Stochastic differential equations with fuzzy drift and diffusion

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Stochastic differential equations with fuzzy drift and diffusion Bj¨orn Sprungk, K. Gerald van den Boogaart

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Cite this article as: Bj¨orn Sprungk and K. Gerald van den Boogaart, Stochastic differential equations with fuzzy drift and diffusion, Fuzzy Sets and Systems, http://dx.do i.org/10.1016/j.fss.2013.02.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Stochastic Differential Equations with fuzzy Drift and Diffusion Bj¨orn Sprungka , K. Gerald van den Boogaartb a

TU Bergakademie Freiberg, Fakult¨ at f¨ ur Mathematik und Informatik, Institut f¨ ur Numerische Mathematik und Optimierung, 09596 Freiberg, Germany b TU Bergakademie Freiberg, Fakult¨ at f¨ ur Mathematik und Informatik, Institut f¨ ur Stochastik, 09596 Freiberg, Germany

Abstract A new framework for the fuzzification of stochastic differential equations is presented. It allows for a detailed description of the model uncertainty and the non-predictable stochastic law of natural systems, e.g. in ecosystems even the probability law of the random dynamic changes due to unobservable influences like anthropogenic disturbances or climate variation. The fuzziness of the stochastic system is modelled by a fuzzy set of stochastic differential equations which is identified with a fuzzy set of initial conditions, timedependent drift and diffusion functions. Using appropriate function spaces the extension principle leads to a consistent theory providing fuzzy solutions in terms of fuzzy sets of processes, fuzzy states, fuzzy moments and fuzzy probabilites. Keywords: Fuzzy stochastic differential equations, fuzzy stochastic processes, model uncertainty

Preprint submitted to Fuzzy Sets and Systems

February 28, 2013

Stochastic Differential Equations with fuzzy Drift and Diffusion Bj¨orn Sprungka , K. Gerald van den Boogaartb a

TU Bergakademie Freiberg, Fakult¨ at f¨ ur Mathematik und Informatik, Institut f¨ ur Numerische Mathematik und Optimierung, 09596 Freiberg, Germany b TU Bergakademie Freiberg, Fakult¨ at f¨ ur Mathematik und Informatik, Institut f¨ ur Stochastik, 09596 Freiberg, Germany

1. Introduction Stochastic differential equations (SDEs) dXt = f (t, Xt ) dt + g(t, Xt ) dWt , X0 = ξ(ω)

(1)

have become a powerful tool to model processes arising in nature, engineering or economic sciences which are not deterministic, but subject to random fluctuation (e.g. [1, 2]). Random disturbances on the modelled system are represented by a stochastic differential term g(t, X) dW , where W denotes a multivariate Brownian motion and g(t, Xt ) dWt is understood in the sense of an Itˆo integral (see [1, 2, 3]). But still there remains a ”structural uncertainty”, fluctuations of the predicted behaviour of the system due to uncertain values or even unknown nonrandom fluctuations of model parameters. For example in stochastic models of ecosystems long term parameters like the capacity for given species might vary due to climate change. In this paper we will present a fuzzy based framework for model and parameter uncertainty in stochastic differential equations. Like for fuzzy (ordinary) differential equations there already exist several approaches. Jung and Kim [4] defined a stochastic integral for set-valued stochastic processes which is used in [5] to define fuzzy set-valued stochastic differential equations. Unfortunately this approach leads to unbounded fuzzy setvalued random variables, since set-valued stochastic integrals are typically unbounded (see [6, 7]). Thereon, Ogura [8], Li and Li [9] and Zhang et al. [7, 10] presented approaches for set-valued and fuzzy set-valued SDEs but with single-valued diffusion terms g(t, Xt ) in order to get bounded results. Preprint submitted to Fuzzy Sets and Systems

February 28, 2013

However, single-valued diffusions of fuzzy-set valued processes seem not to be an appropriate tool for modelling parameter uncertainty, e.g. in economics this would correspond to a single, precisely known volatility of a fuzzy option price, which is indeed the weak point of economic modelling. Another approach to SDEs with set-valued coefficients is the theory of stochastic inclusions (see [11, 12, 13]). There, the set of solutions is given by all processes X whose increments are in the closure of the corresponding Aumann integrals: Z t  Z t Xt − Xs ∈ cl F (r, Xr ) dr + G(r, Xr ) dWr ) . (2) s

s

Here W is one-dimensional, F : [0, T ] × Rn → Kk (Rn ) and G : [0, T ] × Rn → Kk (Rn ) are compact set-valued mappings and the integrals are defined as Aumann integrals. This theory maps set-valued coefficients to sets of processes and avoids the unboundedness-problem of the approaches mentioned before. We will present a much simpler but also more flexible approach, which is in substance the following: go back to the classical extension principle, carry out some sensitivity analysis for SDE models, use fuzzy sets of inputs (or data) and ask for the fuzzy set of outputs (or solutions). If this operation is continuous in a well-defined sense, bounded fuzzy data cannot lead to unbounded fuzzy solutions. In particular, our approach allows to describe more complicated types of uncertainty, e.g. dependencies between drift and the diffusion coefficients, changing model parameters or fuzzy information about range and speed of change of time depending parameters. This article is organized as follows. In section 2 we state the necessary basics of stochastic processes and stochastic differential equations. In section 3 our approach to SDEs with fuzzy initial conditions and coefficients is presented. Therefore, a theory of fuzzy sets of processes is discussed and some properties of the solutions of fuzzy sets of SDEs, so called fuzzy SDEs, are stated. Moreover, it is shown how SDEs depending on fuzzy parameters fit into our concept of fuzzy SDEs. In section 4 the presented approach is applied to an illustrative problem from stochastic population dynamics and in section 5 some conclusions are drawn.

2

2. Stochastic processes and differential equations Throughout this paper (E, dE ) and (E, k · kE ) denote a complete metric and Banach space, respectively, and [0, T ] a finite time interval. Further, k · k stands for the euclidean norm in Rn or the Frobenius norm in Rn×r and ν for the Lebesgue measure in Rn . The Borel-σ-algebras of Rn and [0, T ] are denoted by B(Rn ) and B([0, T ]), respectively. 2.1. Preliminaries In the following we assume a given complete and filtered probability space (Ω, F, Ft , P) where {Ft }t∈[0,T ] is a right-continuous filtration, i.e. a family of sub-σ-algebras of F which fulfills Fs ⊆ Ft for s ≤ t and where F0 contains all P-null sets of F. The usual Lebesgue space of p-integrable, F-measurable random variables (1 ≤ p < ∞) is denoted by Lp (Ω, F, P; Rn ) or shortened Lp (Rn ). We will consider Rn -valued measurable stochastic processes X = {Xt }t∈[0,T ] over the interval [0, T ]. A process is called Ft -adapted, if for all t ∈ [0, T ] the random variable Xt is Ft -measurable. The Lebesgue space of all measurable, Ft -adapted processes X equipped with the norm Z kXkLpT :=

T

1/p E[kXt k ] dt p

(3)

0

is denoted further by LpT (Rn ). A process X is called a c`adl`ag process if its paths X(ω, ·) are P-a.s. c`adl`ag. Subsequently DTp (Rn ) denotes the Banach space of all c`adl`ag processes in LpT (Rn ) equipped with the norm #!1/p

" kXkDTp :=

E

sup kXt kp

.

(4)

t∈[0,T ]

Note, that processes in LpT (Rn ) are identified if they are modifications of each other, i.e. it holds P(Xt = Yt ) = 1 for any t ∈ [0, T ]. Moreover, if they are indistinguishable, i.e. P(∀t ∈ [0, T ] : Xt = Yt ) = 1 holds, they are identified in DTp (Rn ). A process X is called continuous in mean square (resp. continuous in the p-th mean) if it fulfills lims→t E[kXt − Xs k2 ] = 0 (resp. lims→t E[kXt − Xs kp ] = 0) for each t ∈ [0, T ]. Further, W denotes an r-dimensional Brownian motion w.r.t. Ft , i.e. it is an Ft -adapted Brownian motion where Wt −Ws is independent of Fs and W0 = 0 holds P-almost surely.

3

We consider Itˆo-SDEs of the form dXt = f (t, Xt ) dt + g(t, Xt ) dWt , X0 = ξ,

(5)

where f : [0, T ] × Rn → Rn and g : [0, T ] × Rn → Rn×r are measurable functions and ξ ∈ L2 (Rn ) is F0 -measurable. Hence, we will focus on strong solutions of SDEs: given a Brownian motion W w.r.t. Ft , a process X is called a strong solution of (5) if it has P-almost surely continuous paths, belongs to L2T (Rn ) and fulfills P-a.s. Z t Z t Xt = ξ + f (s, Xs ) ds + g(s, Xs ) dWs ∀t ∈ [0, T ]. (6) 0

0

The second integral in (6) is an Itˆo-integral. For a particular introduction into stochastic integration we refer to [2, 3]. Under certain conditions on f and g the existence of a unique strong solution of (5) is ensured. Theorem 2.1 ([2, 3, 14]). Let W be an r-dimensional Brownian motion w.r.t. Ft and let there be constants K, L > 0 such that kf (t, x) − f (t, y)k + kg(t, x) − g(t, y)k ≤ L kx − yk kf (t, x)k + kg(t, x)k ≤ K (1 + kxk) ,

(7) (8)

holds for all t ∈ [0, T ]. Furthermore, let ξ ∈ L2p (Rn ), p ≥ 1, be F0 measurable. Then there exists a unique strong solution X ∈ DT2p (Rn ) of (5). Moreover, X is continuous in mean square and it holds that " #    

E sup kXt k2p ≤ C1 E kξk2p + 1 + E kξk2p T p exp (C2 T ) , (9) t∈[0,T ]

where the constants C1 , C2 depend only on p, K, L and T . By means of (9) and the Gronwall-Bellman inequality the continuity in mean square of solutions of SDEs can be shown. Theorem 2.2 ([1, 14]). Let f and g satisfy the conditions (7) and (8), let ξ ∈ L2p (Rn ) be F0 -measurable and let X be the strong solution of (5). Then, there holds   E kXt − Xs k2p ≤ C|t − s|p ∀s, t ∈ [0, T ], (10) where C depends only on p, K, T and E[kξk2p ]. 4

2.2. Metric spaces of drift and diffusion coefficients In order to develop a framework for modelling uncertain coefficients of an SDE we define the following spaces: Definition 2.3. Let KoT (Rn×r ) be the linear space of measurable functions f : [0, T ] × Rn → Rn×r which are Lipschitz-continuous w.r.t. the second variable and fulfill kf (t, x)k kf kKoT := sup < ∞. (11) t,x 1 + kxk I.e. for each f ∈ KoT (Rn×r ) exists a positive constant Lf < ∞ such that kf (t, x) − f (t, y)k ≤ Lf kx − yk ∀x, y ∈ Rn ∀t ∈ [0, T ]

(12)

holds. Furthermore, for a given L < ∞ let KoT,L (Rn×r ) be the set of all f ∈ KoT (Rn×r ) which fulfill kf (t, x) − f (t, y)k ≤ Lkx − yk ∀x, y ∈ Rn ∀t ∈ [0, T ].

(13)

The linear spaces KoT (Rn ) = KoT (Rn×1 ) and KoT (Rn×r ) consist of all drift and diffusion coefficients, respectively, fulfilling the assumptions of the existence theorem (Theorem 2.1). Therefore, these are appropriate coefficient spaces. However, KoT (Rn×r ) is not complete w.r.t. k · kKoT . The sets KoT,L (Rn×r ) are not linear spaces, but they are complete w.r.t. the metric induced by k · kKoT : dKoT (f, g) := kf − gkKoT . In the following we will work in the metric spaces KoT,L (Rn×r ) rather than in KoT (Rn×r ), but for the sake of clearity we will state results always in terms of k · kKoT . Fuzziness will thus be introduced by fuzzy sets in KoT (Rn×r ). If the fuzzy set only contains functions constant in the first argument, that models non-changing behavior, like having a true unkown parameter. If it contains functions varying over time, this models a non-predictable fuzzy behavior of the system. We identify an SDE (5) by the triple (ξ, f, g) consisting of its initial condition ξ, its drift f and its diffusion coefficient g, and construct the corresponding product space: Definition 2.4. For 2 ≤ p < ∞ the product space DatpT (Rn ) := Lp (Ω, F0 , P ; Rn ) × KoT (Rn ) × KoT (Rn×r )

5

(14)

is called data space and is equipped with the product space norm
1/p k(ξ, f, g)kDatpT := kξkpLp + kf kpKoT + kgkpKoT .

(15)

Further, let the metric space DatpT,L (Rn ) := Lp (Ω, F0 , P ; Rn ) × KoT,L (Rn ) × KoT,L (Rn×r )

(16)

be equipped with the metric induced by k · kDatpT . The space Dat2T (Rn ) consists of all tripel (ξ, f, g) which fulfill the assumptions of Theorem 2.1. Therefore, the data space can be understood as the domain of the following solution operator X: Definition 2.5. Let W be an r-dimensional Brownian motion w.r.t. Ft . The operator X : Dat2T (Rn ) → DT2 (Rn ) (17) is defined by X(ξ, f, g) := X where X is the strong solution of the SDE (ξ, f, g). Subsequently, we will often consider restrictions of the operator X on certain subsets of Dat2T (Rn ). We will do so by stating the respective restricted domains explicitly but still using the symbol X for the restricted operator. The continuous dependence of strong solutions of SDEs on the initial data and the coefficients w.r.t. the L2T -norm is well known. However, one can show in particular the following continuity result: Theorem 2.6. Let L < ∞ and 1 ≤ p < ∞, p ∈ N. Then for each bounded n set S ⊂ Dat2p T,L (R ) it holds that kX(ξ1 , f1 , g1 ) − X(ξ2 , f2 , g2 )kD2p ≤ K k(ξ1 , f1 , g1 ) − (ξ2 , f2 , g2 )kDat2p , (18) T

T

for all (ξi , fi , gi ) ∈ S, i = 1, 2, where the constant K depends only on p, T, L 2p n n and S. In particular, the mapping X : Dat2p T,L (R ) → DT (R ) is locally Lipschitz-continuous. We will omit the proof here, since it is rather technical and can be obtained by adapting the proof of inequality (9) (see [14]) in an appropriate way. Note, that if the coefficients of an SDE are sufficiently smooth, one can even show a differentiable dependence of the strong solution on the coefficients (see [15]). 6

3. Fuzzy SDEs and fuzzy processes In this section we want to present our approach to fuzzy stochastic differential equations. First, we state some notations for fuzzy sets. 3.1. Fuzzy sets Given a complete metric space (E, dE ) a fuzzy set u˜ in E is a mapping u˜ : E → [0, 1] for which we denote the α-cuts by [˜ u]α := {x ∈ E : u˜(x) ≥ α} and the support by supp u˜ := clE {x ∈ E : u˜(x) > 0}. Here, clE A is the closure of A in E. We identify two fuzzy sets u˜ = v˜ if [˜ u]α = [˜ v ]α holds for all α ∈ (0, 1]. A fuzzy set u˜ is called bounded (resp. compact), if the support supp u˜ and the α-cuts [˜ u]α for α > 0 are non-empty, bounded and closed (resp. non-empty and compact). By F(E) and Fk (E) we denote the family of all bounded and compact fuzzy sets in E, respectively. We introduce the following metric in F(E): dF (˜ u, v˜) := sup dH ([˜ u]α , [˜ v ]α ),

(19)

α∈(0,1]

where dH denotes the Hausdorff-metric for subsets A, B ⊆ E:   dH (A, B) := max sup dE (x, B), sup dE (y, A) , x∈A

(20)

y∈B

dE (x, B) := inf y∈B dE (x, y). If (E, k · kE ) is a Banach space, we set further k˜ ukF := supx∈supp u˜ kxkE . Remark 3.1. Other common metrics for fuzzy sets are 1/p Z 1 p α α , p ∈ (1, ∞). dFp (˜ u, v˜) := dH ([˜ u] , [˜ v ] ) dα

(21)

0

It holds that dF (˜ u, v˜) ≥ dFp (˜ u, v˜) for u˜, v˜ ∈ F(E), see [16] for more details. Let now (E1 , dE1 ) and (E2 , dE2 ) be two complete metric spaces, f a mapping between them and u˜ a fuzzy set in E1 . Then, the fuzzy set f (˜ u) is defined by the extension principle ( supx∈E1 ,f (x)=y u˜(x), y ∈ f (supp u˜), f (˜ u)(y) := (22) 0, y∈ / f (supp u˜). Moreover, we denote by cl f (˜ u) the fuzzy set given by [cl f (˜ u)]α = clE2 [f (˜ u)]α for all α ∈ [0, 1]. 7

Theorem 3.2 ([17, Theorem 5]). Let (E1 , dE1 ), (E2 , dE2 ) be two complete metric spaces and let f : E1 → E2 be continuous. Then it holds for a compact fuzzy set u˜ ∈ Fk (E1 ) that f ([˜ u]α ) = [f (˜ u)]α ∀α ∈ (0, 1], f (supp u˜) = supp f (˜ u),

(23)

and therefore f (˜ u) ∈ Fk (E2 ). Barros et al. [17] showed the above theorem for E1 = E2 = Rn , but their proof can be adapted to general metric spaces without any modifications. 3.2. Fuzzy SDEs and fuzzy processes Instead of working with fuzzy set-valued coefficients or processes, the key point of our approach is to consider fuzzy sets of data and map them by the extension principle to fuzzy sets of solutions. Hence, we come up with the following definitions: Definition 3.3. A fuzzy stochastic differential equation ˜s is a bounded fuzzy set in the data space Dat2T (Rn ), i.e. ˜s ∈ F(Dat2T (Rn )). The (strong) solution ˜ of a fuzzy SDE ˜s is the fuzzy set X ˜ = cl X(˜s) in the space of c`adl`ag X 2 n processes DT (R ). ˜ ∈ F(Lp (Rn )) is called a fuzzy (Lp -)process Definition 3.4. A fuzzy set X T p ˜ ∈ F(D (Rn )) is a fuzzy c`adl`ag process. and a fuzzy set X T For a c`adl`ag process X ∈ DTp (Rn ) the projection πt : DTp (Rn ) → Lp (Rn ), πt (X) := Xt , is well-defined and, moreover, continuous and locally bounded w.r.t. X. Hence, we define the state of a fuzzy c`adl`ag process as follows: ˜ ∈ F(Dp (Rn )) and a fixed Definition 3.5. For a fuzzy c`adl`ag process X T ˜ t ∈ F(Lp (R)) is given by X ˜ t := cl πt (X). ˜ A fuzzy t ∈ [0, T ] the fuzzy set X p n p set Y˜ ∈ F(L (R )) is called a fuzzy (L -)random variable. Remark 3.6. Note, that this definition is different from the well-known definition of a fuzzy random variable as a measurable mapping X : Ω → F(Rn ) (see e.g. [18, 19, 20]). In this paper, we mean by fuzzy random variables, fuzzy processes, fuzzy SDEs etc. fuzzy sets of random variables, processes, SDEs et cetera.

8

3.3. Fuzzy random variables By applying the extension principle (22) rigorously the expectation, absolute moments, variance, covariance and conditional expectation of a fuzzy random variable can be constructed. We therefore consider the corresponding operators: E : L1 (Rn ) → Rn , E[k · kp ] : Lp (Rn ) → R, var : L2 (Rn ) → Rn , cov : L2 (Rn ) × L2 (Rn ) → Rn×n , E[· | C] : Lp (Rn ) → Lp (Ω, C, P; Rn ), where C denotes a sub-σ-algebra of F and p ≥ 1. Note, that all these mappings are continuous and locally bounded. ˜ ∈ F(Lp (Rn )), p ≥ 1, be a fuzzy random variable Definition 3.7. Let X ˜ ∈ F(Rn ) and and C a sub-σ-algebra of F. Then the fuzzy sets cl E[X] p ˜ ] ∈ F(R) are the expectation and the p-th absolute moment of X. ˜ cl E[kXk 2 n ˜ ˜ If X ∈ F(L (R )) , the variance and covariance of X are given by the fuzzy ˜ ∈ F(Rn ) and cl cov(X) ˜ ∈ F(Rn×n ). sets cl var (X) ˜ w.r.t. C If C is a sub-σ-algebra of F, then the conditional expectation of X ˜ | C] ∈ F(Lp (Ω, C, P ; Rn )). is the fuzzy random variable cl E[X Subsequently we denote the expectation, the variance etc. for fuzzy ran˜ var (X), ˜ and so on, i.e. we will omit the ”cl” dom variables simply by E[X], in the notation. By the same procedure, probabilites of events A ∈ B(Rn ) can be considered for fuzzy random variables. ˜ ∈ F(L1 (Rn )) be a fuzzy random Definition 3.8. Let A ∈ B(Rn ) and X ˜ ∈ A is the fuzzy set cl P(X ˜ ∈ A) given variable. Then the probability of X by the extension principle applied to the operator P(· ∈ A) : L1 (Rn ) → [0, 1]. Remark 3.9. The operator P(· ∈ A) : L1 (Rn ) → [0, 1] is not continuous in general. But, since convergence in Lp (Rn ) implies convergence in distribution, we get by the Portmanteau theorem (see [21, Theorem 3.25]) continuity of P(· ∈ A) on the subspace of random variables X ∈ L1 (Rn ) whose image measures P ◦ X −1 are absolutely continuous w.r.t. to the Lebesgue measure ν in Rn , as states of solutions of SDEs typically are. 3.4. Properties of solutions of fuzzy SDEs A fuzzy c`adl`ag process can be understood as a fuzzy set-valued mapping ˜ t ∈ F(Lp (Rn )). In order to justify the later use of time-discrete t 7→ X numerical approximations we examine:

9

˜ ∈ F(Dp (Rn )) is called timeDefinition 3.10. A fuzzy c`adl`ag process X T continuous, if for all t ∈ [0, T ]   ˜t, X ˜s = 0 lim dF X (24) s→t

holds. Note, that by the above investigations time-continuity of a fuzzy c`adl`ag ˜ implies the continuous dependence (on t) of the expectations, the process X absolute moments, the variances and the covariances of the fuzzy process ˜t. states X Although we have previously presented a framework for fuzzy processes, we have not shown yet if the solution of a fuzzy SDE is a fuzzy (c`adl`ag) process. n ˜ Theorem 3.11. Let ˜s ∈ F(Dat2p T,L (R )) for p ∈ N and L < ∞, then X = ˜ is moreover time-continuous. cl X(˜s) belongs to F(DT2p (Rn )) and X

˜ = X(˜s) has Proof. It follows from Theorem 2.6 with Theorem 3.2 that X a bounded support and, therefore, it is an element of F(DT2p (Rn )). The ˜ implies further, by (9) and (10), the existence of a boundedness of ˜s and X constant C < ∞ such that   ˜t, X ˜ s ≤ lim sup sup kXt − Xs kL2p lim dF X s→t

s→t α∈[0,1] ˜ α X∈[X]

≤ lim s→t

sup

kXt − Xs kL2p ≤ lim C |t − s|p = 0 s→t

˜ X∈supp X

˜ is time-continuous. holds. Therefore X Combining the continuity result from Theorem 2.6 with Theorem 3.2 leads, moreover, to an α-cut-wise construction of the solution of compact fuzzy SDEs. n Corollary 3.12. Let 1 ≤ p ∈ N, L < ∞ and ˜s ∈ Fk (Dat2p T,L (R )) be compact. ˜ = cl X(˜s) is a compact fuzzy set itself, i.e. X ˜ ∈ Fk (D2p (Rn )) and it Then X T

10

holds in particular ˜α [X] ˜ t ]α [X h iα ˜t] E[X h iα ˜ t kq ] E[kX

= {X : X = X(ξ, f, g), (ξ, f, g) ∈ [˜s]α }, α

(25)

= {Xt : X = X(ξ, f, g), (ξ, f, g) ∈ [˜s] },

(26)

= {E[Xt ] : X = X(ξ, f, g), (ξ, f, g) ∈ [˜s]α },

(27)

= {E[kXt kq ] : X = X(ξ, f, g), (ξ, f, g) ∈ [˜s]α }

(28)

for all α ∈ [0, 1], t ∈ [0, T ] and 1 ≤ q ≤ 2p. If, moreover, for each (ξ, f, g) ∈ supp ˜s it holds that 1. ξ ∈ Rn , f and g are bounded, ∂ 2 (gg T )ij ∂fi ∂(gg T )ij 2. the partial derivatives ∂x , and exist, are bounded and ∂x ∂xi ∂xj i i n H¨older-continuous on [0, T ] × R for i = 1, . . . , n and j = 1, . . . , r, 3. there is a constant c > 0 such that for all (t, x) ∈ [0, T ] × Rn and all y ∈ Rn the inequality y T · gg T (t, x) · y ≥ c kyk holds, then we have for an arbitrary A ∈ B(Rn ) ˜ t ∈ A)]α = {P (Xt ∈ A) : X = X(ξ, f, g), (ξ, f, g) ∈ [˜s]α } [P (X

(29)

for each α ∈ [0, 1]. The corollary above makes use of the classical result for the FokkerPlanck-equation which ensures under the assumptions 1 – 3 absolute continuous (w.r.t. ν in Rn ) image measures of the states Xt of X = X(ξ, f, g), see [3, Section 5.7.B] for details. Remark 3.13. The solutions of compact fuzzy SDEs ˜s ∈ Fk (Dat2T (Rn )) could be interpreted as fuzzy set-valued processes if the pathwise continuous dependence on the initial conditions and coefficients in supp ˜s could be shown. However, up to the authors knowledge, proofs for such a pathwise dependence exist only in special cases of diffusion coefficients (see e.g. [22]). Remark 3.14. There exists a strong connection of fuzzy SDEs and fuzzy stochastic inclusions (fuzzy SI) in the sense of Kisielewicz (see [11, 12, 13]). For each fuzzy SDE ˜s a corresponding fuzzy SI can be constructed such that the α-cuts of its solution set contains the α-cuts of cl X(˜s). Let ˜s ∈ 11

n ˜ ˜ Fk (Dat2p ˜ be the projections of ˜s T,L (R )) be a fuzzy SDE and let ξ, f and g 2p n n n×r onto L (Ω, F0 , P ; R ), KoT,L (R ) and KoT,L (R ), respectively, i.e.

˜ ξ(ξ) =

sup f ∈KoT,L

f˜(f ) = g˜(g) =

g∈KoT,L

sup ξ∈L2p (Ω,F

0

0

(Rn×r )

˜s(ξ, f, g)

sup ,P ;Rn )

sup ξ∈L2p (Ω,F

˜s(ξ, f, g)

sup

(Rn )

g∈KoT,L

∀f ∈ KoT,L (Rn ),

(Rn×r )

˜s(ξ, f, g)

sup ,P ;Rn )

∀ξ ∈ L2p (Ω, F0 , P ; Rn ),

f ∈KoT,L

∀g ∈ KoT,L (Rn×r ).

(Rn )

h i1 Assume that supp ξ˜ = ξ˜ = {ξ} and define fuzzy set-valued coefficients f , g by o \n α β ˜ [f (t, x)] := cl f (t, x) : f ∈ [f ] , β<α α

[g(t, x)]

:= cl

\ g(t, x) : g ∈ [˜ g ]β , β<α

˜ ξ (f , g) of the for any x ∈ Rn and t ∈ [0, T ]. Then the fuzzy set of solutions Λ ˜ = cl X(˜s), i.e. fuzzy stochastic inclusion SI(f , g) includes the fuzzy set X h iα h iα ˜ ⊆ Λ ˜ ξ (f , g) X ∀α ∈ [0, 1]. The converse is not true in general. A counterexample would be dXt = θXt dt + dWt , X(0) = 1, where θ ∈ [0, 1]: the set of solutions of the SI would contain also the solution of dXt = 1[0,T /2] (t)Xt dt + dWt , X(0) = 1, where 1 denotes the indicator function. Thus, SI is thus more inclusive. Moreover in our fuzzy SDE approach the uncertainty in the coefficients is handled more carefully than in the context of fuzzy SI, where only fuzzy set-valued coefficients are considered and no control of the type of uncertainty in the parameter is possible, e.g. of the speed of coefficient changes or constant fuzzy parameters. 3.5. SDEs with fuzzy parameters Let a parameter-dependent SDE dXt = f (t, Xt , θt ) dt + g(t, Xt , θt ) dW, X0 = ξ(θ0 ),

12

(30)

be given, where the parameter θ : [0, T ] → Rq may be time-dependent. Usually the knowledge about model parameters is incomplete, e.g. due to limited possibility to measure certain parameters, unknown behaviour or unmodelled processes effecting these parameters. Hence, parameter uncertainty should be included in the model in a fuzzy manner. This can be done in the framework of fuzzy SDEs. We assume a given metric parameter space (Θ, dΘ ), e.g. the linear space of c`adl`ag functions DT (Rq ) with norm kθkDT = supt∈[0,T ] kθt k. Further, we suppose a bounded fuzzy set θ˜ ∈ F(Θ) describing the possible parameter functions θ. In the following we denote by f (θ) the function (t, x) 7→ f (t, x, θt ). If
1. there exists an L < ∞ such that ξ(θ0 ), f (θ), g(θ) ∈ Dat2T,L (Rn ) holds ˜ for all θ ∈ supp θ,
2. the mapping Θ 3 θ 7→ ξ(θ0 ), f (θ), g(θ) ∈ Dat2T (Rn ) is continuous and ˜ bounded on supp θ, then the fuzzy set θ˜ can be mapped by the extension principle (22) to a ˜ = cl X(˜s ˜) ∈ fuzzy SDE ˜sθ˜ ∈ F(Dat2T,L (Rn )) and, hence, to a fuzzy set X θ F(DT2 (Rn )). If θ˜ is compact, we get again n o
˜α ˜ α = X : X = X ξ(θ0 ), f (θ), g(θ) , θ ∈ [θ] [X] (31) for all α ∈ [0, 1]. Remark 3.15. In the special case of constant parameters, i.e. a finiteˇ dimensional parameter space Θ, the Kolmogorov-Centsov theorem (see [21, Theorem 2.23]) can be applied to show pathwise continuous dependence of the solution of (30) on the parameter θ ∈ Rq . Therefore, the solution of a corresponding fuzzy SDE, which is generated by fuzzy constant parameters θ˜ ∈ F(Rq ), can be understood as a fuzzy set-valued process. We refer also to the work of B. Schmelzer [23], who studied the solution of SDEs depending on uncertain parameters θ ∈ Rq modelled by random sets. His approach constructs rouhly speaking a set-valued process, defined by the set of singe-valued processes which fullfil the SDE for some parameter in the random set. In particular in the case of a crisp set of time-independent parameters, both approaches give the same solution. Also for a random set A(ω) which is generated as the convexd hull of n random points a1 (ω), . . . , an (ω) ∈ Rq , one 13

could consider the crisp set of the P n − 1 dimensional simplex Pn−1 ∆n−1 and the n−1 mapping ∆n−1 3 (θ1 , . . . , θn−1 ) → i=1 θi ai (ω) + (1 − i=1 θi )an (ω) where a fuzzy SDE associated to the crisp set ∆n−1 of parameters would reproduce the solution of the corresponding SDE with parameters from the random set A(ω). 4. Numerical Example We want to conclude our paper with an example for the application of fuzzy SDEs in population dynamics, because the complex math hides how easy this approach can be used in pratice. 4.1. Problem setting Let the Lotka-Volterra system   k − Xt − a Yt Xt dt + σ1 Xt dWt1 , X0 = x0 ∈ R, x0 > 0 dXt = r k   (32) Yt 2 dYt = −m + bt Xt Yt dt + σ2 Yt dWt , Y0 = y0 ∈ R, y0 > 0, z + Yt be given, where W = (W 1 , W 2 ) is a 2-dimensional Brownian motion, r, k, a, σ1 , m, z, σ2 are positive constants and b : [0, T ] → R is a real-valued function. Under certain conditions the corresponding deterministic system admits two asymptotically stable equilibria, one of coexistence and one of predator extinction (Y = 0), as well as two instable equilibria. Therefore, perturbations in the parameters can lead to a totally different asymptotical behaviour of the system. We are further interested in the probability of predator extinction at certain times t. But, since for the solution (X, Y ) of (32) P(Yt = 0) = 0 holds for all t ∈ [0, T ], we consider the event Yt < ye as an adequate description for predator extinction for a sufficiently small ye . In natural systems like ecosystems, parameter may vary over time and are usually not known precisely. Therefore, we model two parameters of (32) as fuzzy: the time-dependent predator reproduction rate b : [0, T ] → R and the constant noise parameter σ2 ∈ R. Remark 4.1. Due to the non-linear terms X 2 , Y 2 , XY the SDE (32) does not fulfill the assumptions of Theorem 2.1. However, since the prey X is bounded by the capacity k, the probability that X and Y take very large 14

values are correspondingly small. Therefore, we can restrict the SDE (resp. its coefficients) to [0, T ] × K where K is a sufficiently large compact subset of R2 . Hence, we can apply Theorem 2.1 and the above presented approach of fuzzy SDEs. 4.2. Modelling of fuzzy parameters We assume a given triangular fuzzy number σ ˜2 for possible values of σ2 and a trapezoidal fuzzy set w˜ for the possible range of b. Natural values as average reproduction rates usually change slowly. Therefore, we restrict the ˜ for its Lipschitz constant possible behaviour of b by a trapezoidal fuzzy set L (see Figure 1). ~ σ

~ L

~ w 1

1

0

1

0 0.025

0.03

0.04

0.045

0 0

0.01

0.015

0.45

0.5

0.55

Figure 1: Fuzzy sets for modelling the uncertain parameters b and σ2 in (32).

We combine the fuzzy information about the range and behaviour of b by defining a fuzzy set ˜b in the space CT (R) of continuous functions f : [0, T ] → R:    |f (t) − f (s)| ˜b(f ) = min min w(f ˜ ˜ (t)), inf L . (33) t∈[0,T ] s6=t∈[0,T ] |t − s| Thus, it holds ˜b ∈ Fk (CT (R)). In particular, we have modelled a fuzzy set ˜b ˜ describing the of coefficient functions b by the two fuzzy numbers w ˜ and L range and the speed of change of the coefficient, respectively. Some coefficient functions b for certain α-cuts of ˜b are displayed in Figure 2. Remark 4.2. Looking at Figure 2 one can think of further ways or information to model the fuzzy set ˜b, e.g. by a third fuzzy number describing the number of turning points for b in the time interval [0, T ]. Using an arbitrary t-norm T we can further define a compact parameter ˜ x) = T(˜b(f ), σ fuzzy set θ(f, ˜2 (x)) in CT (R) × R, as required for applying the 15

b(t)

α=1

α = 0.5

α>0

0.05

0.05

0.05

0.045

0.045

0.045

0.04

0.04

0.04

0.035

0.035

0.035

0.03

0.03

0.03

0.025

0.025

0.025

0.02 0

5

10

0.02 0

5

t

t

10

0.02 0

5

10

t

Figure 2: Some members of different α-cuts of the fuzzy set of time-dependent predator reproduction rates ˜b.

presented approach. Since the coefficient functions of (32) depend continuously on b and σ2 , which can be easily checked, the compact parameter fuzzy set θ˜ can be mapped to a compact fuzzy set of SDEs and solution processes, respectively, and we get by Corollary 3.12 h  iα n o
˜α . P Y˜t < ye = P(Yt < ye ) : (X, Y ) = X ξ, f (b), g(σ2 ) , (b, σ2 ) ∈ [θ] (34) Remark 4.3. It can be checked, that the SDE (32) with fuzzy parameters ˜b, σ ˜2 fulfills the assumptions of Corollary 3.12 if supp σ ˜2 ⊂ (0, +∞) and σ1 > 0, taking into account that then a.s. (Xt , Yt ) ∈ (0, +∞)2 holds. 4.3. Results Figure 3 shows approximated upper and lower bounds of certain α-cuts of the fuzzy probability P(Y˜ < ye ). In this case we have used the minimum˜ The corresponding values of the t-norm to define the parameter fuzzy set θ. crisp and fuzzy model parameters are given in Remark 4.4. For computation we have drawn by Monte Carlo methods 1000 parameter pairs (b, σ2 ) from each of M = 11 ”α-strips” ˜ θ] ˜ α1 , S1 = [θ] ˜ α1 \[θ] ˜ α2 , . . . , SM −2 = [θ] ˜ αM −2 \[θ] ˜ 1 , SM −1 = [θ] ˜ 1, S0 = supp θ\[ where αj = j/(M − 1). For each of these parameter pairs we simulated the corresponding SDEs by the Milstein scheme [14] (10000 paths, time stepsize 0.01) and estimated the probabilities Pi = Pi (b, σ2 ) of Yti < ye , where the {ti }i=1,...,N were given by the time discretization of the Milstein scheme. The 16

range of results was than used as an approximation for the α-cuts of P(Y˜ti < ye ): ( ) h i αj [ min P(Y˜t < ye ) ≈ min Pi = Pi (b, σ2 ) : (b, σ2 ) ∈ Sk , (35) i

k≥j

analogously for the upper bound of [P(Y˜ti < ye )]αj . 0.7

Probability

0.6 0.5

α=1 α = 0.5 α>0

0.4 0.3 0.2 0.1 0 0

2

4

6

8

10

Time

Figure 3: Estimated upper and lower bounds for several α-cuts of P(Y˜t < ye ).

Remark 4.4. The results in Figure 3 have been computed for the values x0 = 10, y0 = 5, k = 25, r = 1, a = 0.2, σ1 = 0.25, m = 0.3, z = 0.5, ye = 0.5, T = 10 and the fuzzy data given in Figure 1. Remark 4.5. The presented Monte Carlo approach is just a first attempt for solving fuzzy SDEs. The computations have taken around 2 days on our workstation, but could probably be decreased significantly by massive parallelization. However, since for some quantities of interest, like expected values E[Yt ], computing the upper and lower bounds of the fuzzy sets E[Y˜t ] can be formulated as a stochastic control problem, a more advanced numerical method for solving fuzzy SDEs would be to apply algorithms from stochastic control theory. 5. Conclusions In this paper we have presented a new approach to fuzzy SDEs using fuzzy sets of functions, stochastic processes, random variables etc. instead 17

of fuzzy set-valued functions, processes and variables. This approach avoids like stochastic inclusions the problem of unbounded set-valued stochastic integrals. But it allows also for a more precise control of the fuzziness to be introduced to the stochastic model. For example, fuzzy information about the temporal behaviour of the dynamic of the system or dependency structures between the coefficients can be taken into account, since fuzzy sets in the product space Dat2T (Rn ) are considered. Despite the complicated spaces we have to work on, the mathematical arguments needed to create a complete consistent theory including fuzzy objects for the equations, for the solutions, for the states, for moments and for probabilities are simple and robust continuity arguments. Thus the approach provides an elegant and flexible way of modelling nonstochastic uncertainty about stochastic systems in a clean fuzzy framework. There remain questions about efficient numerical methods for fuzzy SDEs. The Monte Carlo technique, which has been used for the purpose of this paper, should just be seen as a first, basic idea. Further investigations on the relationships between fuzzy SDEs and other approaches such as fuzzy set-valued processes could also be fertile. Acknowledgements The authors would like to thank Wolfgang N¨ather and Andreas W¨ unsche for inspiring discussions and the reviewers for their very helpful remarks. [1] E. Allen, Modeling with Itˆo Stochastic Differential Equations, Springer, Dordrecht, 2007. [2] B. Øksendal, Stochastic Differential Equations, Springer, Berlin Heidelberg, sixth edition, 2003. [3] I. Karatzas, S. E. Shreve, Brownian Motion And Stochastic Calculus, Springer, New York, second edition, 1991. [4] E. J. Jung, J. H. Kim, On set-valued stochastic integrals, Stoch. Anal. and Appl. 21 (2003) 401–418. [5] J. H. Kim, On fuzzy stochastic differential equations, J. Korean Math. Soc. 42 (2005) 153–169.

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[6] Y. Ogura, On stochastic differential equations with set coefficients and the Black-Scholes model, in: Proceedings of the Eighth International Conference on Intelligent Technologies, Sydney, pp. 300–304. [7] J. Zhang, et al., On set-valued stochastic integrals in an m-type 2 Banach space, J. Math. Anal. Appl. 350 (2009) 216–233. [8] Y. Ogura, On stochastic differential equations with fuzzy set coefficients, in: D. Dubois, et. al. (Eds.), Soft Methods for Handling Variability and Imprecision, Selected papers from the 4th International Conference on Soft Methods in Probability and Statistics, volume 48 of Advances in Soft Computing, Springer, Toulouse, France, 2008. [9] J. Li, S. Li, Itˆo type set-valued stochastic differential equation, J. Uncertain Systems 3 (2009) 52–63. [10] J. Zhang, et al., On the solutions of set-valued stochastic differential equations in m-type 2 Banach spaces, Tohoku Math. J. 61 (2009) 417– 440. [11] M. Kisielewicz, Properties of solution set of stochastic inclusions, J. Appl. Math. and Stoch. Anal. 5 (1993) 217–236. [12] M. Kisielewicz, Existence theorem for nonconvex stochastic inclusions, J. Appl. Math. and Stoch. Anal. 7 (1994) 151–159. [13] M. Kisielewicz, Set-valued stochastic integrals and stochastic inclusions, Stoch. Anal. and Appl. 15 (1997) 783–800. [14] P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin Heidelberg, 1992. [15] T. S. Knudson, Mean square differentiability with respect to the coefficients of solutions of stochastic differential equations, Stochastics An Int. Journal of Prob. and Stoch. Proc. 58 (1996) 93–113. [16] P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific Publishing, Singapore, 1994. [17] L. C. Barros, R. C. Bassanezi, P. A. Tonelli, On the continuity of the Zadeh’s extension, in: Proc. IFSA ’97 Congress, Prague. 19

[18] S. Li, Y. Ogura, V. Kreinovich, Limit Theorems and Applications of Set-valued and Fuzzy Set-vauled Random Variables, Kluwer Academic Publishers, Dordrecht, 2002. [19] M. L. Puri, D. A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409–422. [20] V. Kr¨atschmer, A unified approach to fuzzy random variables, Fuzzy Sets and Systems 123 (2001) 1–9. [21] O. Kallenberg, Foundations of Modern Probability, Springer, New York, 1997. [22] T. S. Knudson, Continuity in a pathwise sense with respect to the coefficients of solutions of stochastic differential equations, Stochastic Processes and their Applications 68 (1997) 155–179. [23] B. Schmelzer, On solutions of stochastic differential equations with parameters modeled by random sets, Int. J. Approx. Reason 51 (2010) 1159–1171.

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