brownian motion 𝐵 can be defined as ⟨𝐵⟩(𝑡)=lim_{𝜇(𝜋𝑁𝑡)→0}^𝑁−1_𝑘=0𝐵𝑡^𝑁_𝑘+1−𝐵𝑡^𝑁_𝑘²=𝐵²(𝑡)−2^𝑡₀𝐵(𝑠)𝑑𝐵(𝑠) (27) in 𝕃²_𝐺.

The function 𝑡↦⟨𝐵⟩(𝑡) is continuous and increasing outside a polar set. We can define the integral ^𝑇₀𝜂(𝑡)𝑑⟨𝐵⟩(𝑡)∶=^𝑁_𝑗=1𝜉_𝑗⟨𝐵⟩𝑡_𝑗−⟨𝐵⟩𝑡_𝑗−1 (28) as a map from 𝑀_𝐺^1,0([0,𝑇]) into 𝕃¹_𝐺, and the map is linear and continuous, so it can be extended continuously to 𝑀¹_𝐺([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). Let 𝐹∈𝐶^2,1(ℝ×[0,∞)). One then has 𝐹(𝐵(𝑡),𝑡)=𝐹(𝐵(0),0)+^𝑡₀𝜕𝐹(𝐵(𝑠))(𝑠)+𝜕𝑥,𝑠𝑑𝐵_𝑡0𝜕𝐹(𝐵(𝑠))+1𝜕𝑡,𝑠𝑑𝑠2^𝑡₀𝜕²𝜕𝑥²𝐹(𝐵(𝑠),𝑠)𝑑⟨𝐵⟩(𝑠) (29) for all 𝑡≥0.

Lemma 12 (see [14]). If there exists an 𝜀₀>0 such that 𝔼1exp2+𝜀₀^𝑇₀𝐻²(𝑠,𝜔)𝑑⟨𝐵⟩(𝑠)<∞ (30) then the process 𝐵(𝑡)−^𝑡₀𝐻(𝑠,𝜔)𝑑⟨𝐵⟩(𝑠),0≤𝑡≤𝑇 (31) is a 𝐺-Brownian motion under some 𝐺-expectation.

Lemma 13 (see [19]). Let 𝑝≥2 and 𝜂={𝜂(𝑡),𝑡∈[0,𝑇]}∈𝑀^𝑝_𝐺([0,𝑇]). Then 𝔼sup_{𝑠≤𝑢≤𝑡}^𝑢_𝑠𝜂(𝑟)𝑑𝐵(𝑟)^𝑝≤𝐶_𝑝𝔼^𝑡_𝑠𝜂(𝑟)²𝑑𝑟^𝑝/2 (32) for all 0≤𝑠≤𝑡≤𝑇, where 𝐶_𝑝 is a positive constant depending only on 𝑝.

Lemma 14 (see [19]). Let 𝑝≥1 and 𝜂∈𝑀^𝑝_𝐺([0,𝑇]). Then we have 𝔼sup_{𝑠≤𝑢≤𝑡}^𝑢_𝑠𝜂(𝑟)𝑑⟨𝐵⟩(𝑟)^𝑝≤(𝑡−𝑠)^𝑝−1𝔼^𝑡_𝑠𝜂(𝑟)^𝑝𝑑𝑟 (33) 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.

Let 𝜏>0 and denote by 𝐶([−𝜏,0];ℝ) the family of continuous functions 𝜑 from [−𝜏,0] to ℝ with the norm 𝜑=sup_{−𝜏≤𝜃≤0}𝜑(𝜃). (34) Given 0≤𝑡₀<𝑇<∞ let 𝑏,ℎ,𝜎∶𝑡₀,𝑇×𝐶([−𝜏,0];ℝ)⟶ℝ (35) be three Borel measurable functions satisfying 𝑏(⋅,𝑥),ℎ(⋅,𝑥),𝜎(⋅,𝑥)∈𝑀²_𝐺𝑡₀,𝑇 (36) for any 𝑥∈𝐶([−𝜏,0];ℝ). Consider the following SFDE with finite delay of the form 𝑑𝑋(𝑡)=𝑏𝑡,𝑋_𝑡𝑑𝑡+ℎ𝑡,𝑋_𝑡𝑑⟨𝐵⟩(𝑡)+𝜎𝑡,𝑋_𝑡𝑑𝐵(𝑡), (37) where 𝑋_𝑡={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)𝑋_𝑡0=𝜉_𝑡0={𝜉_𝑡0(𝜃);−𝜏≤𝜃≤0} is an ℱ_𝑡0-measurable, 𝐶([−𝜏,0];ℝ)-valued random variable such that 𝔼‖𝜉‖²<∞. Thus we can get the following equivalent form: 𝑋(𝑡)=𝜉_𝑡0(0)+^𝑡_𝑡0𝑏𝑠,𝑋_𝑠𝑑𝑠+^𝑡_𝑡0ℎ𝑠,𝑋_𝑠𝑑⟨𝐵𝑠)+⟩(_𝑡𝑡0𝜎𝑠,𝑋_𝑠𝑑𝐵(𝑠) (38) with 𝑡∈[𝑡₀,𝑇]. In order to give the existence and uniqueness we now propose the following assumptions:(H1)There exists a function 𝐻(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 constant 𝐾>0 the deterministic ODE 𝑑𝑢()𝑡𝑑𝑡=𝐾𝐻𝑡,𝑢,𝑡∈₀,𝑇, (39) has a global solution for any initial value.(H2)There exists a function Γ(𝑡,𝑢)∶[𝑡₀,𝑇]×ℝ₊→ℝ₊ such that(2a)𝑏(𝑡,𝑥)−𝑏𝑡,𝑦²∨ℎ(𝑡,𝑥)−ℎ𝑡,𝑦²∨𝜎(𝑡,𝑥)−𝜎𝑡,𝑦²≤Γ𝑡,𝑥−𝑦² (40) 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 𝑍(𝑡)≤𝐷^𝑡_𝑡0Γ(𝑠,𝑍(𝑠))𝑑𝑠,𝑡∈𝑡₀,𝑇, (41) where 𝐷>0 is a positive constant, then 𝑍(𝑡)≡0 for all 𝑡∈[𝑡₀,𝑇].

Example 15 (see [25]). Let Γ(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, ^𝑢(𝑡1)_𝑢(𝑡)1𝛼(𝑢)𝑑𝑢=^𝑡1_𝑡𝜆(𝑠)𝑑𝑠, (42) 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∈𝕃²_𝐺.

Proof. Define 𝑋⁰_𝑡0=𝜉_𝑡0, 𝑋⁰(𝑡)=𝜉_𝑡0(0) for 𝑡∈[𝑡₀,𝑇], and let 𝑋^𝑛_𝑡0=𝜉_𝑡0 for each 𝑛=1,2,…. Consider the following Picard iterations: 𝑋^𝑛(𝑡)=𝜉_𝑡0(0)+^𝑡_𝑡0𝑏𝑠,𝑋_𝑠^𝑛−1𝑑𝑠+^𝑡_𝑡0ℎ𝑠,𝑋_𝑠^𝑛−1𝑑⟨𝐵𝑠)+⟩(_𝑡𝑡0𝜎𝑠,𝑋_𝑠^𝑛−1𝑑𝐵(𝑠) (43) for 𝑛=0,1,2,…. Then we get 𝑋¹(𝑡)=𝜉_𝑡0(0)+^𝑡_𝑡0𝑏𝑠,𝑋⁰_𝑠𝑑𝑠+^𝑡_𝑡0ℎ𝑠,𝑋⁰_𝑠𝑑⟨𝐵𝑠)+⟩(_𝑡𝑡0𝜎𝑠,𝑋⁰_𝑠𝑑𝐵(𝑠). (44)
Step  1. We claim that 𝔼𝑋^𝑛(𝑡)²(𝑡),𝔼𝑋≤𝑢_𝑛(𝑡)−𝑋⁰(𝑡)²≤𝐶 (45) 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 𝔼𝑋^𝑛(𝑡)²≤𝑅 (46) 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 Lemmas 5, 13, and 14 and assumptions (1a), (1b) that there exist three positive constants 𝑘₁, 𝑘₂, and 𝑘₃ independent of 𝑡 such that 𝔼𝑋¹(𝑡)²≤4𝔼𝜉_𝑡0(0)²+4𝑘₁𝑡𝔼^𝑡_𝑡0𝑏𝑠,𝑋⁰_𝑠²𝑑𝑠+4𝑘₂𝑡𝔼^𝑡_𝑡0ℎ𝑠,𝑋⁰_𝑠²𝑑𝑠+4𝑘₃𝔼^𝑡_𝑡0𝜎𝑠,𝑋⁰_𝑠²𝔼𝜉𝑑𝑠≤4²+4𝑘₁𝑇+𝑘₂𝑇+𝑘₃𝔼^𝑡_𝑡0𝐻𝑠,𝑋⁰_𝑠²𝔼𝜉𝑑𝑠≤4²+4𝑘₁𝑇+𝑘₂𝑇+𝑘₃^𝑡_𝑡0𝐻𝑠,𝔼𝑋⁰_𝑠²𝑑𝑠 (47) 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 𝑢(𝑡)−𝔼𝑋¹(𝑡)²>4𝑘₁𝑇+𝑘₂𝑇+𝑘₃×^𝑡_𝑡0𝐻(𝑠,𝑢(𝑠))−𝐻𝑠,𝔼𝑋⁰_𝑠²𝑑𝑠≥0. (48)
Denote that 𝑝₀=max{𝑢(𝑡)∣𝑡∈[𝑡₀,𝑇]}. Then continuity of 𝑢(𝑡) on [𝑡₀,𝑇] deduces 𝑝₀<∞ and 𝐻(𝑠,𝑢(𝑠))≤𝐻𝑠,𝑝₀ (49) for each 𝑠∈[𝑡₀,𝑇]. Therefore, there exists a positive constant 𝐶 such that 𝔼𝑋¹(𝑡)−𝑋⁰(𝑡)²≤3𝑘₁𝑇+𝑘₂𝑇+𝑘₃^𝑡_𝑡0𝐻𝑠,𝑝₀𝑑𝑠≤𝐶. (50) We now suppose that the inequalities 𝔼𝑋^𝑘(𝑡)²(𝑡),𝔼𝑋≤𝑢_𝑘(𝑡)−𝑋⁰(𝑡)²≤𝐶 (51) hold for all 𝑡∈[𝑡₀,𝑇] and 𝑘≥1. Then we have 𝑋^𝑘+1(𝑡)=𝜉_𝑡0(0)+^𝑡_𝑡0𝑏𝑠,𝑋^𝑘_𝑠+𝑑𝑠_𝑡𝑡0ℎ𝑠,𝑋^𝑘_𝑠𝑑⟨𝐵⟩(𝑠)+^𝑡_𝑡0𝜎𝑠,𝑋^𝑘_𝑠(𝑠),𝔼𝑋𝑑𝐵_𝑘+1(𝑡)²≤4𝔼𝜉²+4𝑘₁𝑇+𝑘₂𝑇+𝑘₃×^𝑡_𝑡0𝐻𝑠,𝔼𝑋^𝑘_𝑠²𝑑𝑠. (52) Since 𝑢₀>4𝔼‖𝜉‖², we obtain that 𝑢(𝑡)−𝔼𝑋^𝑘+1(𝑡)²>4𝑘₁𝑇+𝑘₂𝑇+𝑘₃×^𝑡_𝑡0𝐻(𝑠,𝑢(𝑠))−𝐻𝑠,𝔼𝑋^𝑘_𝑠²𝑑𝑠≥0; (53) then 𝔼|𝑋^𝑘+1(𝑡)|²≤𝑢(𝑡) for all 𝑡∈[𝑡₀,𝑇], and 𝔼𝑋^𝑘+1(𝑡)−𝑋⁰(𝑡)²≤3𝑘₁𝑇+𝑘₂𝑇+𝑘₃×^𝑡_𝑡0𝐻𝑠,𝑝₀𝑑𝑠≤𝐶 (54) for all 𝑡∈[𝑡₀,𝑇]. Thus, we obtain our assertion by induction.
Step  2. We claim that {𝑋^𝑛(𝑡)} is a Cauchy sequence in 𝕃²_𝐺. Thus, 𝑋^𝑛(𝑡)→𝑋(𝑡) in 𝕃²_𝐺 as 𝑛→∞. From Lemmas 5, 13, and 14 and assumptions (2a) and (2b), we get, for all 𝑚≥𝑛≥0, 𝔼sup_{𝑡0≤𝑣≤𝑡}𝑋^𝑚(𝑣)−𝑋^𝑛(𝑣)²=𝔼sup_{𝑡0≤𝑣≤𝑡}^𝑣_𝑡0𝑏𝑠,𝑋_𝑠^𝑚−1−𝑏𝑠,𝑋_𝑠^𝑛−1+𝑑𝑠_𝑣𝑡0ℎ𝑠,𝑋_𝑠^𝑚−1−ℎ𝑠,𝑋_𝑠^𝑛−1𝑑⟨𝐵𝑠)+⟩(_𝑣𝑡0𝜎𝑠,𝑋_𝑠^𝑚−1−𝜎𝑠,𝑋_𝑠^𝑛−1𝑑𝐵(𝑠)²≤3𝑘₁𝑇𝔼^𝑡_𝑡0𝑏𝑣,𝑋_𝑣^𝑚−1−𝑏𝑣,𝑋_𝑣^𝑛−12𝑑𝑣+3𝑘₂𝑇𝔼^𝑇_𝑡0ℎ𝑣,𝑋_𝑣^𝑚−1−ℎ𝑣,𝑋_𝑣^𝑛−12𝑑𝑣+3𝑘₃𝔼^𝑡_𝑡0𝜎𝑣,𝑋_𝑣^𝑚−1−𝜎𝑣,𝑋_𝑣^𝑛−12≤𝑑𝑣3𝑘₁𝑇+3𝑘₂𝑇+3𝑘₃×^𝑡_𝑡0Γ𝑣,𝔼𝑋_𝑣^𝑚−1−𝑋_𝑣^𝑛−12≤𝑑𝑣3𝑘₁𝑇+3𝑘₂𝑇+3𝑘₃×^𝑡_𝑡0Γ𝑣,𝔼sup_{𝑡0≤𝑠≤𝑣}𝑋^𝑚−1(𝑠)−𝑋^𝑛−1(𝑠)²𝑑𝑣. (55) Let 𝑎(𝑡)=lim_{𝑚,𝑛→∞}𝔼(sup_{𝑡0≤𝑣≤𝑡}|𝑋^𝑚(𝑣)−𝑋^𝑛(𝑣)|²); then we get 𝑎(𝑡)≤3𝑘₁𝑇+3𝑘₂𝑇+3𝑘₃⋅^𝑡_𝑡0lim_{𝑚,𝑛→∞}Γ𝑣,𝔼sup_{𝑡0≤𝑠≤𝑣}𝑋^𝑚−1(𝑠)−𝑋^𝑛−1(𝑠)²=𝑑𝑣3𝑘₁𝑇+3𝑘₂𝑇+3𝑘₃×^𝑡_𝑡0Γ𝑣,lim_{𝑚,𝑛→∞}𝔼sup_{𝑡0≤𝑠≤𝑣}𝑋^𝑚−1(𝑠)−𝑋^𝑛−1(𝑠)²𝑑𝑣, (56) which implies that 𝑎(𝑡)≤3𝑘₁𝑇+3𝑘₂𝑇+3𝑘₃^𝑡_𝑡0Γ(𝑠,𝑎(𝑠))𝑑𝑠. (57) It follows that 𝑎(𝑡)≡0 for all 𝑡∈[𝑡₀,𝑇]. That is, lim_{𝑚,𝑛→∞}𝔼sup_{𝑡0≤𝑡≤𝑇}𝑋^𝑚(𝑡)−𝑋^𝑛(𝑡)²=0. (58) This shows that the sequence {𝑋^𝑛(⋅)} is a Cauchy sequence in 𝕃²_𝐺, which deduces 𝑋^𝑛(⋅)→𝑋(⋅) in 𝕃²_𝐺 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 𝔼^𝑡_𝑡0𝑏𝑠,𝑋^𝑛_𝑠−𝑏𝑠,𝑋_𝑠𝑑𝑠²+𝔼^𝑡_𝑡0ℎ𝑠,𝑋^𝑛_𝑠−ℎ𝑠,𝑋_𝑠𝑑⟨𝐵⟩(𝑠)²+𝔼^𝑡_𝑡0𝜎𝑠,𝑋^𝑛_𝑠−𝜎𝑠,𝑋_𝑠𝑑𝐵(𝑠)²≤𝑘₁𝑇𝔼^𝑡_𝑡0𝑏𝑠,𝑋^𝑛_𝑠−𝑏𝑠,𝑋_𝑠²𝑑s+𝑘₂𝑇𝔼^𝑡_𝑡0ℎ𝑠,𝑋^𝑛_𝑠−ℎ𝑠,𝑋_𝑠²𝑑𝑠+𝑘₃𝔼^𝑡_𝑡0𝜎𝑠,𝑋^𝑛_𝑠−𝜎𝑠,𝑋_𝑠²≤𝑘𝑑𝑠₁𝑇+𝑘₂𝑇+𝑘₃^𝑡_𝑡0Γ𝑠,𝔼𝑋^𝑛_𝑠−𝑋_𝑠²𝑑𝑠 (59) for all 𝑡∈[𝑡₀,𝑇] and 𝑛≥0. Noting that the sequence {𝑋^𝑛(𝑡)} uniformly converges on [𝑡₀,𝑇], we get ^𝑡_𝑡0𝑏𝑠,𝑋^𝑛_𝑠𝑑𝑠⟶^𝑡_𝑡0𝑏𝑠,𝑋_𝑠𝑑𝑠,_𝑡𝑡0ℎ𝑠,𝑋^𝑛_𝑠𝑑⟨𝐵⟩(𝑠)⟶^𝑡_𝑡0ℎ𝑠,𝑋_𝑠𝑑⟨𝐵⟩(_𝑡𝑡0𝑠),𝜎𝑠,𝑋^𝑛_𝑠𝑑𝐵(𝑠)⟶^𝑡_𝑡0𝜎𝑠,𝑋_𝑠𝑑𝐵(𝑠) (60) in 𝕃²_𝐺, as 𝑛→∞, which deduces 𝑋(𝑡)=𝜉_𝑡0(0)+^𝑡_𝑡0𝑏𝑠,𝑋_𝑠𝑑𝑠+^𝑡_𝑡0ℎ𝑠,𝑋_𝑠(𝑠)+𝑑⟨𝐵⟩_𝑡𝑡0𝜎𝑠,𝑋_𝑠𝑑𝐵(𝑠). (61) Thanks to Lemmas 13, 14, and 5 and assumptions (1a), (1b), and (1c), we can also get 𝑋(𝑡)∈𝕃²_𝐺 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)=𝜉_𝑡0(0). Then, we have 𝔼sup_{𝑡0≤𝑣≤𝑡}|𝑋(𝑣)−𝑌(𝑣)|²=𝔼sup_{𝑡0≤𝑣≤𝑡}^𝑣_𝑡0𝑏𝑠,𝑋_𝑠−𝑏𝑠,𝑌_𝑠+𝑑𝑠_𝑣𝑡0ℎ𝑠,𝑋_𝑠−ℎ𝑠,𝑌_𝑠(𝑠)+𝑑⟨𝐵⟩_𝑣𝑡0(𝜎(𝑠,𝑋_𝑠)−𝜎(𝑠,𝑌_𝑠))𝑑𝐵(𝑠)²≤3𝑘₁𝑇+3𝑘₂𝑇+3𝑘₃^𝑡_𝑡0Γ𝑠,𝔼𝑋_𝑠−𝑌_𝑠²≤𝑑𝑠3𝑘₁𝑇+3𝑘₂𝑇+3𝑘₃×^𝑡_𝑡0Γ𝑠,𝔼sup_{𝑡0≤𝑣≤𝑠}|𝑋(𝑣)−𝑌(𝑣)|²𝑑𝑠, (62) 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: 𝑑𝑆(𝑡)=𝑏𝑆_𝑡𝑆(𝑡)𝑑𝑡+𝜎𝑆_𝑡𝑆(𝑡)(𝑡)𝑆𝑑𝐵+ℎ_𝑡𝑆(𝑡)𝑑⟨𝐵𝑡),𝑆⟩(₀=𝜉={𝜉(𝜃);−𝜏≤𝜃≤0}, (63) where 𝑆_𝑡={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 𝑏,𝜎,ℎ∶𝐶([−𝜏,0];ℝ)⟶ℝ (64) satisfy the following conditions: (1)there exists a 𝜀₁>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 a 𝑀>0, such that |𝑏(𝑥)|∨|𝜎(𝑥)|∨|ℎ(𝑥)|≤𝑀.

Corollary 17. Under the above assumption, the above SDDE brings a unique solution and 𝑆(𝑡)=𝜉(0)exp^𝑡₀𝑏𝑆_𝑣𝑑𝑣+^𝑡₀𝜎𝑆_𝑣(𝑣)+𝑑𝐵_𝑡0ℎ𝑆_𝑣²−12𝜎𝑆_𝑣²𝑑⟨𝐵⟩(𝑣). (65) Furthermore, if 𝜉(0)≥0(𝜉(0)>0) q.s., then 𝑆(𝑡)≥0(𝑆(𝑡)>0) for all 𝑡≥0 q.s.

Proof. Let 𝜃₀∈[−𝜏,0]. Consider the case 𝑡∈[0,−𝜃₀], and in this situation, we have 𝜃₀≤𝑡+𝜃₀≤0 and 𝑆_𝑡=𝑆𝑡+𝜃₀=𝜉𝑡+𝜃₀, (66) which deduces 𝑑𝑆(𝑡)=𝑏𝜉𝑡+𝜃₀𝑆(𝑡)𝑑𝑡+𝜎𝜉𝑡+𝜃₀𝑆(𝑡)(𝑡)𝜉𝑑𝐵+ℎ𝑡+𝜃₀𝑆(𝑡)𝑑⟨𝐵𝑡),𝑆(0)(0).⟩(=𝜉 (67) It follows from Theorem 16 and Itô’s formula that the unique solution of (67) 𝑆(𝑡)=𝜉(0)exp^𝑡₀𝑏𝜉𝑣+𝜃₀𝑑𝑣+^𝑡₀𝜎𝜉𝑣+𝜃₀(𝑣)+𝑑𝐵_𝑡0ℎ𝜉𝑣+𝜃₀²−12𝜎𝜉𝑣+𝜃₀²×𝑑⟨𝐵⟩(𝑣) (68) 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: 𝑆(𝑡)=𝜉(0)exp^𝑡₀𝑏𝑆𝑣+𝜃₀𝑑𝑣+^𝑡₀𝜎𝑆𝑣+𝜃₀(𝑣)+𝑑𝐵_𝑡0ℎ𝑆𝑣+𝜃₀²−12𝜎𝑆𝑣+𝜃₀²×𝑑⟨𝐵⟩(𝑣) (69) for 𝑡∈[0,𝑇]. Since that equality is true for all −𝜏≤𝜃₀≤0, we get our result.

We denote the wealth process by 𝑉(𝑡) and units of the stock that we hold by 𝛼(𝑡). Thanks to the assumption of self-financing, we can get (𝑡)(𝑉(𝑡)(𝑡)𝑆(𝑡(𝑡)(𝑡)(𝑉(𝑡)(𝑡)𝑆(𝑡(𝑡𝑏𝑆𝑑𝑉=𝑟−𝛼))𝑑𝑡+𝛼𝑑𝑆=𝑟−𝛼))𝑑𝑡+𝛼)_𝑡𝑆(𝑡)𝑑𝑡+𝜎𝑆_𝑡𝑆(𝑡)(𝑡)𝑆𝑑𝐵+ℎ_𝑡𝑆(𝑡)𝑑⟨𝐵⟩(𝑡) (70) for all 𝑡∈[0,𝑇]. Let 𝐻()=𝑆𝑡,𝜔𝑟−𝑏_𝑡/𝜎²₀−ℎ𝑆_𝑡𝜎𝑆_𝑡. (71) Then there exists a 𝜀₀>0 such that 1𝔼exp2+𝜀₀^𝑇₀𝐻²(𝑡,𝜔)𝑑⟨𝐵⟩(𝑡)<∞, (72) which shows that 𝐵(𝑡)=𝐵(𝑡)−^𝑡₀𝐻(𝑠,𝜔)𝑑⟨𝐵⟩(𝑠) (73) is 𝐺-Brownian motion under some 𝐺-expectation 𝔼 by 𝐺-Girsanov theorem. Let now Γ(𝑡)=𝑒^−𝑟𝑡; then (𝑡)(𝑡)(𝑡)𝑉(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)𝑉(𝑡)(𝑡)(𝑡)(𝑡)Γ(𝑡)𝑆(𝑡𝑏𝑆𝑑Γ=−𝑟Γ𝑑𝑡,𝑑Γ=Γ𝑑𝑉+𝑉𝑑Γ+𝑑Γ𝑑𝑉=−𝑟Γ𝑑𝑡+Γ𝑑𝑉=𝛼)_𝑡𝑆−𝑟𝑑𝑡+𝜎_𝑡(𝑡)𝑆𝑑𝐵+ℎ_𝑡𝑑⟨𝐵𝑡(𝑡)Γ(𝑡)𝑆(𝑡𝑏𝑆⟩()=𝛼)_𝑡𝑆−𝑟𝑑𝑡+𝜎_𝑡𝑑𝐵(𝑡)+𝑟−𝑏𝑆_𝑡𝜎²₀𝑑⟨𝐵𝑡(𝑡)Γ(𝑡)𝐷(𝑡)𝑆(𝑡𝑑𝜎⟩()=𝛼)²₀𝑡⟨𝐵𝑡(𝑡)Γ(𝑡)𝑆(𝑡)𝜎𝑆−𝑑⟩()+𝛼_𝑡𝑑𝐵(𝑡), (74) where 𝐷(𝑡)=(𝑏(𝑆_𝑡)−𝑟)/𝜎²₀; 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 𝐶(0)≤𝔼[𝑉(𝑇)Γ(𝑇)]. (75)

Acknowledgments

This work is supported by the Project sponsored by NSFC (no. 11171062), Innovation Program of Shanghai Municipal Education Commission (no. 12ZZ063), and the Project sponsored by CUSF-DH-D-2013038.