Multi-dimensional Fuzzy Euler Approximation

Transcription

1 Mathematica Aeterna, Vol 7, 2017, no 2, Multi-dimenional Fuzzy Euler Approximation Yangyang Hao College of Mathematic and Information Science Hebei Univerity, Baoding , China Cuilian You College of Mathematic and Information Science Hebei Univerity, Baoding , China Abtract Multi-dimenional Fuzzy differential equation driven by multi-dimenional Liu proce, have been intenively applied in many field However, we can not obtain the analytic olution of every multi-dimenional fuzzy differential equation Then, it i neceary for u to dicu the numerical reult in mot ituation Thi paper focue on the numerical method of multi-dimenional fuzzy differential equation The multi-dimenional fuzzy Taylor expanion i given, baed on thi expanion, a numerical method which i deigned for giving the olution of multi-dimenional fuzzy differential equation via multi-dimenional Euler method will be preented, and it local convergence alo will be dicued Mathematic Subject Claification: 34A07 Keyword: fuzzy proce; fuzzy differential equation; method method; Taylor expanion 1 Introduction A a elf-dual meaure, credibility meaure wa preented by Liu and Liu [6] in 2002 Later, credibility theory which i an efficient way for u to tudy the behavior of fuzzy phenomena wa founded by Liu [7] in 2004 and wa refined by Liu [8] in 2007 Baed on credibility theory, ome definition and theorem have been provided, uch a fuzzy variable wa propoed by Liu [11], it i a function from credibility pace to the et of real number The concept of fuzzy proce wa provided by Liu [11] to decribe dynamic fuzzy phenomena A a pecial fuzzy proce with tationary and independent normal fuzzy proce, Liu proce wa deigned by Liu [10] Jut like the aumption that tock price

2 164 Yangyang Hao and Cuilian You follow geometric Brownian motion, an alternative aumption that tock price follow geometric Liu proce wa introduced by Liu [10], and the European option pricing formula for Liu tock model wa given by Qin and Li [13] Gao [4] initiated a new tock model incorporating the mean reverion In order learn more about Liu proce, Dai [2] proved that all ample path of Liu proce are Lipchitz continuou A reflection principle related to Liu proce wa given by Dai [3] Qin and Wen [12] extended ome propertie of Liu proce to the cae of complex Liu proce You, Huo and Wang [14] conidered multi-dimenional Liu proce After the concept of Liu proce wa put forward, many mathematical tool bae on Liu proce have been developed A fuzzy counterpart of Ito integral and Ito formula, Liu integral and Liu formula were introduced by Liu [10] Multi-dimenional Liu integral and multi-dimenional Liu formal which are the tool to deal with the problem with everal dynamic factor were tudied by You, Wang and Huo [14] Later, You and Wang [16] conidered the propertie of Liu integral, and the propertie of complex Liu differential wa dicued by You and Wang [17] The definition of generalized Liu integral wa preented by You, Ma and Huo [18] In You, Huo and Wang [14], Liu integral and Liu formula were extended to the cae of multi-dimenional Liu integral and multi-dimenional Liu formula Fuzzy differential equation with fuzzy parameter or fuzzy initial condition have already got a better development in the pat year For example, the Cauchy problem for fuzzy differential equation wa howed by Kaleva [5] In 2008, Liu [10] introduced the concept of fuzzy differential equation driven by Liu proce Different from fuzzy differential equation with fuzzy parameter or fuzzy initial condition, the fuzzine of fuzzy differential equation driven by Liu proce i not only in parameter, in fuzzy initial condition, but alo in the driven proce, thu fuzzy differential equation driven by Liu proce i a general cae of fuzzy differential equation Over the pat year, many cholar have conducted a lot of reearche on fuzzy differential equation driven by Liu proce A exitence and uniquene theorem for homogeneou fuzzy differential equation wa tudied in You, Wang and Huo [15] Chen and Qin [1] conidered a new exitence and uniquene theorem that can deal with more general cae of fuzzy differential equation Zhu [21] dicued the tability for fuzzy differential equation driven by Liu proce Stability in ditance for fuzzy differential equation wa propoed by You and Hao [19] You, Wang and Huo [16] figured out the analytic olution of linear fuzzy differential equation However, we can not obtain the analytic olution of every fuzzy differential equation Then, it i neceary for u to dicu the numerical reult in mot ituation A numerical method that wa deigned for giving the olution of fuzzy differential equation via Euler method wa introduced by You and Hao [20]

3 Multi-dimenional Fuzzy Euler Approximation 165 In thi paper, we will extend the numerical method to the cae of multidimenional fuzzy differential equation The ret of the paper i organized a follow In Section 2, ome baic knowledge on credibility theory will be introduced In Section 3, we will give the multi-dimenional fuzzy Taylor expanion, thi expanion i the key to the multi-dimenional fuzzy numerical analyi The multi-dimenional fuzzy Euler method will appear in Section 4, and it local convergence will be alo proved in thi ection 2 Preliminary Note In thi ection, we will firt review ome baic knowledge on credibility theory for further undertanding Let T be an index et and (Θ, P, Cr) be a credibility pace, where Θ i a nonempty et, P i the power et of Θ, and Cr i a credibility meaure A fuzzy proce X(t, θ) i a function of two variable from T (Θ, P, Cr) to the et of real number Thi fuzzy proce X(t, θ) i a fuzzy variable for each fixed t For each fixed θ, the function X(t, θ ) i called a ample path of the fuzzy proce Intead of longer notation X(t, θ), we will ue the ymbol X t in the following ection Definition 21 (Liu [9]) Suppoe that ξ, ξ 1, ξ 2, are fuzzy variable defined on credibility pace (Θ, P, Cr) The equence ξ i i aid to be convergent a to ξ if and only if there exit an event A with Cr{A} = 1 uch that lim ξ i(θ) ξ(θ) = 0 i Definition 22 (Liu [10]) A fuzzy proce C t i aid to be a Liu proce if (i) C 0 = 0, (ii) C t ha tationary and independent increment, (iii) C t C i a normally ditributed fuzzy variable with expected value et and variance σ 2 t 2 Note that Liu proce i aid to be a tandard Liu proce if e = 0 and σ = 1 Definition 23 (You, Wang and Huo [14]) If C it, i = 1, 2,, m are Liu procee, then C t = (C 1t, C 2t,, C mt ) T i called an m-dimenional Liu proce Theorem 21 (Dai [2]) Let C t be a Liu proce For any given θ Θ with Cr{θ} > 0, the path C t (θ) i Lipchitz continuou, ie there exit a Lipchitz contant K(θ) atifying C t C K(θ) t

4 166 Yangyang Hao and Cuilian You Theorem 22 (Liu Formula, Liu [10]) Let C t be a tandard Liu proce, and let h(t, c) be a continuouly differentiable function Define X t = h(t, C t ) Then we have the following chain rule dx t = h t (t, C t)dt h c (t, C t)dc t Theorem 23 (Multi-dimenional Liu Formula, You, Wang and Huo [14]) Let C t = (C 1t, C 2t,, C mt ) T be an m-dimenional tandard Liu proce, and let h(t, x 1,, x n ) = (h 1 (t, x 1,, x n ), h 2 (t, x 1,, x n ), h p (t, x 1,, x n )) T, where h i (t, x 1,, x n ), i = 1, 2, p are multivariate continuouly differentiable function If fuzzy proce (X 1t, X 2t,, X nt ) T i given by dx 1t = u 1t dt v 11t dc 1t v 1mt dc mt, dx nt = u nt dt v n1t dc 1t v nmt dc mt where u it, v ijt are abolutely integrable fuzzy procee and Liu integrable fuzzy proce, repectively Define Y t = h(t, X 1t, X 2t,, X nt ) Then dy t = dy 1t dy 2t dy pt h 1 (t, X t 1t, X 2t,, X nt )dt n h 1 i=1 x i (t, X 1t, X 2t,, X nt )dx it h 2 = t (t, X 1t, X 2t,, X nt )dt n h 2 i=1 x i (t, X 1t, X 2t,, X nt )dx it h p (t, X t 1t, X 2t,, X nt )dt n h p i=1 x i (t, X 1t, X 2t,, X nt )dx it Definition 24 (Liu Integral, Liu [10]) Let C t be a fuzzy proce and let C t be a tandard Liu proce For any partition of cloed interval[a, b] with a = t 1 < t 2 < < t k1 = b, the meh i written a = max 1 i k t i1 t i Then the fuzzy integral of fuzzy proce X t with repect to C t i b a X t dc t = lim 0 k X ti (C ti1 C ti ), provided that the limit exit almot urely and i a fuzzy variable i=1 Definition 25 (Multi-dimenional Liu Integral, You, Wang and Huo [14]) Let C t = (C 1t, C 2t,, C mt ) T be an m-dimenional tandard Liu proce, and

5 Multi-dimenional Fuzzy Euler Approximation 167 let ν n m denote the et of n m matrice v t = [v ijt ], where each entry v ijt i a Liu integrable fuzzy proce Suppoe a < b If v t ν n m, the m-dimenional Liu integral i defined, uing matrix notation v 11t v 12t v 1mt dc 1t b b v 21t v 22t v 2mt dc 2t v t dc t = a a v n1t v n2t v nmt dc mt a the n 1 matrix whoe ith component i the following um of Liu integral: m i=1 b a v ijt dc jt Definition 26 (Fuzzy Differential Equation, Liu [10]) Suppoe C t i a tandard Liu fuzzy proce, and f and g are ome given function Then dx t = f(t, X t )dt g(t, X t )dc t (1) i called a fuzzy differential equation driven by Liu proce A olution i a fuzzy proce X t that atifie above equality identically in t Definition 27 (Multi-dimenional Fuzzy Differential Equation, Zhu [21]) Let C t be a tandard n-dimenional Liu proce Then dx t = f(t, X t )dt g(t, X t )dc t, X t0 = x 0 (2) i called an n-dimenional fuzzy differential equation driven by a n-dimenional tandard Liu proce, where X t i the n-dimenional tate vector, x 0 i the crip n-dimenional initial tate vector, f(t, X t ) i ome given vector function of time t and tate X t, and g(t, X t ) i ome given matrix-valued function of time t and tate X t in C n n 3 Multi-dimenional fuzzy Taylor expanion Fuzzy Tayor expanion i the neceary tool to contruct numerical method of fuzzy differential equation In order to obtain Euler method of multidimenional fuzzy differential equation, we will give the multi-dimenional fuzzy Tayor expanion in thi ection Conider the following multi-dimenional fuzzy differential equation dx t = f(x t )dt g(x t )dc t, (1)

6 168 Yangyang Hao and Cuilian You where and f = (f 1 (x), f 2 (x),, f n (x)) T g 11 (x) g 12 (x) g 1n (x) g 21 (x) g 22 (x) g 2n (x) g(x) = g n1 (x) g n2 (x) g nn (x) are ome given continuouly differentiable function, X t = (X 1t, X 2t,, X nt ) i a multi-dimenional fuzzy proce, C t = (C 1, C 2,, C n ) T i a multi-dimenional tandard Liu proce Then the olution X t i to be interpreted a X t = X 1t X nt = X 1t0 f 1 (X )d n X nt0 f n (X )d n g 1j (X )dc j (2) g nj (X )dc j According to multi-dimenional Liu formula, the ith component X it of olution X t of the fuzzy differential equation can be rewritten a the following form a(x t ) = a(x t0 ) L 0 a(x )d L j a(x )dc j, (3) where L 0 a(x) = a x i f i, L j a(x) = a x j g ij, a(x) i a function with n variable (i) Take a = f i, a = g ij, j = 1, 2,, n, repectively, we have f i (X t ) = f i (X t0 ) g ij (X t ) = g ij (X t0 ) L 0 f i (X )d L 0 g ij (X )d l=1 L j f i (X )dc j, (4) L l g ij (X )dc l (5) Subtituting (4) and (5) into the ith component of (2), the component of n- dimenional fuzzy Taylor expanion a follow X it = X it0 {f i (X t0 ) L 0 f i (X 1 )d 1 L j f i (X 1 )dc j1 }d {g ij (X t0 ) L 0 g ij (X 1 )d 1 L l g ij (X 1 )dc l1 }dc j = X it0 f i (X t0 ) d l=1 g ij (X t0 ) dc j R 1, (6)

7 Multi-dimenional Fuzzy Euler Approximation 169 where R 1 = L 0 f i (X 1 )d 1 d L 0 g ij (X 1 )d 1 dc j L j f i (X 1 )dc j1 d l=1 L l g ij (X 1 )dc l1 dc j Thi i the implet multi-dimenional fuzzy Taylor expanion (ii) Let a = f i xl f l, a = g ij xl f l, a = f i xl g il, a = g ij xp g ip, repectively, then we have f i xl (X t )f l (X t ) =f i xl (X t0 )f l (X t0 ) k=1 L 0 f i xl (X )f l (X )d L k f i xl (X )f l (X )dc k, (7) g ij xl (X t )f l (X t ) =g ij xl (X t0 )f l (X t0 ) k=1 f i xl (X t )g il (X t ) =f i xl (X t0 )g il (X t0 ) k=1 L 0 g ij xl (X )f l (X )d L k g ij xl (X )f l (X )dc k, L 0 f i xl (X )g il (X )d L k f i xl (X )g il (X )dc k, (8) (9) g ij xp (X t )g ip (X t ) =g ij xp (X t0 )g ip (X t0 ) L 0 g ij xp (X )g ip (X )d (10) L m g ij xp (X )g ip (X )dc m m=1 Subtituting (7), (8), (9) and (10) into (6), we have X it =X it0 f i (X t0 ) d g ij (X t0 ) dc j f i xl (X t0 )f l (X t0 ) g ij xl (X t0 )f l (X t0 ) dc j1 d f i xl (X t0 )g il (X t0 ) g ij xp (X t0 )g ip (X t0 ) dc l1 dc j R 2, l=1 d 1 d d 1 dc j

8 170 Yangyang Hao and Cuilian You where R 2 = 1 L 0 f i xl (X 2 )f l (X 2 )d 2 d 1 d 1 1 k=1 1 1 k=1 1 l=1 l=1 m=1 k=1 L 0 g ij xl (X 2 )f l (X 2 )d 2 dc j1 d 1 L k g ij xl (X 2 )f l (X 2 )dc k2 dc j1 d L 0 f i xl (X 2 )g il (X 2 )d 2 d 1 dc j 1 L k f i xl (X 2 )g il (X 2 )dc k2 d 1 dc j L 0 g ij xp (X 2 )g ip (X 2 )d 2 dc l1 dc j L m g ij xp (X 2 )g ip (X 2 )dc m2 dc l1 dc j Thi i the econd-order multi-dimenional fuzzy Taylor expanion Continuing thi proce, thu the high order multi-dimenional fuzzy Taylor expanion can be obtained 4 Multi-dimenional fuzzy Euler method A a numerical method for etimating the olution of differential function, Euler method play an important role in differential function When the fuzzy Taylor expanion of multi-dimenional fuzzy differential function i given, we can obtain the Euler method by truncating multiple integral a the following form X tm1 = X 1tm1 X ntm1 = X 1tm f 1 (X tm )h n g 1j(X tm ) C jtm X ntm f n (X tm )h, n g nj(x tm ) C jtm where h = t m1 t m, C jtm = C jtm1 C jtm Example 41 In order to illutrate multi-dimenional fuzzy differential equation, let u conider the 2-dimenional fuzzy differential equation ( ) ( ) a1 X dx t = t b1 X dt dc a 2 X b 2 X t t L k f i xl (X 2 )f l (X 2 )dc k2 d 1 d

9 Multi-dimenional Fuzzy Euler Approximation 171 where X t = (X 1t, X 2t ) T, C t = (C 1t, C 2t ) T When the interval [, t] wa divided into n part, we can obtain the numerical olution of the 2-dimenional fuzzy equation eaily via multi-dimenional fuzzy Euler method ( ) ( X1t X1tn 1 a X t = = 1 X h b 1tn 1 1X C ) tn 1 1t, X 2t X 2tn 1 a 2 X 1tn 1 h b 2 X tn 1 C 2t where h = t n t n 1, C 1t = C 1tn C 1tn 1, C 2t = C 2tn C 2tn 1, X itn 1 i ith component of analytic olution of the 2-dimenional fuzzy differential equation when t = t j 1, X tn 1 i the analytic olution of the 2-dimenional fuzzy differential equation when t = t j 1 Theorem 41 Conidering mutil-dimenional fuzzy differential equation (2) Let f i and g ij be given function that atify the Linear growth condition f i (x) 2 g ij (x) 2 L 1 (1 x 2 ), for i = 1, 2,, n, j = 1, 2,, n Then there exit contant N 1 (θ) uch that X 2 it 3(n 2)N 1 (θ) for the ith component X it of olution of (2), where N 1 (θ) jut relie on integrating range and initial value only Proof: We hall rewrite the component X it of olution of (2) a the integral form X it = X it0 f i (X )d g i1 (X )dc 1 g in (X )dc n (11) Taking quare of both ide and applying the inequality (a 1 a 2 a n ) 2 na 2 1 na 2 2 na 2 n, we have Xit 2 (n 2)[Xit 2 0 ( f i (X )d) 2 ( Then it follow from Lipchitz continuity of Liu proce that i=0 n 1 g ij (X )dc j = lim g ij (X ti )(C jti1 C jti ) t 0 0 g ij (X )dc j ) 2 ] (12) n 1 K j (θ) lim g ij (X tk )(t k1 t k ) 0 k=0 K j (θ) g ij (X )d

10 172 Yangyang Hao and Cuilian You It i eay to rewrite (12) a the following inequality Xit 2 (n 2)[Xit 2 0 ( f i (X )d) 2 (K j (θ)) 2 ( By Hölder inequality and Linear growth condition, we have ( f i (X )d) 2 (t ) In a imilar proof, we have K 2 j (θ)( Thu (t ) f 2 i (X )d L 1 (1 X 2 )d = (t ) 2 L 1 (t )L 1 g ij (X )d) 2 ] X 2 d g ij (X )d) 2 K 2 j (θ)(t ) 2 L 1 K 2 j (θ)(t )L 1 X 2 d Xit 2 (n 2)[Xit 2 0 (t ) 2 L 1 L 1 (t ) X 2 d [Kj 2 (θ)(t ) 2 L 1 (t )L 1 Kj 2 (θ) X 2 d] According to the Grownwall inequality, we obtain X 2 it (n 2)[X 2 i M 0 exp[m 0 (t )] M j (θ)exp[m j (θ)(t )] Let N 1 (θ) = max{x 2 i, M 0 exp[m 0 (t )], n M j(θ)[m j (θ)(t )]} The inequality i obtained Theorem 42 Conidering mutil-dimenional fuzzy differential equation (2) Let f i and g ij be given function that atify the Linear growth condition f i (x) 2 g ij (x) 2 L 1 (1 x 2 ), for i = 1, 2,, n, j = 1, 2,, n Then for 0 t T, there exit a contant N 2 (θ) uch that (X it X i ) 2 N 2 (θ)(t ) 2, where X it, X i are the ith component of olution of (2) and N 2 (θ) jut relie on, t and initial value only

11 Multi-dimenional Fuzzy Euler Approximation 173 Proof: For any 0 t T, X it X i = f i (X τ )dτ g ij (X τ )dc jτ Taking quare of both ide and applying the inequality (a 1 a 2 a n ) 2 na 2 1 na 2 2 na 2 n, we have (X it X i ) 2 (n 1)[ f i (X τ )dτ) 2 ( g ij (X τ )dc jτ ) 2 ] Then it follow from the Hölder inequality and Linear growth condition that ( In a imilar proof, we have K 2 j (θ)( f i (X τ )dτ) 2 (t ) (t ) f 2 i (X τ )dτ L 1 (1 X τ 2 )dτ = L 1 (t ) 2 L 1 (t ) X τ 2 dτ g ij (X τ )dc jτ ) 2 L 1 (t ) 2 K 2 j (θ) L 1 K 2 j (θ)(t ) According to Theorem 41, we have (X it X j ) 2 L 1 (n1)(t ) 2 {13(n2)N 1 (θ) Let N 2 (θ) = L 1 (n 1){1 3(n 2)N 1 (θ) Then we have (X it X i ) 2 N 2 (θ)(t ) 2 The theorem i proved X τ 2 dτ [Kj 2 (θ)3kj 2 (θ)(n2)n 1 (θ)]} [Kj 2 (θ) 3Kj 2 (θ)(n 2)N 1 (θ)]} Theorem 43 Conidering mutil-dimenional fuzzy differential equation (2) Let f i and g ij be given function and atify the Linear growth condition and Global Lipchitz condition f i (x) 2 g ij (x) 2 L 1 (1 x 2 ), f i (x) f i (y) g ij (x) g ij (y) L 2 x y, for i = 1, 2,, n, j = 1, 2,, n Then the mutil-dimenional Euler approximation converge almot urely locally

12 174 Yangyang Hao and Cuilian You Proof: According to the Euler approximation, the local error of the olution X t can be written a δ n1 = X tn1 X tn1 = n (X itn1 X itn1 ) 2 i=1 where X itn1 i analytic olution of (11) when t = t n1, Xitn1 i numerical olution that jut ued Euler approximation once, ie n1 X itn1 = X itn t n f i (X n )d n1 t n g ij (X n )dc jn, where X itn i the analytic olution of (11) when t = t n Applying Global Lipchitz condition, Theorem 41 and Theorem 42, we have lim δ n1 = lim n (X itn1 X itn1 ) 2 h 0 h 0 i=1 n1 n1 = lim { [f i (X ) f i (X tn )]d [g ij (X ) g ij (X tn )]dc j } 2 h 0 t n t n h 0 i=1 lim [L 2 nn2 (θ)h 2 L 2 K j (θ) nn 2 (θ)h 2 ] 2 lim = 0 h 0 h 2 i=1 [L 2 nn2 (θ) L 2 K j (θ) nn 2 (θ)] 2 i=1 Then the theorem i proved The effectivene of Theorem 43 can be illutrated by the following example Example 42 Conider Example 41, ince f i (x) = a i x and g i (x) = b i (x) atify the Linear growth condition and Global Lipchitz condition f i (x) 2 g i (x) 2 max{a 2 i, b 2 i }(1 x 2 ), f i (x) f i (y) g i (x) g i (y) max{a i, b i } x y, for i = 1, 2 Then the numerical olution converge almot urely locally according to Theorem 43

13 Multi-dimenional Fuzzy Euler Approximation Concluion Multi-dimenional fuzzy differential equation driven by multi-dimenional Liu proce, i an important tool to deal with multi-dimenional dynamic ytem in fuzzy environment In many cae, unfortunately, it i difficult to find analytic olution of multi-dimenional fuzzy differential equation Thi paper gave a multi-dimenional Euler method which wa obtained by truncating multiple integral of multi-dimenional Taylor expanion for olving multidimenional fuzzy differential equation Furthermore, the local convergence of multi-dimenional Euler method wa proved ACKNOWLEDGEMENTS Thi work wa upported by Natural Science Foundation of China Grant No Reference [1] Chen X, Qin Z, A new exitence and uniquene theorem for fuzzy differential equation, International Journal of Fuzzy Sytem, 2011, 13(2) : [2] Dai W, Lipchitz continuity of Liu proce, /080831pdf [3] Dai W, Reflection principle of Liu proce, /071110pdf, 2007 [4] Gao J, Gao X, A new tock model for credibilitic option pricing, Journal of Uncertain Sytem, 2008, 4(2) : [5] Kaleva O, The Cauchy problem for fuzzy differential equation, Fuzzy Set and Sytem, 1990, 35 : [6] Liu B, Liu YK, Expected value of fuzzy variable and fuzzy expected value model, IEEE Tranaction on Fuzzy Sytem, 2002, 10(4) : [7] Liu B, Uncertainty Theory, Springer-Verlag, Berlin, 2004 [8] Liu B, Uncertainty Theory, 2nd ed, Springer-Verlag, Berlin, 2007 [9] Liu B, Inequalitie and convergence concept of fuzzy and rough variable, Fuzzy Optimization and Deciion Marking, 2003, 2(2) : [10] Liu B, Fuzzy proce, hybrid proce and uncertain proce, Journal of Uncertain Sytem, 2008, 2(1) : 3 16 [11] Liu B, Uncertainty Theory, Third Edition,

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