مجلة الكوت للعلوم االقتصادية واالدارية تصدرعن كلية اإلدارة واالقتصاد/جامعة واسط العدد) 23 ( 2016

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1 اخلالصة المعادالث التفاضليت العشىائيت هي حقل مهمت في مجال االحتماالث وتطبيقاتها في السىىاث االخيزة, لذلك قام الباحث بذراست المعادالث التفاضليت العشىائيت المساق بعمليت Levy بذال مه عمليت Brownian باستخذام المحاكاة مه خالل اوىاع مه عمليت Levy لبيان تأثيزها احصائيا مه خالل بعض المقاييس االحصائيت وكذلك مسار المعادلت التفاضليت العشىائيت. الللمات االفتتاحية دراسة احملاكاة للمعادالت التفاضلية العشوائية املشاق بعملية Levy A Simulation Study on Stochastic Differential Equation Driven by Levy Process م.د.مهند فائز الشعدون قشم االحصاء /كلية االدارة واالقتصاد /جامعة القادسية عمليتLevy,معادالث تفاضليت عشىائيت, عمليت تبايه كاما, عمليت Normal-Inverse Gaussian,عمليتLevy Hyperbolic Abstract A Stochastic Differential Equation (SDE) is an important field in both Probability theory and its application in recent years. We will study the Stochastic Differential Equation (SDE) driven by Levy Processes using different types of this process. Levy processes implement in SDE instead of Brownian motion, which is a special case of it. The aim of this article is to show different types of Levy processes and affect on the path of SDE using some descriptive statistics and the graph of path for SDE in simulation framework. Keywords: Levy process, Stochastic Differential Equation, Variance-Gamma Process, Normal-Inverse Gaussian Process, Hyperbolic Levy Process. 1 Introduction Levy process is introduced by Paul Levy 1950 [1, 10 and 11]. Our motivate is to study the path of Stochastic Differential Equation (SDE) driven by Levy Process using (Compound Poisson, Variance Gamma, Normal Inverse Gaussian and Hyperbolic) Process, instead of Brownian motion to show some graphics and descriptive statistics tools. The key of Levy process is to deal as a small or a big jump on continuous time stochastic processes. The aim of this article is to study some types of Levy Processes and affect on the path of in SDE. In literature review, S.M.Iacus [8], [9] and F.C. Klebaner [11] show the simulation of Stochastic Differential Equations in discrete time and explain the most popular processes of these equations such as Ornstein-Uhlenbeck, Geometric Brownian and Vasicek Interest Rate Processes and some algorithms for them with application in financial aspect. Omer, Onalan [15] presents the Ornstein- Uhlenbeck driven by Levy Processes. The application of Levy

2 Process is in Finance, storage issue and Insurance as shown in [12] whom present some Monte Carlo methods for this process. In pure probability, [11] presents Stochastic Calculus to deal with Stochastic Differential Equation driven by Levy processes. Finally, the statistical methods for Stochastic Differential Equation will be presented by [10]. The paper is organized as follows: Section 2 we replace Brownian motion by Levy processes in Ornstein-Uhlenbeck (OU) and Geometric Brownian (GB) Processes as example of SDEs. In Section 3 shows some definitions, properties and types of Levy Processes. In Section 4 we apply these SDEs are driven by Levy Processes using simulation study. Finally, In Section 5 we include some conclusions. 2 Stochastic Differential Equation driven by Levy Process The compounded of stochastic Differential Equation is a deterministic term and a noise term that represents a stochastic Process. The general form of SDE is [7]: dxt = μdt + ζdwt, (1) where Xt is a stochastic process in time t; μ is a drift parameter; ζ is a volatility parameter; Wt is a Brownian motion. In this article, we will replace Brownian motion by Levy Process. Then, we will rewrite equation (1) as follows [2, 3, 4 and 14]: where Lt is a Levy Process in time t. dxt = μdt + ζdlt, (2) We will present two examples of SDE. These examples are Ornstein- Uhlenbeck (OU) and Geometric Brownian (GB) processes. Now we will study them as follows: 2.1 Ornstein-Uhlenbeck (OU) Process It is introduced by Leonard Ornstein and George Eugene Uhlenbeck (1930) [2] which is a popular process in financial mathematics. It can be written as SDE as follows [17]: dxt = (θ1 θ2xt)dt + θ3dlt, (3) where Xt is a stochastic process in time t. θ1, θ2 and θ3 are parameters. Using Itoˆ Lemma [3], the solution of this process is as:

3 X0 is an initial value of Xt. We will use this process in discrete time as shown in [6] and [15]. 2.2 Geometric Brownian (GB) Process It is introduced by Samulson (1965) [9]. This process deals with exponential formula for Brownian motion. It can be written as SDE: This formula is a popular process in financial mathematics in particular, stock price in European option [12]. Using Ito lemma [15] to solve equation (5), we can write it as follows: As we mentioned above, we will this process in discrete time as shown in [6] using Euler scheme. Now, we will present Levy process and implement in our processes. 3 Levy Processes The main idea of Levy processes is to work with a jump in continuous time stochastic process. The simplest example of Levy process is a compound Poisson process. In general, a Levy process is the natural generalization of both case Brownian motion and the Poisson process [12]. Let us start with some definitions and properties of Levy processes as shown in many books [1], [9] and [12]: Definition 3.1 A Levy Process L is a stochastic process that define on probability space (ω, F, P), if we have L is a Levy process then the properties L are: 1- The path of L are almost surely right continuous with left limits; 2- L has stationary increments, i.e., Lt Ls is equal in distribution to Lt s; 3- The increments are independent of the past: Lt Ls is independent of ζ(lu : u s) where 0 s t.

4 In pure probability, there are some definitions of Levy processes as follows: Definition 3.2 Levy filtration is letting L be a Levy process, then L x fulfills the used conditions of completeness and right continuity. Definition 3.3 Levy measure: if L is a Levy process that define a set v on R by setting for every Borel set Γ which does not contain zero. and v(0) = 0. hen v is a positive measure, called the Le vy measure. hen it has the properties: 1- v(γ) < is a Borel set, and bounded away from zero; 2- where v(γ) is the expected number of jumps per unit time whose sizes fall in the set Γ, shown in [4 and 16]. Definition 3.4 Levy- Khintchine formula: Let L be a Le vy process. hen, there exists a triple (γ, ζ 2, v) such that the cumulate function can be written for Z D as where γ and ζ 2 are constant, and v is the Le vy measure L(γ, ζ 2, v) is called the generating triple of the Le vy process L. either ζ 2 nor v depend on the choice of truncation function h, but γ does. Now, We will explain some popular types of Le vy rocesses as follows

5 3.1 Compound Poisson Process If Lt is a continuous time stochastic process which has a compound poisson process, then: Where Lt is a compound poisson process, N (t) is a poisson distribution with intensity parameter λ and zi is a probability density which is independent of N (t). The continuous time of Lt is distributed as exponential process. We will rewrite Equations (4) and (6) as follows: and, is very to compute the mean, variance, skewness and kurtosis from Equations (7) and (8). It easy 3.2 The Variance Gamma Process A Variance Gamma process (VG) is a stochastic process that is also defined as Laplace motion [12]. It considers as a pure jump in probability theory. The process L(t; ζ, v, θ) is a Variance Gamma with parameters ζ, v and θ. Therefore, we can rewrite Equation (4) and (6) as follows: And where, Lt is VG process, μ is the drift parameter and ζ is the volatility

6 parameter. To compute the mean, variance, skewness and kurtosis of these processes we can use Equations (9) and (10). 3.3 Normal Inverse Gaussian Process N I (t) is stochastic process that has Normal Inverse Gaussian NIG. The density function of NIG can be written as: where α, β and δ are the parameters of IG process, and k1 is the modified Bessel function third kind [5]. Our Equations (3) and (6) will be written as follows: and where E(exp(ζ I(t))) represents the martingale measure [10] and [13]. We can compute the mean, variance, skewness and kurtosis using Equations (11) and (12). 3.4 The Hyperbolic Levy Process Now, the Levy process represents as Hyperbolic distribution [12]. Then, the density function of Hyperbolic, when t = 1, is: where ε and δ are real constants and k1(ε) is the modified Bessel function of the third kind and the formula is [9] :

7 Therefore, we can rewrite Equation (4) and (6) as follows: To compute the mean, variance, skewness and kurtosis, we can use Equations (13) and (14). 4 Simulation Study We use the simulation study to show our processes that are driven by Levy Process such as Compound Poisson, Variance Gamma, Normal Inverse Gaussian and Hyperbolic Process. We use R language, which is free software, dealing with our processes. We set up some parameters for our processes. The number observation of simulation study is = he parameters of OU rocess are θ1 = 0,θ2 = 5 and θ3 = 3.5. he parameters of GB rocess are μ = 0.3 and ζ = 0.7. he parameter of Compound oisson rocess is λ = 2. he parameters of Variance Gamma process are ε = 2 and δ = 2. he parameters of ormal Inverse Gaussian are α = 1, β = 0.5 and δ = 1. he parameters of Hyperbolic rocess are ε = 1 and δ = 0.5. We will present the results of our processes are follows 4.1 Compound Poisson Process Table 1 shows some important descriptive statistics, i.e., (Minimum and Maximum) values, 1 st and 3 rd quartile, Median, Mean, Skewness, Kurtosis and Number of Jump in our processes. Interestingly, the skewness, we have known, is the measure of symmetry. In Table 1 we can say that our processes are approximately symmetry and left side. Kurtosis, as we have known, is measure of data that is the heavy or light tail. In Table 1 we can say that our processes are heavy-tail distribution. Figure 1 represents the path of OU and GB Process driven by Compound Poisson Process using Equations (7) and (8), respectively.

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9 4.2 Variance Gamma Process Table 2 shows some important descriptive statistics, i.e., (Minimum and Maximum) values, 1 st and 3 rd quartile, Median, Mean, Skewness, Kurtosis and Number of Jump in our processes. The value of skewness in Table 2 are approximately symmetry and right side in our processes. The value of Kurtosis in Table 2 is the heavy-tail distribution. Figure 2 represents the path of OU and GB Process driven by Variance Gamma Process using Equations (9) and (10), respectively.

10 4.3 Normal Inverse Gaussian Process Table 3 shows some important descriptive statistics, i.e., (Minimum and Maximum) values, 1 st and 3 rd quartile, Median, Mean, Skewness, Kurtosis and Number of Jump in our processes. The value of skewness in Table 3 are clearly not symmetry with left side for OU Process and approximately symmetry for GB Process with left side as well. The value of Kurtosis in Table 3 is the heavytail distribution. Figure 3 represents the path of OU and GB Process driven by Normal Inverse Gaussian Process using Equations (11) and (12), respectively.

11 4.4 Hyperbolic Process Table 4 shows some important descriptive statistics, i.e., (Minimum and Maximum) values, 1 st and 3 rd quartile, Median, Mean, Skewness, Kurtosis and Number of Jump in our processes. The value of skewness in Table 4 are approximately symmetry with left side for our processes. The value of Kurtosis in Table 4 is the heavy-tail distribution. Figure 4 represents the path of OU and GB Process driven Hyperbolic Process using Equations (13) and (14), respectively.

12 5 Conclusion We have presented some popular SDEs such as Ornstein-Ulhecbeck and Geometric Brownian equation driven by Levy Process. We have used Compound Poisson, Variance Gamma, Normal Inverse Gaussian and Hyperbolic to present the Levy Process. In general, our processes are almost surely symmetry and heavy tail distribution.

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