Abstract and Applied Analysis

Volume 2013, Article ID 637106, 9 pages

http://dx.doi.org/10.1155/2013/637106

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

^{1}Glorious Sun School of Business and Management, Donghua University, 1882 West Yan-an Rood, Shanghai 200051, China^{2}Department of Mathematics, Donghua University, 2999 North Renmin Rood, Songjiang, Shanghai 201620, China

Received 11 November 2013; Accepted 5 December 2013

Academic Editor: Yaozhong Hu

Copyright © 2013 Litan Yan and Qinghua Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider the stochastic functional differential equations with finite delay driven by -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 -expectation (see also Peng et al. [2–4]), which is associated with the following nonlinear heat equation:
where is Laplacian and the sublinear function is defined as
with two given constants . Together with the notion of -expectations Peng also introduced the related -normal distribution, the -Brownian motion, and related stochastic calculus *under *-expectation, and moreover an Itô’s formula for the -Brownian motion was established. -Brownian motion has a very rich and interesting new structure which nontrivially generalizes the classical one. Briefly speaking, a -Brownian motion is a continuous process with independent stationary increments being -normally distributed under a given sublinear expectation . A very interesting new phenomenon of -Brownian motion is that its quadratic process 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 -Brownian motion. We consider call options that can be exercised only at the maturity date, namely, European call options. Let be the -Brownian motion with the quadratic variation process . We consider SFDE with finite delay of the form where (i) and is a -valued stochastic process; (ii) , , and are three given functions satisfying some satiable conditions; (iii) and 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 -Brownian motion . 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 -Brownian motion.

#### 2. Preliminaries

In this section, we briefly recall some basic notations and results for -Brownian motion under -framework. For more aspects on these material we refer to Peng [4], Denis et al. [8], and Hu and Peng [9]. More works for -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 be a given set and let be a linear space of real-valued functions defined on such that and for all . It is important to note that we can suppose that if , , for all , where denotes the space of all bounded and Lipschitz functions on .

*Definition 1. *A sublinear expectation on is a functional with the following properties: for all , , we have (i)monotonicity: if , then ;(ii)constant preserving: , for all ;(iii)subadditivity: ;(iv)positive homogeneity: , for all . The triple is called a sublinear expectation space, and is considered as the space of random variables on .

In this paper we throughout let be the space of all real-valued continuous functions on with initial value , equipped with the distance We denote by the Borel-algebra on . We also denote, for each , and , where . Let be the closure of with respect to the norm with . Clearly, the space is a Banach space and the space of bounded continuous functions on is a subset of , and moreover, there exists a weakly compact family of probability measures on such that So we can introduce the Choquet capacity by

*Definition 2. *A set is called polar if . A property is said to hold “quasi sure” (q.s.) if it holds outside a polar set.

The above family of probability measures allows characterizing the space as follows:

Lemma 3 (Denis et al. [8]). *Let . Consider the sets and , where
**
Then *(i)* is a Banach space with respect to the norm ;*(ii)* is the completion of with respect to the norm .*

Lemma 4 (Li and Peng [17]). *For a given , If the sequence converges to in , then there exists a subsequence such that converges to quasi surely.*

Lemma 5 (see [18]). *Let be an increasing and concave function. Then the inequality
**
holds for all -measurable real-valued functions on .*

For , the functional defined by is called the distribution of under . In a sublinear expectation space , a random vector , , is said to be independent under from another random vector , , if for each test function we have

##### 2.2. -Brownian Motion

Let , be two real numbers with .

A random variable in a sublinear expectation space is called -normal distributed, denoted by , if, for each , the function defined by is the unique viscosity solution of the following nonlinear heat equation: where is Laplacian and the sublinear function is defined as

In particular, is the distribution of .

*Example 6 (Peng [1]). *Let . We then have
for all convex functions and
for all concave functions .

*Definition 7. *A process in a sublinear expectation space is called a -Brownian motion if the following properties are satisfied: (i);(ii)for each , the increment is -distributed and is independent of , for all and .

Recall that a process is called a -martingale if, for each , we have and for all , and moreover, is called a symmetric -martingale if both and are -martingales.

*Remark 8. *For simplicity throughout this paper we let , and .

##### 2.3. Itô’s Integral

We now recall the definition of Itô’s integral and quadratic variation process of the -Brownian motion. In Li and Peng [17], a generalized Itô integral and a generalized Itô formula with respect to the -Brownian motion are discussed as follows. For arbitrarily fixed and , we first denote by the set of step processes with . Moreover, we denote by the completion of under the norm Now we can define the stochastic integral for .

*Definition 9. *For every with the form (21), we define Itô’s integral

One can show that and the linear mapping (see Li and Peng [17]) is continuous. Moreover, it can be continuously extended to And, for all , we have We now recall the definition of quadratic variation process of the -Brownian motion .

*Definition 10 (quadratic variation). *Let be a partition of for , such that as . The quadratic variation of -Brownian motion can be defined as
in .

The function is continuous and increasing outside a polar set. We can define the integral as a map from into , and the map is linear and continuous, so it can be extended continuously to .

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 . One then has
**
for all .*

Lemma 12 (see [14]). *If there exists an such that
**
then the process
**
is a -Brownian motion under some -expectation.*

Lemma 13 (see [19]). *Let and . Then
**
for all , where is a positive constant depending only on .*

Lemma 14 (see [19]). *Let and . Then we have
**
for all .*

#### 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 -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 and denote by the family of continuous functions from to with the norm Given let be three Borel measurable functions satisfying for any . Consider the following SFDE with finite delay of the form where and is regarded as a -valued stochastic process. Then we can impose the initial data: (i) is an -measurable, -valued random variable such that . Thus we can get the following equivalent form: with . In order to give the existence and uniqueness we now propose the following assumptions:(H1)There exists a function , such that(1a)(1b) is integrable in for each fixed , and is continuous, concave, nondecreasing in for each fixed ,(1c)for any constant the deterministic ODE has a global solution for any initial value.(H2)There exists a function such that(2a) for all and ;(2b) is integrable in for each fixed and is continuous, concave, nondecreasing in for each fixed . Moreover, , and if there exists a nonnegative continuous function , satisfies where is a positive constant, then for all .

*Example 15 (see [25]). *Let , , , where is integrable and is a continuous, concave, monotone nondecreasing function with such that . Let be a solution of . If , then we get for all . If not, we can suppose without loss of generality that there exists such that is positive for all . Let . Thus,
which implies a contradiction letting . Therefore, for all .

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 with the above initial data and .*

*Proof. *Define , for , and let for each . Consider the following Picard iterations:
for . Then we get
*Step 1.* We claim that
for all and all , where is a positive constant, which points out that there exists a real number depending only on and such that
for all and all , since is continuous on .

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 , , and independent of such that
for all . Notice that has a global solution with any initial value . We can take , such that , and may assume that is the solution of the above ODE with the initial value , which implies that

Denote that . Then continuity of on deduces and
for each . Therefore, there exists a positive constant such that
We now suppose that the inequalities
hold for all and . Then we have
Since , we obtain that
then for all , and
for all . Thus, we obtain our assertion by induction.*Step 2.* We claim that is a Cauchy sequence in . Thus, in as . From Lemmas 5, 13, and 14 and assumptions (2a) and (2b), we get, for all ,
Let ; then we get
which implies that
It follows that for all . That is,
This shows that the sequence is a Cauchy sequence in , which deduces in as . Moreover, it is uniformly convergent on , and therefore, is continuous.*Step 3.* We show that the stochastic process given above is a solution of (37). We have
for all and . Noting that the sequence uniformly converges on , we get
in , as , which deduces
Thanks to Lemmas 13, 14, and 5 and assumptions (1a), (1b), and (1c), we can also get for all .*Step 4.* Finally, we will show that the uniqueness of the solution. Let and be two solutions existing on and . Then, we have
which deduces for all 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: where is regarded as a -valued stochastic process. is the quadratic variation process of -Brownian motion .

In the rest of this section, we want to consider a special case of the above equation. We let the functions satisfy the following conditions: (1)there exists a such that ;(2), where denotes the rate of return on a riskless asset, which is compounded continuously;(3)there exists a , such that .

Corollary 17. *Under the above assumption, the above SDDE brings a unique solution and
**
Furthermore, if q.s., then for all q.s.*

*Proof. *Let . Consider the case , and in this situation, we have and
which deduces
It follows from Theorem 16 and Itô’s formula that the unique solution of (67)
for all . This implies that for all , when q.s. By a similar argument, it follows that for all q.s. Therefore for all q.s. By induction, we can also get the solution to (67) as follows:
for . Since that equality is true for all , we get our result.

We denote the wealth process by and units of the stock that we hold by . Thanks to the assumption of self-financing, we can get for all . Let Then there exists a such that which shows that is -Brownian motion under some -expectation by -Girsanov theorem. Let now ; then where ; we can get that the discounted wealth process is -martingale under some -expectation. Thus, we can obtain that the price of European call option is

#### 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.

#### References

- S. Peng, “
*G*-expectation,*G*-Brownian motion and related stochastic calculus of Itô type,” in*Stochastic Analysis and Applications*, vol. 2 of*Abel Symposia*, pp. 541–567, Springer, Berlin, Germany, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - S. Peng, “Multi-dimensional
*G*-Brownian motion and related stochastic calculus under*G*-expectation,”*Stochastic Processes and Their Applications*, vol. 118, no. 12, pp. 2223–2253, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - S. Peng, “
*G*-Brownian motion and dynamic risk measure under volatility uncertainty,” http://arxiv.org/abs/0711.2834. - S. Peng, “Nonlinear expections and stochastic calculus under uncertainty,” http://arxiv.org/abs/1002.4546.
- F. Black and M. Scholes, “The pricing of options and corporate liabilities,”
*Journal of Political Economy*, vol. 81, pp. 637–659, 1973. View at Google Scholar - R. C. Merton, “Theory of rational option pricing,”
*The Bell Journal of Economics and Management Science*, vol. 4, no. 1, pp. 141–183, 1973. View at Google Scholar · View at MathSciNet - M. Arriojas, Y. Hu, S.-E. Mohammed, and G. Pap, “A delayed black and scholes formula,”
*Stochastic Analysis and Applications*, vol. 25, no. 2, pp. 471–492, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - L. Denis, M. Hu, and S. Peng, “Function spaces and capacity related to a sublinear expectation: application to
*G*-Brownian motion paths,”*Potential Analysis*, vol. 34, no. 2, pp. 139–161, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - M.-S. Hu and S.-G. Peng, “On representation theorem of
*G*-expectations and paths of*G*-Brownian motion,”*Acta Mathematica Sinica*, vol. 25, no. 3, pp. 539–546, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - M. Hu and X. Li, “Independence under the
*G*-expectation framework,”*Journal of Theoretical Probability*, 2012. View at Publisher · View at Google Scholar - Q. Lin, “Local time and tanaka formula for the
*G*-Brownian motion,”*Journal of Mathematical Analysis and Applications*, vol. 398, no. 1, pp. 315–334, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - S. Peng, Y. Song, and J. Zhang, “A complete representation theorem for
*G*-martingales,” http://arxiv.org/abs/1201.2629. - Y. Song, “Some properties on
*G*-evaluation and its applications to*G*-martingale decomposition,”*Science China Mathematics*, vol. 54, no. 2, pp. 287–300, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - J. Xu, H. Shang, and B. Zhang, “A Girsanov type theorem under
*G*-framework,”*Stochastic Analysis and Applications*, vol. 29, no. 3, pp. 386–406, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - L. Yan, B. Gao, and Q. Zhang, “The fractional derivatives for
*G*-Brownian local time”. - L. Yan, X. Sun, and B. Gao, “Integral with respect to the
*G*-Brownian local time,” http://arxiv.org/abs/1212.6353. - X. Li and S. Peng, “Stopping times and related Itô's calculus with
*G*-Brownian motion,”*Stochastic Processes and Their Applications*, vol. 121, no. 7, pp. 1492–1508, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - X. Bai and Y. Lin, “On the existence and uniqueness of solutions to stochastic differential equations driven by
*G*-Brownian motion with integral-lipschitz coefficients,” http://arxiv.org/abs/1002.1046v3. - F. Gao, “Pathwise properties and homeomorphic flows for stochastic differential equations driven by
*G*-Brownian motion,”*Stochastic Processes and Their Applications*, vol. 119, no. 10, pp. 3356–3382, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Chen and D. Zhang, “Exponential stability for stochastic differential equation driven by
*G*-Brownian motion,”*Applied Mathematics Letters*, vol. 25, no. 11, pp. 1906–1910, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Q. Lin, “Properties of solutions of stochastic differential equations driven by the
*G*-Brownian motion,” http://arxiv.org/abs/1010.3158. - Y. Lin, “Stochastic differential equations driven by
*G*-Brownian motion with reflecting boundary conditions,”*Electronic Journal of Probability*, vol. 18, article 9, pp. 1–23, 2013. View at Publisher · View at Google Scholar - Y. Ren, Q. Bi, and R. Sakthivel, “Stochastic functional differential equations with infinite delay driven by
*G*-Brownian motion,”*Mathematical Methods in the Applied Sciences*, vol. 36, no. 13, pp. 1746–1759, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Ren and L. Hu, “A note on the stochastic differential equations driven by
*G*-Brownian motion,”*Statistics & Probability Letters*, vol. 81, no. 5, pp. 580–585, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - T. Taniguchi, “Successive approximations to solutions of stochastic differential equations,”
*Journal of Differential Equations*, vol. 96, no. 1, pp. 152–169, 1992. View at Publisher · View at Google Scholar · View at MathSciNet