Fuzzy Process, Hybrid Process and Uncertain Process

Transcription

1 Journl of Uncertin Systems Vol.2, No.1, pp.3-16, 28 Online t: Fuzzy Process, Hybrid Process nd Uncertin Process Boding Liu Deprtment of Mthemticl Sciences Tsinghu University, Beijing 184, Chin Received 7 July 27; Revised 1 August 27; Accepted 25 September 27 Abstrct This pper first reviews different types of uncertinty. In order to construct fuzzy counterprts of Brownin motion nd stochstic clculus, this pper proposes some bsic concepts of fuzzy process, including fuzzy clculus nd fuzzy differentil eqution. Those new concepts re lso extended to hybrid process nd uncertin process. A bsic stock model is presented, thus opening up wy to fuzzy finncil mthemtics. Keywords: fuzzy process, fuzzy clculus, fuzzy differentil eqution, finncil mthemtics 1 Introduction Rndomness is bsic type of objective uncertinty, nd probbility theory is brnch of mthemtics for studying the behvior of rndom phenomen. The study of probbility theory ws strted by Pscl nd Fermt (1654), nd n xiomtic foundtion of probbility theory ws given by Kolmogoroff (1933) in his Foundtions of Probbility Theory. The concept of fuzzy set ws initited by Zdeh [13] vi membership function in In order to mesure fuzzy event, Liu nd Liu [8] introduced the concept of credibility mesure in 22. Li nd Liu [5] gve sufficient nd necessry condition for credibility mesure in 26. Credibility theory ws founded by Liu [9] in 24 nd refined by Liu [11] in 27 s brnch of mthemtics for studying the behvior of fuzzy phenomen. Credibility theory is deduced from the normlity, monotonicity, self-dulity, nd mximlity xioms. Fuzziness nd rndomness re two bsic types of uncertinty. In mny cses, fuzziness nd rndomness simultneously pper in system. In order to describe this phenomen, fuzzy rndom vrible ws introduced by Kwkernk [3][4] s rndom element tking fuzzy vrible vlues. In ddition, rndom fuzzy vrible ws proposed by Liu [7] s fuzzy element tking rndom vrible vlues. More generlly, hybrid vrible ws introduced by Liu [1] s tool to describe the quntities with fuzziness nd rndomness. Fuzzy rndom vrible nd rndom fuzzy vrible re instnces of hybrid vrible. In order to mesure hybrid events, concept of chnce mesure ws introduced by Li nd Liu [6]. A clssicl mesure is essentilly set function (i.e., function whose rgument is set) stisfying nonnegtivity nd countble dditivity xioms. Clssicl mesure theory, developed by Borel nd Lebesgue round 19, hs been widely pplied in both theory nd prctice. However, the dditivity xiom of clssicl mesure theory hs been chllenged by mny mthemticins. The erliest chllenge ws from the theory of cpcities by Choquet [2] in which monotonicity nd continuity xioms were ssumed. Sugeno [12] generlized clssicl mesure theory to fuzzy mesure theory by replcing dditivity xiom with monotonicity nd semicontinuity xioms. In order to del with generl uncertinty, self-dulity plus countble subdditivity is much more importnt thn continuity nd semicontinuity. For this reson, Liu [11] founded n uncertinty theory tht is brnch of mthemtics bsed on normlity, monotonicity, self-dulity, nd countble subdditivity xioms. This pper will review different types of uncertinty, including rndomness nd fuzziness. This pper lso provides some bsic concepts of fuzzy process, hybrid process nd uncertin process. We develop fuzzy clculus nd propose fuzzy differentil eqution. A bsic stock model is lso presented, thus opening up wy to fuzzy finncil mthemtics.

2 4 B. Liu: Fuzzy Process, Hybrid Process nd Uncertin Process 2 Preliminries In this section, we will introduce some useful definitions nd properties bout rndom vrible, fuzzy vrible, hybrid vrible nd uncertin vrible. 2.1 Rndom Vrible Let Ω be nonempty set, nd A σ-lgebr over Ω. Ech element in A is clled n event. In order to present n xiomtic definition of probbility, it is necessry to ssign to ech event A number Pr{A} which indictes the probbility tht A will occur. In order to ensure tht the number Pr{A} hs certin mthemticl properties which we intuitively expect probbility to hve, the following three xioms must be stisfied: Axiom 1. (Normlity) Pr{Ω} = 1. Axiom 2. (Nonnegtivity) Pr{A} for ny A A. Axiom 3. (Countble Additivity) For every countble sequence of mutully disjoint events {A i }, we hve { } Pr A i = Pr{A i }. (1) Definition 1 The set function Pr is clled probbility mesure if it stisfies the normlity, nonnegtivity, nd countble dditivity xioms. Definition 2 A rndom vrible is mesurble function from probbility spce (Ω, A, Pr) to the set of rel numbers, i.e., for ny Borel set B of rel numbers, the set {ω Ω ξ(ω) B} is n event. 2.2 Fuzzy Vrible Let Θ be nonempty set, nd let P be the power set of Θ (i.e., ll subsets of Θ). Ech element in P is clled n event. In order to present n xiomtic definition of credibility, we ccept the following four xioms: Axiom 1. (Normlity) Cr{Θ} = 1. Axiom 2. (Monotonicity) Cr{A} Cr{B} whenever A B. Axiom 3. (Self-Dulity) Cr{A} + Cr{A c } = 1 for ny A P. Axiom 4. (Mximlity) Cr { i A i } = sup i Cr{A i } for ny events {A i } with sup i Cr{A i } <.5. Definition 3 (Liu nd Liu [8]) The set function Cr is clled credibility mesure if it stisfies the normlity, monotonicity, self-dulity nd mximlity xioms. Now we define fuzzy vrible s function on credibility spce just s rndom vrible is defined s mesurble function on probbility spce. Definition 4 A fuzzy vrible is function from credibility spce (Θ, P, Cr) to the set of rel numbers. If fuzzy vrible ξ is defined s function on credibility spce, then we my get its membership function vi µ(x) = (2Cr{ξ = x}) 1, x R. (2) Conversely, if fuzzy vrible ξ is given by membership function µ, then we my get the credibility vlue vi Cr{ξ B} = 1 ( ) sup µ(x) + 1 sup µ(x) (3) 2 x B x B c where B is set of rel numbers.

3 Journl of Uncertin Systems, Vol.2, No.1, pp.3-16, Hybrid Vrible Definition 5 (Liu [1]) Suppose tht (Θ, P, Cr) is credibility spce nd (Ω, A, Pr) is probbility spce. The product (Θ, P, Cr) (Ω, A, Pr) is clled chnce spce. The universl set Θ Ω is clerly the set of ll ordered pirs of the form (θ, ω), where θ Θ nd ω Ω. Wht is the product σ-lgebr P A? Wht is the product mesure Cr Pr? Let us discuss these two bsic problems. Definition 6 (Liu [11]) Let (Θ, P, Cr) (Ω, A, Pr) be chnce spce. A subset Λ Θ Ω is clled n event if Λ(θ) = {ω Ω (θ, ω) Λ} A for ech θ Θ. It hs been proved by Liu [11] tht the clss of ll events is σ-lgebr over Θ Ω, nd denoted by P A. Definition 7 (Li nd Liu [6]) Let (Θ, P, Cr) (Ω, A, Pr) be chnce spce. Then chnce mesure of n event Λ is defined s Ch{Λ} = sup(cr{θ} Pr{Λ(θ)}), θ Θ 1 sup(cr{θ} Pr{Λ c (θ)}), θ Θ if sup(cr{θ} Pr{Λ(θ)}) <.5 θ Θ if sup(cr{θ} Pr{Λ(θ)}).5. θ Θ It is proved tht chnce mesure is norml, incresing, nd countbly subdditive. Definition 8 (Liu [1]) A hybrid vrible is mesurble function from chnce spce (Θ, P, Cr) (Ω, A, Pr) to the set of rel numbers, i.e., for ny Borel set B of rel numbers, the set {(θ, ω) Θ Ω ξ(θ, ω) B} is n event. (4) 2.4 Uncertin Vrible Let Γ be nonempty set, nd let L be σ-lgebr over Γ. Ech element Λ L is clled n event. In order to present n xiomtic definition of uncertin mesure, it is necessry to ssign to ech event Λ number M{Λ} which indictes the level tht Λ will occur. In order to ensure tht the number M{Λ} hs certin mthemticl properties, Liu [11] proposed the following four xioms: Axiom 1. (Normlity) M{Γ} = 1. Axiom 2. (Monotonicity) M{Λ 1 } M{Λ 2 } whenever Λ 1 Λ 2. Axiom 3. (Self-Dulity) M{Λ} + M{Λ c } = 1 for ny event Λ. Axiom 4. (Countble Subdditivity) For every countble sequence of events {Λ i }, we hve. { } M Λ i M{Λ i }. (5) Definition 9 (Liu [11]) The set function M is clled n uncertin mesure if it stisfies the normlity, monotonicity, self-dulity, nd countble subdditivity xioms. Definition 1 (Liu [11]) An uncertin vrible is mesurble function from n uncertinty spce (Γ, L, M) to the set of rel numbers, i.e., for ny Borel set B of rel numbers, the set {γ Γ ξ(γ) B} is n event. 2.5 Reltions Probbility theory is brnch of mthemtics bsed on the normlity, nonnegtivity, nd countble dditivity xioms. In fct, those three xioms my be replced with four xioms: normlity, monotonicity, self-dulity, nd countble dditivity. Thus ll of probbility, credibility, chnce, nd uncertin mesures meet the normlity, monotonicity nd self-dulity xioms. The essentil difference mong those mesures is how to determine

4 6 B. Liu: Fuzzy Process, Hybrid Process nd Uncertin Process the mesure of union. For ny mutully disjoint events {A i } with sup i π{a i } <.5, if π stisfies the countble dditivity xiom, i.e., { } π A i = π{a i }, (6) then π is probbility mesure; if π stisfies the mximlity xiom, i.e., { } π A i = sup π{a i }, (7) 1 i< then π is credibility mesure; if π stisfies the countble subdditivity xiom, i.e., { } π A i π{a i }, (8) then π is n uncertin mesure. Since dditivity nd mximlity re specil cses of subdditivity, probbility nd credibility re specil cses of chnce mesure, nd three of them re in the ctegory of uncertin mesure. This fct lso implies tht rndom vrible nd fuzzy vrible re specil cses of hybrid vribles, nd three of them re instnces of uncertin vribles. Credibility Model. Hybrid Model Probbility Model. Uncertinty Model. Figure 1: Reltions mong Uncertinties 3 Fuzzy Process Definition 11 Let T be n index set nd let (Θ, P, Cr) be credibility spce. A fuzzy process is function from T (Θ, P, Cr) to the set of rel numbers. Tht is, fuzzy process X(t, θ) is function of two vribles such tht the function X(t, θ) is fuzzy vrible for ech t. For ech fixed θ, the function X(t, θ ) is clled smple pth of the fuzzy process. A fuzzy process X(t, θ) is sid to be smple-continuous if the smple pth is continuous for lmost ll θ. Insted of longer nottion X(t, θ), sometimes we use the symbol X t. Definition 12 A fuzzy process X t is sid to hve independent increments if X t1 X t, X t2 X t1,, X tk X tk 1 re independent fuzzy vribles for ny times t < t 1 < < t k. A fuzzy process X t is sid to hve sttionry increments if, for ny given t >, the X s+t X s re identiclly distributed fuzzy vribles for ll s >. Exmple 1: Assume (i) X =, (ii) X t hs sttionry nd independent increments, nd (iii) every increment X s+t X s is tringulr fuzzy vrible (t, bt, ct). Then X t is fuzzy process. Exmple 2: Assume (i) X =, (ii) X t hs sttionry nd independent increments, nd (iii) every increment X s+t X s is trpezoidl fuzzy vrible (t, bt, ct, dt). Then X t is fuzzy process.

5 Journl of Uncertin Systems, Vol.2, No.1, pp.3-16, 28 7 Exmple 3: Assume (i) X =, (ii) X t hs sttionry nd independent increments, nd (iii) every increment X s+t X s is n exponentilly distributed fuzzy vrible with second moment m 2 t 2 whose membership function is ( ( )) 1 πx µ(x) = exp, x. 6mt Then X t is fuzzy process. Exmple 4: (Fuzzy Renewl Process) Let iid positive fuzzy vribles ξ 1, ξ 2, denote the interrrivl times of successive events. Define S = nd S n = ξ 1 + ξ ξ n for n 1. Then S n cn be regrded s the witing time until the occurrence of the nth event. For ny t >, let N t be the number of renewls in (, t], i.e., { N t = mx n Sn t }. (9) n It is cler tht N t is fuzzy process, nd we cll it fuzzy renewl process. Ech smple pth of N t is right-continuous nd incresing step function tking only integer vlues. Furthermore, the size of ech jump of N t is lwys 1. In other words, N t hs t most one renewl t ech time. In prticulr, N t does not jump t time. Since N t n if nd only if S n t, we hve { Cr{N t n} = Cr{S n t} = Cr ξ 1 t }. (1) n It is esy to verify tht E[N t ] = Zho nd Liu [14] proved the following formul, 3.1 C Process n=1 E[N t ] lim = E t t { Cr ξ 1 t }. (11) n [ 1 ξ 1 ]. (12) Definition 13 A fuzzy process C t is sid to be C process if (i) C =, (ii) C t hs sttionry nd independent increments, (iii) every increment C s+t C s is normlly distributed fuzzy vrible with expected vlue et nd vrince σ 2 t 2, whose membership function is ( ( )) 1 π x et µ(x) = exp, x R. 6σt The prmeters e nd σ re clled the drift nd diffusion coefficients, respectively. The C process is sid to be stndrd if e = nd σ = 1. The C process plys the role of Brownin motion. Perhps the reders would like to know why the increment is normlly distributed fuzzy vrible. The reson is tht normlly distributed fuzzy vrible hs mximum entropy when its expected vlue nd vrince re given, just like normlly distributed rndom vrible. Theorem 1 (Existence Theorem) There is C process tht is smple-continuous. Sketch of Proof: Without loss of generlity, we only prove tht there is stndrd C process on the rnge of t [, 1]. Let { ξ(r) r represents rtionl numbers in [, 1] } be countble sequence of independently nd normlly distributed fuzzy vribles with expected vlue zero nd vrince one. For ech integer n, we define fuzzy process X n (t) = 1 n k ( i ξ n liner, ), if t = k n otherwise. (k =, 1,, n) Since the limit lim n X n (t) exists lmost surely, we my verify tht the limit meets the conditions of C process nd is smple-continuous. Hence there is stndrd C process.

6 8 B. Liu: Fuzzy Process, Hybrid Process nd Uncertin Process 3.2 Fuzzy Clculus Let C t be stndrd C process, nd dt n infinitesiml time intervl. Then dc t = C t+dt C t is fuzzy process such tht, for ech t, the dc t is normlly distributed fuzzy vrible with E[dC t ] =, V [dc t ] = dt 2, E[dC 2 t ] = dt 2, V [dc 2 t ] 7dt 4. Definition 14 Let X t be fuzzy process nd let C t be stndrd C process. For ny prtition of closed intervl [, b] with = t 1 < t 2 < < t k+1 = b, the mesh is written s Then the fuzzy integrl of X t with respect to C t is = mx 1 i k t i+1 t i. X t dc t = lim provided tht the limit exists lmost surely nd is fuzzy vrible. k X ti (C ti+1 C ti ) (13) Remrk 1: Note tht the subscript of X ti is the left end point of intervl [t i, t i+1 ]. The integrl does not remin unchnged if the subscript tkes other points, sy, the right end point t i+1 or the middle point (t i + t i+1 )/2. Remrk 2: Here the fuzzy integrl is defined in the sense of convergence lmost surely. In fct, it my lso be defined in sense of convergence in men or in men squre. Exmple 5: If C t is stndrd C process, then dc t = C s, tdc t = sc s C t dt, C t dc t = 1 2 C2 s, C 2 t dc t = 1 3 C3 s. Theorem 2 Let C t be stndrd C process, nd let h(t, c) be continuously differentible function. Define X t = h(t, C t ). Then we hve the following chin rule dx t = h t (t, C t)dt + h c (t, C t)dc t. (14) Proof: Since the function h is continuously differentible, by using Tylor series expnsion, the infinitesiml increment of X t hs first-order pproximtion Hence we obtin the chin rule becuse it mkes for ny s. X t = h t (t, C t) t + h c (t, C t) C t. X s = X + h t (t, C t)dt + h c (t, C t)dc t Remrk 3: The infinitesiml increment dc t in (14) my be replced with the derived C process where u t nd v t re bsolutely integrble fuzzy processes, thus producing dy t = u t dt + v t dc t (15) dh(t, Y t ) = h t (t, Y t)dt + h c (t, Y t)dy t. (16)

7 Journl of Uncertin Systems, Vol.2, No.1, pp.3-16, 28 9 Theorem 3 (Integrtion by Prts) Suppose tht C t is stndrd C process nd F (t) is n bsolutely continuous function. Then F (t)dc t = F (s)c s C t df (t). (17) Proof: By defining h(t, C t ) = F (t)c t nd using the chin rule, we get d(f (t)c t ) = C t df (t) + F (t)dc t. Thus F (s)c s = which is just (17). d(f (t)c t ) = C t df (t) + F (t)dc t 3.3 Fuzzy Differentil Eqution Definition 15 Suppose C t is stndrd C process, nd f nd g re some given functions. Then dx t = f(t, X t )dt + g(t, X t )dc t (18) is clled fuzzy differentil eqution. A solution is fuzzy process X t tht stisfies (18) identiclly in t. Exmple 6: Let C t be stndrd C process. Then the fuzzy differentil eqution dx t = dt + bdc t hs solution X t = t + bc t which is just C process with drift coefficient nd diffusion coefficient b. Exmple 7: Let C t be stndrd C process. Then the fuzzy differentil eqution dx t = X t dt + bx t dc t hs solution X t = exp (t + bc t ) which is just geometric C process. Exmple 8: Let C t be stndrd C process. Then the fuzzy differentil equtions { dxt = Y t dc t dy t = X t dc t hve solution (X t, Y t ) = (cos C t, sin C t ) which is clled C process on unit circle since X 2 t + Y 2 t A Bsic Stock Model It ws ssumed tht stock price follows geometric Brownin motion, nd stochstic finncil mthemtics ws then founded bsed on this ssumption. This pper presents n lterntive ssumption tht stock price follows geometric C process. Bsed on this ssumption, it is expected to reconsider option pricing, optiml stopping, portfolio selection nd so on, thus producing totlly new fuzzy finncil mthemtics. This pper proposes bsic stock model for fuzzy finncil mrket in which the bond price X t nd the stock price Y t follow { Xt = X exp(rt) (19) Y t = Y exp(et + σc t ) or equivlently { dxt = rx t dt dy t = ey t dt + σy t dc t (2) where r is the riskless interest rte, e is the stock drift, σ is the stock diffusion, nd C t is stndrd C process. It is just fuzzy counterprt of Blck-Scholes stock model [1]. This model my lso be extended to the cses of multifctor nd multi-stock.

8 1 B. Liu: Fuzzy Process, Hybrid Process nd Uncertin Process 3.5 Fuzzy Differentil Eqution with Jumps In mny cses the stock price is not continuous becuse of economic crisis or wr. In order to incorporte those into stock model, we should develop fuzzy clculus with jump process. For mny pplictions, fuzzy renewl process N t is sufficient. The fuzzy integrl of fuzzy process X t with respect to N t is X t dn t = lim k X ti (N ti+1 N ti ) = X t (N t N t ). (21) Definition 16 Suppose C t is stndrd C process, N t is fuzzy renewl process, nd f, g, λ re some given functions. Then dx t = f(t, X t )dt + g(t, X t )dc t + λ(t, X t )dn t (22) is clled fuzzy differentil eqution with jumps. A solution is fuzzy process X t tht stisfies (22) identiclly in t. t b Exmple 9: Let C t be stndrd C process nd N t fuzzy renewl process. Then the fuzzy differentil eqution with jumps dx t = dt + bdc t + cdn t hs solution X t = t + bc t + cn t which is just jump process. Exmple 1: Let C t be stndrd C process nd N t fuzzy renewl process. Then the fuzzy differentil eqution with jumps dx t = X t dt + bx t dc t + cx t dn t hs solution X t = exp (t + bc t + cn t ) which my be employed to model stock price with jumps. 4 Hybrid Process Definition 17 Let T be n index set, nd (Θ, P, Cr) (Ω, A, Pr) chnce spce. A hybrid process is mesurble function from T (Θ, P, Cr) (Ω, A, Pr) to the set of rel numbers, i.e., for ech t T nd ny Borel set B of rel numbers, the set {(θ, ω) Θ Ω X(t, θ, ω) B} is n event. Tht is, hybrid process X(t, θ, ω) is function of three vribles such tht the function X(t, θ, ω) is hybrid vrible for ech t. For ech fixed (θ, ω ), the function X(t, θ, ω ) is clled smple pth of the hybrid process. A hybrid process X(t, θ, ω) is sid to be smple-continuous if the smple pth is continuous for lmost ll (θ, ω). Insted of longer nottion X(t, θ, ω), sometimes we use the symbol X t. Definition 18 A hybrid process X t is sid to hve independent increments if X t1 X t, X t2 X t1,, X tk X tk 1 re independent hybrid vribles for ny times t < t 1 < < t k. A hybrid process X t is sid to hve sttionry increments if, for ny given t >, the X s+t X s re identiclly distributed hybrid vribles for ll s >. Exmple 11: Let X t be fuzzy process nd let Y t be stochstic process. Then X t + Y t is hybrid process. Exmple 12: (Hybrid Renewl Process) Let iid positive hybrid vribles ξ 1, ξ 2, denote the interrrivl times of successive events. Define S = nd S n = ξ 1 + ξ ξ n for n 1. Then S n cn be regrded s the witing time until the occurrence of the nth event. For ny t >, let N t be the number of renewls in (, t], i.e., { N t = mx n Sn t }. (23) n It is cler tht N t is hybrid process, nd we cll it hybrid renewl process. Ech smple pth of N t is right-continuous nd incresing step function tking only integer vlues. Since N t n if nd only if S n t, we hve Ch{N t n} = Ch{S n t}, E[N t ] = Ch{S n t}. (24) n=1

9 Journl of Uncertin Systems, Vol.2, No.1, pp.3-16, D Process Definition 19 Let B t be Brownin motion, nd let C t be C process. Then D t = (B t, C t ) is clled D process. The D process is sid to be stndrd if both B t nd C t re stndrd. Definition 2 Let B t be stndrd Brownin motion, nd let C t be stndrd C process. Then the hybrid process X t = et + σ 1 B t + σ 2 C t (25) is clled sclr D process. The prmeter e is clled the drift coefficient, σ 1 is clled the rndom diffusion coefficient, nd σ 2 is clled the fuzzy diffusion coefficient. Definition 21 Let B t be stndrd Brownin motion, nd let C t be stndrd C process. Then the hybrid process X t = exp(et + σ 1 B t + σ 2 C t ) is clled geometric D process. 4.2 Hybrid Clculus Let D t be stndrd D process, nd dt n infinitesiml time intervl. Then dd t = D t+dt D t = (db t, dc t ) is hybrid process. Definition 22 Let X t = (Y t, Z t ) where Y t nd Z t re sclr hybrid processes, nd let D t = (B t, C t ) be stndrd D process. For ny prtition of closed intervl [, b] with = t 1 < t 2 < < t k+1 = b, the mesh is written s = mx 1 i k t i+1 t i. Then the hybrid integrl of X t with respect to D t is X t dd t = lim provided tht the limit exists in men squre nd is hybrid vrible. k ( Yti (B ti+1 B ti ) + Z ti (C ti+1 C ti ) ) (26) Remrk 4: The hybrid integrl my lso be written s follows, X t dd t = (Y t db t + Z t dc t ). (27) Exmple 13: Let B t be stndrd Brownin motion, nd C t stndrd C process. Then (σ 1 db t + σ 2 dc t ) = σ 1 B s + σ 2 C s where σ 1 nd σ 2 re constnts, rndom vribles, fuzzy vribles, or hybrid vribles. Exmple 14: Let B t be stndrd Brownin motion, nd C t stndrd C process. Then (B t db t + C t dc t ) = 1 2 (B2 s s + C 2 s ), (C t db t + B t dc t ) = B s C s. Theorem 4 Let B t be stndrd Brownin motion, C t stndrd C process, nd h(t, b, c) twice continuously differentible function. Define X t = h(t, B t, C t ). Then we hve the following chin rule dx t = h t (t, B t, C t )dt + h b (t, B t, C t )db t + h c (t, B t, C t )dc t h 2 b 2 (t, B t, C t )dt. (28)

10 12 B. Liu: Fuzzy Process, Hybrid Process nd Uncertin Process Proof: Since the function h is twice continuously differentible, by using Tylor series expnsion, the infinitesiml increment of X t hs second-order pproximtion X t = h t (t, B t, C t ) t + h b (t, B t, C t ) B t + h c (t, B t, C t ) C t h 2 t 2 (t, B t, C t )( t) h 2 b 2 (t, B t, C t )( B t ) h 2 c 2 (t, B t, C t )( C t ) h t b (t, B t, C t ) t B t + 2 h t c (t, B t, C t ) t C t + 2 h b c (t, B t, C t ) B t C t. Since we cn ignore the terms ( t) 2, ( C t ) 2, t B t, t C t, B t C t nd replce ( B t ) 2 with t, the chin rule is obtined becuse it mkes h s X s = X + t dt + h s b db h t + c dc t h 2 b 2 dt for ny s. Remrk 5: The infinitesiml increments db t nd dc t in (28) my be replced with the derived D process dy t = u t dt + v 1t db t + v 2t dc t (29) where u t nd v 2t re bsolutely integrble hybrid processes, nd v 1t is squre integrble hybrid process, thus producing dh(t, Y t ) = h t (t, Y t)dt + h b (t, Y t)dy t h 2 b 2 (t, Y t)v1tdt. 2 (3) 4.3 Hybrid Differentil Eqution Definition 23 Suppose B t is stndrd Brownin motion, C t is stndrd C process, nd f, g 1, g 2 re some given functions. Then dx t = f(t, X t )dt + g 1 (t, X t )db t + g 2 (t, X t )dc t (31) is clled hybrid differentil eqution. A solution is hybrid process X t tht stisfies (31) identiclly in t. Exmple 15: Let B t be stndrd Brownin motion, nd let ã nd b be two fuzzy vribles. Then the hybrid differentil eqution dx t = ãdt + bdb t hs solution X t = ãt + bb t. The hybrid differentil eqution dx t = ãx t dt + bx t db t hs solution (( ) ) t + bb t X t = exp ã b 2 2. Exmple 16: Let C t be stndrd C process, nd let ξ nd η be two rndom vribles. Then the hybrid differentil eqution dx t = ξdt + ηdc t hs solution X t = ξt + ηc t. The hybrid differentil eqution dx t = ξx t dt + ηx t dc t hs solution X t = exp (ξt + ηc t ). Exmple 17: Let B t be stndrd Brownin motion, nd C t stndrd C process. Then the hybrid differentil eqution dx t = dt + bdb t + cdc t hs solution X t = t + bb t + cc t which is just sclr D process. Exmple 18: Let B t be stndrd Brownin motion, nd C t stndrd C process. Then the hybrid differentil eqution dx t = X t dt + bx t db t + cx t dc t hs solution which is just geometric D process. (( ) ) X t = exp b2 t + bb t + cc t 2

11 Journl of Uncertin Systems, Vol.2, No.1, pp.3-16, A Bsic Stock Model This pper ssumes tht stock price follows geometric D process, nd presents bsic stock model in which the bond price X t nd the stock price Y t re determined by { dxt = rx t dt dy t = ey t dt + σ 1 Y t db t + σ 2 Y t dc t (32) where r is the riskless interest rte, e is the stock drift, σ 1 is the rndom stock diffusion, σ 2 is the fuzzy stock diffusion, B t is stndrd Brownin motion, nd C t is stndrd C process. This model my lso be extended to the cses of multifctor nd multi-stock. 4.5 Hybrid Differentil Eqution with Jumps Let N t be hybrid renewl process. Then the hybrid integrl of hybrid process X t with respect to N t is X t dn t = lim k X ti (N ti+1 N ti ) = X t (N t N t ). (33) Definition 24 Suppose B t is stndrd Brownin motion, C t is stndrd C process, N t is hybrid renewl process, nd f, g 1, g 2, λ re some given functions. Then t b dx t = f(t, X t )dt + g 1 (t, X t )db t + g 2 (t, X t )dc t + λ(t, X t )dn t (34) is clled hybrid differentil eqution with jumps. A solution is hybrid process X t tht stisfies (34) identiclly in t. Exmple 19: Let B t be stndrd Brownin motion, C t stndrd C process, nd N t hybrid renewl process. Then the hybrid differentil eqution dx t = dt + bdb t + cdc t + λdn t hs solution X t = t + bb t + cc t + λn t which is just jump process. Exmple 2: Let B t be stndrd Brownin motion, C t stndrd C process, nd N t hybrid renewl process. Then the hybrid differentil eqution dx t = X t dt + bx t db t + cx t dc t + λn t (( ) ) hs solution X t = exp b2 2 t + bb t + cc t + λn t which my be employed to model stock price with jumps. 5 Uncertin Process Definition 25 Let T be n index set nd let (Γ, L, M) be n uncertinty spce. An uncertin process is mesurble function from T (Γ, L, M) to the set of rel numbers, i.e., for ech t T nd ny Borel set B of rel numbers, the set {γ Γ X(t, γ) B} is n event. Tht is, n uncertin process X(t, γ) is function of two vribles such tht the function X(t, γ) is n uncertin vrible for ech t. For ech fixed γ, the function X(t, γ ) is clled smple pth of the uncertin process. An uncertin process X(t, γ) is sid to be smple-continuous if the smple pth is continuous for lmost ll γ. Insted of longer nottion X(t, γ), sometimes we use the symbol X t. Definition 26 An uncertin process X t is sid to hve independent increments if X t1 X t, X t2 X t1,, X tk X tk 1 re independent uncertin vribles for ny times t < t 1 < < t k. An uncertin process X t is sid to hve sttionry increments if, for ny given t >, the increments X s+t X s re identiclly distributed uncertin vribles for ll s >.

12 14 B. Liu: Fuzzy Process, Hybrid Process nd Uncertin Process Exmple 21: (Uncertin Renewl Process) Let iid positive uncertin vribles ξ 1, ξ 2, denote the interrrivl times of successive events. Define S = nd S n = ξ 1 + ξ ξ n for n 1. Then S n cn be regrded s the witing time until the occurrence of the nth event. For ny t >, let N t be the number of renewls in (, t], i.e., { N t = mx n Sn t }. (35) n It is cler tht N t is n uncertin process, nd we cll it n uncertin renewl process. Ech smple pth of N t is right-continuous nd incresing step function tking only integer vlues. It is esy to verify tht 5.1 Cnonicl Process M{N t n} = M{S n t}, E[N t ] = Definition 27 An uncertin process W t is sid to be cnonicl process if (i) W = nd W t is smple-continuous, (ii) W t hs sttionry nd independent increments, (iii) W 1 is n uncertin vrible with expected vlue nd vrince 1. Theorem 5 (Existence Theorem) There is cnonicl process. M{S n t}. (36) Proof: In fct, stndrd Brownin motion nd stndrd C process re instnces of cnonicl process. Theorem 6 Let W t be cnonicl process. Then E[W t ]. Proof: Let f(t) = E[W t ]. Then for ny times t 1 nd t 2, by using the property of sttionry nd independent increments, we hve f(t 1 + t 2 ) = E[W t1+t 2 ] = E[W t1+t 2 W t2 + W t2 W ] = E[W t1 ] + E[W t2 ] = f(t 1 ) + f(t 2 ) which implies tht there is constnt e such tht f(t) = et. The theorem is proved vi f(1) =. Definition 28 Let W t be cnonicl process. Then et + σw t is clled derived cnonicl process, nd the uncertin process X t = exp(et + σw t ) is clled geometric cnonicl process. 5.2 Uncertin Clculus Let W t be cnonicl process, nd dt n infinitesiml time intervl. Then dw t = W t+dt W t is n uncertin process with E[dW t ] = nd dt 2 E[dW 2 t ] dt. Definition 29 Let X t be n uncertin process nd let W t be cnonicl process. For ny prtition of closed intervl [, b] with = t 1 < t 2 < < t k+1 = b, the mesh is written s = mx 1 i k t i+1 t i. Then the uncertin integrl of uncertin process X t with respect to W t is X t dw t = lim provided tht the limit exists in men squre nd is n uncertin vrible. n=1 k X ti (W ti+1 W ti ) (37) Remrk 6: Note tht the subscript of X ti is the left end point of intervl [t i, t i+1 ]. The integrl does not remin unchnged if the subscript tkes other points, sy, the right end point t i+1 or the middle point (t i + t i+1 )/2.

13 Journl of Uncertin Systems, Vol.2, No.1, pp.3-16, Theorem 7 Let W t be cnonicl process, nd let h(t, w) be twice continuously differentible function. Define X t = h(t, W t ). Then we hve the following chin rule dx t = h t (t, W t)dt + h w (t, W t)dw t h w 2 (t, W t)dwt 2. (38) Proof: Since the function h is twice continuously differentible, by using Tylor series expnsion, the infinitesiml increment of X t hs second-order pproximtion X t = h t (t, W t) t + h w (t, W t) W t h w 2 (t, W t)( W t ) h t 2 (t, W t)( t) h t w (t, W t) t W t. Since we cn ignore the terms ( t) 2 nd t W t, the chin rule is proved becuse it mkes for ny s. h s X s = X + t (t, W h t)dt + w (t, W t)dw t h 2 w 2 (t, W t)dwt 2 Remrk 7: The infinitesiml increment dw t in (38) my be replced with the derived cnonicl process dy t = u t dt + v t dw t (39) where u t is n bsolutely integrble uncertin process, nd v t is squre integrble uncertin process, thus producing dh(t, Y t ) = h t (t, Y t)dt + h w (t, Y t)dy t h 2 w 2 (t, Y t)vt 2 dwt 2. (4) Theorem 8 (Integrtion by Prts) Suppose tht W t is cnonicl process nd F (t) is n bsolutely continuous function. Then F (t)dw t = F (s)w s Proof: By defining h(t, W t ) = F (t)w t nd using the chin rule, we get d(f (t)w t ) = W t df (t) + F (t)dw t. W t df (t). (41) Thus which is just (41). F (s)w s = d(f (t)w t ) = W t df (t) + F (t)dw t 5.3 Uncertin Differentil Eqution Definition 3 Suppose W t is cnonicl process, nd f nd g re some given functions. Then dx t = f(t, X t )dt + g(t, X t )dw t (42) is clled n uncertin differentil eqution. A solution is n uncertin process X t tht stisfies (42) identiclly in t. Exmple 22: Let W t be smple-continuous uncertin process. Then the uncertin differentil eqution dx t = dt + bdw t hs solution X t = t + bw t.

14 16 B. Liu: Fuzzy Process, Hybrid Process nd Uncertin Process 5.4 Uncertin Differentil Eqution with Jumps Let N t be n uncertin renewl process. Then the uncertin integrl of uncertin process X t with respect to N t is k X t dn t = lim X ti (N ti+1 N ti ) = X t (N t N t ). (43) Definition 31 Suppose W t is cnonicl process, N t is n uncertin renewl process, nd f, g, λ re some given functions. Then dx t = f(t, X t )dt + g(t, X t )dw t + λ(t, X t )dn t (44) is clled n uncertin differentil eqution with jumps. A solution is n uncertin process X t tht stisfies (44) identiclly in t. We my ssume tht stock price follows geometric cnonicl process with jumps, nd obtin the following stock model in which the bond price X t nd the stock price Y t re determined by { dxt = rx t dt dy t = ey t dt + σy t dw t + λy t dn t (45) where r is the riskless interest rte, e is the stock drift, σ is the stock diffusion, nd λ is the jump coefficient. Acknowledgments t b This work ws supported by Ntionl Nturl Science Foundtion of Chin Grnt No References [1] Blck, F. nd M. Scholes, The pricing of option nd corporte libilities, Journl of Politicl Economy, vol.81, pp , [2] Choquet, G., Theory of cpcities, Annls de l Institute Fourier, vol.5, pp , [3] Kwkernk, H., Fuzzy rndom vribles I: Definitions nd theorems, Informtion Sciences, vol.15, pp.1-29, [4] Kwkernk, H., Fuzzy rndom vribles II: Algorithms nd exmples for the discrete cse, Informtion Sciences, vol.17, pp , [5] Li, X. nd B. Liu, A sufficient nd necessry condition for credibility mesures, Interntionl Journl of Uncertinty, Fuzziness & Knowledge-Bsed Systems, vol.14, no.5, pp , 26. [6] Li, X. nd B. Liu, Chnce mesure for hybrid events with fuzziness nd rndomness, Soft Computing, to be published. [7] Liu, B., Theory nd Prctice of Uncertin Progrmming, Physic-Verlg, Heidelberg, 22. [8] Liu, B. nd Y. K. Liu, Expected vlue of fuzzy vrible nd fuzzy expected vlue models, IEEE Trnsctions on Fuzzy Systems, vol.1, no.4, pp , 22. [9] Liu, B., Uncertinty Theory, Springer-Verlg, Berlin, 24. [1] Liu, B., A survey of credibility theory, Fuzzy Optimiztion nd Decision Mking, vol.5, no.4, pp , 26. [11] Liu, B., Uncertinty Theory, 2nd ed., Springer-Verlg, Berlin, 27. [12] Sugeno, M., Theory of Fuzzy Integrls nd its Applictions, Ph.D. Disserttion, Tokyo Institute of Technology, [13] Zdeh, L. A., Fuzzy sets, Informtion nd Control, vol.8, pp , [14] Zho, R. nd B. Liu, Renewl process with fuzzy interrrivl times nd rewrds, Interntionl Journl of Uncertinty, Fuzziness & Knowledge-Bsed Systems, vol.11, no.5, pp , 23.