A numerical method for solving uncertain differential equations

Transcription

1 Journal of Intelligent & Fuzzy Systems 25 ( DOI:1.3233/IFS IOS Press 825 A numerical method for solving uncertain differential equations Kai Yao a and Xiaowei Chen b, a Department of Mathematical Sciences, Tsinghua University, Beijing, China b Department of Risk Management and Insurance, Nankai University, Tianjin, China Abstract. Uncertain differential equation is a type of differential equation driven by canonical process. In this paper, a concept of α-path to uncertain differential equation is first introduced, which is a type of deterministic function that solves an associate ordinary differential equation. Then, a numerical method is designed for solving uncertain differential equations, which essentially solves each α-path and produces an inverse uncertainty distribution of the solution. To illustrate the efficiency of the numerical method, several examples are given. Keywords: Uncertain differential equation, numerical solution, uncertainty theory 1. Introduction Wiener process, defined mathematically by Norbert Wiener in 1931, is a continuous stochastic process with stationary and independent increments that are normal uncertain variables. Based on Wiener process, Kiyosi Ito founded stochastic calculus in mid-twentieth century. After that, stochastic differential equation, a type of differential equation driven by Wiener process, was studied and applied widely. In order to solve stochastic differential equations numerically, the usual strategy is first to select a recursion formula of solution. From this recursion formula an attempt is made to construct a spectrum of sample paths. Then the probability distribution is obtained from these sample paths via some methods of statistics. The numerical methods based upon this strategy can be classified as Euler-Maruyama approximation (Maruyama [12], Milstein approximation (Milstein [13] and Runge-Kutta approximation Corresponding author. Xiaowei Chen, Department of Risk Management and Insurance, Nankai University, Tianjin 371, China. Tel./Fax: ; chenx@nankai.edu.cn. (Rumelin [15] according to the types of recursion formula. Euler-Maruyama approximation uses first-order Taylor expansion as the recursion formula, Milstein approximation uses second-order Taylor expansion as the recursion formula, and Runge-Kutta approximation uses Runge-Kutta expansion as the recursion formula. Although stochastic process, stochastic calculus and stochastic differential equation find many applications in daily life, the evolution of some undetermined phenomena behaves not like randomness but like uncertainty, which was first pointed out by Liu [5]. As an uncertain counterpart of Wiener process, Liu [7] designed canonical process, which is a Lipschitz continuous uncertain process with stationary and independent increments that are normal uncertain variables rather than normal random variables. Uncertain calculus, proposed by Liu [7] in 29, is a branch of mathematics that deals with differentiation and integration of functions of uncertain processes. In 212, Yao [17] proposed uncertain calculus with respect to renewal process. Uncertain differential equation, proposed by Liu [6] in 28, is a type of differential equation driven by canonical process. It is an important tool to deal with /13/$ IOS Press and the authors. All rights reserved

2 826 K. Yao and X. Chen / A numerical method for solving uncertain differential equations dynamic systems in uncertain environment. Uncertain differential equation was first introduced to finance by Liu [7] to model stock price in uncertain market. After that, Peng and Yao [14] proposed another stock model to describe the stock price in long-run, and Liu and Chen [11] proposed a currency model in uncertain environment. Besides, Zhu [21] applied uncertain differential equation to optimal control, and obtained a necessary condition for the uncertain optimal control model. The existence and uniqueness of solution of uncertain differential equations were proved by Chen and Liu [1], and stability analysis of uncertain differential equations was given by Yao et al. [18]. Many researchers managed to find analytic solutions for some special types of uncertain differential equations such as Chen and Liu [1], and Liu [1]. However, it is difficult to find analytic solutions of uncertain differential equations in many cases. Thus we have to design some numerical methods to solve uncertain differential equations. This paper aims at giving a numerical method to solve uncertain differential equations. The remainder of this paper is organized as follows. Some basic concepts and properties are reviewed in Section 2, regarding uncertain process, uncertain integral and uncertain differential equation. Section 3 presents a concept of α-path, and Section 4 designs a 99-method to solve uncertain differential equations. Then the 99-method is demonstrated by some numerical experiments in Section 5. After that, the 99-method is applied to pricing European option, and the numerical result is compared with the analytic result to illustrate the efficiency of 99-method in Section Preliminary Definition 2.1. (Liu [5]Let L be a σ-algebra on a nonempty set Ɣ. A set function M : L [, 1] is called an uncertain measure if it satisfies the following axioms: Axiom 1: (Normality Axiom M{Ɣ =1 for the universal set Ɣ. Axiom 2: (Duality Axiom M{ +M{ c =1 for any event. Axiom 3: (Subadditivity Axiom For every countable sequence of events 1, 2,, we have { M i M { i. i=1 i=1 Besides, the product uncertain measure on the product σ-algebra L was defined by Liu [7] as follows, Axiom 4: (Product Axiom Let (Ɣ k, L k, M k be uncertainty spaces for k = 1, 2, Then the product uncertain measure M is an uncertain measure satisfying { M k = M k { k i=1 k=1 where k are arbitrarily chosen events from L k for k = 1, 2,, respectively. The triplet (Ɣ, L, M is called an uncertainty space. Based on the axioms of uncertain measure, an uncertainty theory was founded by Liu [5] in 27 and refined by Liu [8] in 21. Many researchers have done some theoretical work such as [2 4, 9, 16, 19, 2]. Definition 2.2. (Liu [5] An uncertain variable is a function from an uncertainty space (Ɣ, L, M to the set of real numbers, i.e., for any Borel set B of real numbers, the set is an event. {ξ B ={γ Ɣ ξ(γ B Definition 2.3. (Liu [5] The uncertainty distribution of an uncertain variable ξ is defined by for any real number x. (x = M{ξ x Theorem 2.1. (Liu [8] Let ξ be an uncertain variable with an uncertainty distribution. Iff is a strictly increasing function, then η = f (ξ is an uncertain variable 1 (α = f ( 1 (α. If f is a strictly decreasing function, then η = f (ξ is an uncertain variable with an inverse uncertainty distribution 1 (α = f ( 1 (1 α. Definition 2.4. (Liu [5] The expected value of an uncertain variable ξ is defined by E[ξ] = + M{ξ xdx M{ξ xdx provided that at least one of the two integrals exists.

3 K. Yao and X. Chen / A numerical method for solving uncertain differential equations 827 In order to calculate the expected value via inverse uncertainty distribution, Liu [8] gave the following theorem. Theorem 2.2. (Liu [8] Let ξ be an uncertain variable with an uncertainty distribution. If the expected value exists, then E[ξ] = 1 (αdα. An uncertain process is essentially a sequence of uncertain variables indexed by time. The study of uncertain process was started by Liu [6] in 28. Definition 2.5. (Liu [6] Let T be an index set and let (Ɣ, L, M be an uncertainty space. An uncertain process is a measurable function from T (Ɣ, L, M to the set of real numbers, i.e., for each t T and any Borel set B of real numbers, the set is an event. {X t B ={γ Ɣ X t (γ B Definition 2.6. (Liu [7] An uncertain process C t (t is said to be a canonical process if (i C = and almost all sample paths are Lipschitz continuous, (ii C t has stationary and independent increments, (iii every increment C s + t C s is a normal uncertain variable with expected value and variance t 2, whose uncertainty distribution is ( ( πx 1 (x = 1 + exp,x R. 3t Based on canonical process, uncertain integral and uncertain differential were then defined by Liu [7], thus offering a theory of uncertain calculus. Definition 2.7. (Liu [7] Let X t be an uncertain process and C t be a canonical process. For any partition of closed interval [a, b] with a = t 1 <t 2 < <t k+1 = b, the mesh is written as = max t i+1 t i. 1 i k Then Liu integral of X t is defined by b a X t dc t = lim k X ti (C ti+1 C ti i=1 provided that the limit exists almost surely and is finite. Definition 2.8. (Liu [7] Let C t be a canonical process and Z t be an uncertain process. If there exist uncertain processes µ s and σ s such that Z t = Z + t µ s ds + t σ s dc s for any t, then Z t is said to be differentiable and have an uncertain differential dz t = µ t dt + σ t dc t. Liu [7] verified the fundamental theorem of uncertain calculus, i.e., for a canonical process C t and a continuous differentiable function h(t, c, the uncertain process Z t = h(t, C t has an uncertain differential dz t = h t (t, C tdt + h c (t, C tdc t. Based on the fundamental theorem, Liu proved the integration by parts theorem, i.e., for two differentiable uncertain process X t and Y t, the uncertain process X t Y t has an uncertain differential d(x t Y t = Y t dx t + X t dy t. Uncertain calculus provides a theoretical foundation for constructing uncertain differential equations. Instead of using Wiener process, a differential equation driven by canonical process is defined as follows. Definition 2.9. (Liu [6] Suppose C t is a canonical process, and f and g are some given functions. Given an initial value X, then dx t = f (t, X t dt + g(t, X t dc t (1 is called an uncertain differential equation with an initial value X. A solution is an uncertain process X t that satisfies Equation (1 identically in t. Theorem 2.3. (Existence and Uniqueness Theorem, Chen and Liu [1] The uncertain differential equation (1 has a unique solution if the coefficients f (x, t and g(x, t satisfy the Lipschitz condition f (x, t f (y, t + g(x, t g(y, t L x y for all x, y R,t and linear growth condition f (x, t + g(x, t L(1 + x for all x R,t for some constants L. Moreover, the solution is sample-continuous.

4 828 K. Yao and X. Chen / A numerical method for solving uncertain differential equations It has been proved by Chen and Liu [1] that the linear uncertain differential equation dx t = (u 1t X t + u 2t dt + (v 1t X t + v 2t dc t t u t 2s v 2s X t = U t (X + ds + dc s U s U s where ( t t U t = exp u 1r dr + v 1r dc r. Liu [1] gave a analytic method to solve uncertain differential equations. We will introduce it as follows. Theorem 2.4. (Liu [1] Let f be a function of two variables and let σ t be an integrable uncertain process. Then the uncertain differential equation dx t = f (t, X t dt + σ t X t dc t X t = Yt 1 Z t where ( t Y t = exp σ s dc s and Z t is the solution of uncertain differential equation dz t = Y t f (t, Yt 1 Z t dt with initial value Z = X. 3. Concept of α-path For solving an uncertain differential equation dx t = f (t, X t dt + g(t, X t dc t, a key point is to obtain a 99- table of its solution X s. In order to do so, we first introduce a concept of α-path, then we will discuss the relationship between the 99-table and α-path in next section. Definition 3.1. The α-path ( <α<1 of an uncertain differential equation dx t = f (t, X t dt + g(t, X t dc t with initial value X is a deterministic function X α t with respect to t that solves the corresponding ordinary differential equation dx α t = f (t, X α t dt + g(t, Xα t 1 (αdt where 1 (α is the inverse uncertainty distribution of standard normal uncertain variable, i.e., 3 1 (α = π ln α 1 α, <α<1. Example 3.1. The uncertain differential equation has an α-path dx t = adt + bdc t, X = X α t = at + b 1 (αt. Example 3.2. The uncertain differential equation has an α-path dx t = ax t dt + bx t dc t, X = 1 X α t = exp ( at + b 1 (αt. Example 3.3. The uncertain differential equation dx t = atx t dt + btx t dc t, X = 1 has an α-path ( 1 Xt α = exp 2 at b 1 (αt 2. Example 3.4. The uncertain differential equation has an α-path dx t = ax t dt + b exp(atdc t, X = 1 X α t = ( 1 + b 1 (αt exp (at Method for solving uncertain differential equation Different from generating a large quantity of sample paths to get an approximate probability distribution of a stochastic differential equation, an uncertainty distribution of an uncertain differential equation can be obtained exactly by a spectrum of α-paths. In order to prove this, we first introduce two basic results in ordinary differential equations. Lemma 4.1. Assume that f (t, x and g(t, x are continuous functions. Let φ(t be a solution of the ordinary differential equation dx dt = f (t, x + K g(t, x, x( = x

5 K. Yao and X. Chen / A numerical method for solving uncertain differential equations 829 where K is a real number. Let ψ(t be a solution of the ordinary differential equation dx dt = f (t, x + k(tg(t, x, x( = x where k(t is a real function. i If k(tg(t, x K g(t, x for t [, T], then ψ(t φ(t. ii If k(tg(t, x > K g(t, x for t [, T], then ψ(t >φ(t. Theorem 4.1. Assume that f (t, x and g(t, x are continuous functions. Let Xt α be an α-path of the uncertain differential equation dx t = f (t, X t dt + g(t, X t dc t, t [,s]. Then M{X s Xs α=α, i.e., X s has an inverse uncertainty distribution s 1 (α = Xs α, <α<1. Proof. Write A = { t [,s] g ( t, Xt α, B = { t [,s] g ( t, Xt α <. Then it is obvious that A B = [,s]. Write { 1 = γ dc t (γ 1 (α for t [,u], dt { 2 = γ dc t (γ dt 1 (1 α for t (u, s] where 1 is the inverse uncertainty distribution of N(, 1. Since C t is an independent increment process, we get M{ 1 2 =α. For any γ 1 2, we have g(t, X t (γ dc t(γ g(t, Xt α dt 1 (α, t [,s]. It follows from Lemma 1 that X t (γ Xt α, so 1 2 {X t Xt α, and M{X s Xs α M{ 1 2 =α. (2 Write { 3 = γ dc t (γ dt { 4 = γ dc t (γ dt > 1 (α for t [,u], < 1 (1 α for t (u, s]. Then we get M{ 3 4 =1 α. For any γ 3 4, we have g(t, X t (γ dc t(γ > g(t, Xt α dt 1 (α, t [,s]. It follows from Lemma 1 that X t (γ >Xt α, so 3 4 {X t >Xt α, and M{X s Xs α =1 M{X s >Xs α 1 M{ 3 4 =α. (3 By Equations (2 and (3, we have M{X s X α s =α, which means that X s has an inverse uncertainty distribution s 1 (α = Xs α, <α<1. The theorem is verified. Example 4.1. The uncertain differential equation dx t = adt + bdc t, X = (4 X t = at + bc t t 1 (α = at + b 1 (αt, Equation (4. Example 4.2. The uncertain differential equation dx t = ax t dt + bx t dc t, X = 1 (5 X t = exp (at + bc t ( t 1 (α = exp at + b 1 (αt, equation (5. Example 4.3. The uncertain differential equation dx t = atx t dt + btx t dc t, X = 1 (6

6 83 K. Yao and X. Chen / A numerical method for solving uncertain differential equations ( t X t = exp 2 at2 + b sdc t t 1 (α = exp ( 1 2 at b 1 (αt 2, equation (6. Example 4.4. The uncertain differential equation dx t = ax t dt + b exp (atdc t, X = 1 (7 X t = exp(at(1 + bc t ( t 1 (α = 1 + b 1 (αt exp (at, equation (7. Example 4.5. The uncertain differential equation dx t = (m ax t dt + σdc t, X = (8 X t = m t a (1 exp ( at + σ exp (a(s tdc s ( m 1 (α = a + σ a 1 (α (1 exp ( at, Equation (8. Based on the previous theorem, a 99-method for solving an uncertain differential equation is designed as below. Step : Fix a time s and set α =. Step 1: Set α α +.1. Step 2: Solve the corresponding ordinary differential equation dx α t = f (t, X α t dt + g(t, Xα t 1 (αdt and obtain Xs α. It is suggested to employ a numerical method to solve the equation when an analytic solution is unavailable. Step 3: Repeat Step 1 and Step 2 for 99 times. Fig. 1. The 99-Table of dx t = (1 X t dt + X t dc t with X = 2 at t = 1. Step 4: The solution X s has a 99-table, Xs.1 Xs.2... Xs.99 This table gives an approximate uncertainty distribution of X s, i.e., for any α = i/1,i= 1, 2,...,99, we can find Xs α from the table such that M{X s Xs α=α. If α/= i/1,i= 1, 2,...,99, then it is suggested to employ a numerical interpolation method to get an approximate Xs α. The 99-method can be extended to 999-method if a more precise uncertainty distribution is needed. 5. Numerical experiments In this section, we illustrate the efficiency of 99- method by three uncertain differential equations which cannot be solved explicitly. Example 5.1. Consider the nonlinear uncertain differential equation dx t = (1 X t dt + X t dc t, X = 2. The 99-method obtains a 99-table of X t at time t = 1 shown in Fig. 1. Example 5.2. Consider the nonlinear uncertain differential equation dx t = X t 1dt + (1 tdc t, X = 2. (9 Note that 1 t when <t 1, and 1 t< when t>1. The α-path Xt α of (9 is a solution of the associated ordinary differential equation

7 K. Yao and X. Chen / A numerical method for solving uncertain differential equations 831 Fig. 2. The 99-Table of dx t = X t 1dt + (1 tdc t with X = 2 at t = 3/2. dx α t = X α t 1dt + 1 t 1 (αdt, X = 2. By 99-method, we obtain a 99-table of X t at time t = 3/2 shown in Fig. 2. Example 5.3. The uncertain differential equation dx t = rx t (K X t dt + βx t dc t can be used to model the size of a population in an uncertain, crowded environment. The constant K> is called the carrying capacity of the environment, the constant r R measures the quality of the environment, and the constant β R measures the uncertainty in the environment. Assume that r = 1,K = 1,β = 1 and X = 1. Then the 99-method obtains a 99-table of X t at time t = 1 shown in Fig Application In this section, we apply the 99-method to pricing European option, and the numerical result is compared with the analytic result to illustrate the efficiency of 99-method. First, we give the following theorem. Theorem 6.1. Let X t and X α t be the solution and α- path of the uncertain differential equation dx t = f (t, X t dt + g(t, X t dc t, respectively. Then for the monotone function J, we have E[J(X t ] = J(X α t. Fig. 3. The 99-Table of dx t = X t (1 X t dt + 1X t dc t with X = 1 at t = 1. Proof. At first, it follows from Theorem 4.1. that X t has an uncertainty distribution s 1 (α = Xs α. When J is a strictly increasing function, it follows from Theorem 2.1. that J(X t has an inverse uncertainty distribution Thus we have E[J(Xt] = 1 (α = J(X α t. 1 (αdα = J(X α t dα. When J is a strictly decreasing function, it follows from Theorem 2.1. that J(X t has an inverse uncertainty distribution 1 (α = J(Xt 1 α Thus we have E[J(Xt] = = 1 (αdα = J(X α t dα. The theorem is thus proved. J(Xt 1 α dα Example 6.4. Assume that the stock price follows geometric canonical process. Liu [7] presented a stock model in which the bond price X t and the stock price Y t are determined by { dxt = rx t dt dy t = ey t dt + σy t dc t (1

8 832 K. Yao and X. Chen / A numerical method for solving uncertain differential equations where r is the riskless interest rate, e is the stock drift and σ is the stock diffusion. European call option price f c for the stock model (1 is defined as f c = E[exp ( rt (Y T K + ] where K is the strike price at the exercise time T. Liu [7] gave the European call option pricing formula + ( ( π(ln y et 1 exp ( rt Y 1 + exp dy K/Y 3σT in 29. Assuming a stock is presently selling for an initial price Y = 3, the riskless interest rate is 8% per annum, the stock drift e is.6 and the stock diffusion σ is.25. Calculate an European call option that expires in three months and has a strike price of K = 29 by the European call option pricing formula, we have f c = This means the approximate call option price is 169 cents. The 99-method can solve the uncertain differential Equation (1 successfully, and obtain an inverse uncertainty distribution of Y T. With the inverse uncertainty distribution, we can calculate European call option price f c by Theorem 6.1., because f c is a increasing function of Y T. Calculate the European call option price by the 99-method, and the result shows that f c = Thus the error rate of the call option is ( / <.85%, which means the 99-method works well in calculating the European call option. 7. Conclusions Uncertain differential equation is an important tool to deal with dynamic systems in uncertain environment. In this paper, we proposed a numerical method to solve uncertain differential equation. A concept of α-path, which essentially is the solution of an ordinary differential equation, was first introduced. Then its solution was proved to be just the α-critical value of the solution of uncertain differential equation. Based on this, a 99- method for solving an uncertain differential equation was designed. After that, we illustrated the efficiency of 99-method by some numerical experiments. However, technical difficulty arises in solving the family of corresponding ordinary differential equations numerically, which brings about the error of 99-method. Our further research point is to design more efficient numerical method to solve a family of ordinary differential equations. Acknowledgments This work was supported by National Natural Science Foundation of China Grant No References [1] X. Chen and B. Liu, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making 9 (21, [2] X. Chen, S. Kar and D.A. Ralescu, Cross-entropy measure of uncertain variables, Information Sciences 21 (212, [3] X. Gao, Y. Gao and D.A. Ralescu, On Liu s inference rule for uncertain systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 18 (21, [4] X. Li and B. Liu, On distance between fuzzy variables, Journal of Intelligent & Fuzzy Systems 19 (28, [5] B. Liu, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 27. [6] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems 2 (28, [7] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems 3 (29, 3 1. [8] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 21. [9] Y.H. Liu and M. Ha, Expected value of function of uncertain variables, Journal of Uncertain Systems 4 (21, [1] Y.H. Liu, An analytic method for solving uncertain differential equations, Journal of Uncertain Systems 6 (212, [11] Y.H. Liu and X. Chen, Uncertain currency model and currency pricing formula, [12] G. Maruyama, Continuous Markov processes and stochastic equations, Rendiconti del Circolo Matematico di Palermo 4 (1955, [13] G.N. Milstein, Approximate integration of stochastic differential equations, Theory of Probability and its Applications 19 (1974, [14] J. Peng and K. Yao, A new option pricing model for stocks in uncertainty markets, International Journal of Operations Research 8 (211, [15] W. Rumelin, Numerical treatment of stochastic differential equations, SIAM Journal on Numerical Analysis 19 (1982, [16] L. Yang and B. Liu, On continuity theorem for characteristic function of fuzzy variable, Journal of Intelligent & Fuzzy Systems 17 (26, [17] K. Yao, Uncertain calculus with renewal process, Fuzzy Optimaztion and Decision Making 11 (212, [18] K. Yao, J. Gao and Y. Gao, Some stability theorems of uncertain differential equation, Fuzzy Optimization and Decision Making 12 (213, DOI: 1.17/s [19] C. You, Some convergence theorems of uncertain sequences, Mathematical and Computer Modelling 49 (29, [2] Y. Zhu and X. Ji, Expected values of functions of fuzzy variables, Journal of Intelligent & Fuzzy Systems 17 (26, [21] Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems 41 (21,