On Fliess Operators Driven by L2-Ito Random Processes - CiteSeerX

Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009

FrB05.1

On Fliess Operators Driven by L2 -Itˆo Random Processes Luis A. Duffaut Espinosa W. Steven Gray Oscar R. Gonz´alez

Abstract— Fliess operators with deterministic inputs have been studied since the late 1970’s and are well understood. When the inputs are stochastic processes the theory is less developed. There have been several interesting approaches for Wiener process inputs. But the interconnection of systems is not well-posed in this context, and this limits their use in applications. This paper has two specific goals. The first goal is to describe the theoretical framework under which a Fliess operator can be driven by a class of L2 -Itˆo random processes. The second goal is to derive a sufficient condition for the stochastic convergence of the series which defines the corresponding output process.

I. I NTRODUCTION Functional series expansions of nonlinear input-output operators have been utilized since the early 1900’s in engineering, mathematics and physics. Among the more representative approaches are those of M. Fliess [6]–[11], V. Volterra [25] and N. Wiener [26]. From a deterministic point of view, a broad class of nonlinear systems can be described by Fliess operators, which are input-output maps constructed using the Chen-Fliess formalism [3], [7]. Such an operator is comprised of a summation of Lebesgue iterated integrals codified using the theory of noncommutative formal power series. Specifically, let X = {x0 , x1 , . . . , xm } be an alphabet and X ∗ the free monoid comprised of all words over X (including the empty word ∅) under the catenation product. A formal power series in X is any mapping of the form X ∗ → Rℓ , and the set of all such mappings will be denoted by Rℓ hhXii. For each c ∈ Rℓ hhXii, one can formally associate an m-input, ℓ-output operator Fc in the following manner. Let p ≥ 1 and a < b be given. For a measurable function u : [a, b] → Rm , define kukp , max{kui kp : 1 ≤ i ≤ m}, where kui kp is the usual Lp -norm for a measurable real-valued function, ui , defined on [a, b]. Let Lm p [a, b] denote the set of all measurable functions defined on [a, b] having a finite k·kp -norm and Bpm (R)[a, b] , {u ∈ Lm p [a, b] : kukp ≤ R}. With t0 , T ∈ R fixed and T > 0, define recursively for each η ∈ X ∗ the mapping Eη : Lm 1 [t0 , t0 + T ] → C[t0 , t0 + T ] by E∅ = 1, and Zt (1) Exi η′ [u](t) , ui (s)Eη′ [u](s) ds, t0

This research was supported by the NASA Langley Research Center under grant NNX07AD52A. The authors are affiliated with the Department of Electrical and Computer Engineering, Old Dominion University, Norfolk, Virginia 23529-0246, USA. [email protected], {gray,gonzalez}@ece.odu.edu

978-1-4244-3872-3/09/$25.00 ©2009 IEEE

where xi ∈ X, η ′ ∈ X ∗ and u0 = 1. The input-output operator corresponding to c is then X (c, η) Eη [u](t), Fc [u](t) , η∈X ∗

which is called a Fliess operator. All Volterra operators with analytic kernels, for example, are Fliess operators. In the classical literature where these operators first appeared, it was normally assumed that there exist real numbers K, M > 0 such that |(c, η)| ≤ KM |η| |η|!, ∀ η ∈ X ∗ , where |z| = max{|z1 | , |z2 | , . . . , |zℓ |} when z ∈ Rℓ , and |η| denotes the number of symbols in η [7], [9], [10], [23]. This growth condition on the coefficients of c ensures that there exist positive real numbers R and T such that for all measurable u with kuk∞ ≤ R the series defining Fc converges uniformly and absolutely on [t0 , t0 +T ]. Such a power series c is said to be locally convergent. More recently, Gray and Wang showed in [14] that m Y Uiαi (t) , (2) |Eη [u](t)| ≤ αi ! i=0 Rt where for each xi , Ui (t) = t0 |ui (s)| ds, and αi = |η|xi is the number of times the letter xi appears in η. This bound can be used to show that Fc constitutes a well-defined operator from Bpm (R)[t0 , t0 + T ] into Bqℓ (S)[t0 , t0 + T ] for sufficiently small R, S, T > 0, where the numbers p, q ∈ [1, ∞] are conjugate exponents, i.e., 1/p + 1/q = 1 with (1, ∞) being a conjugate pair by convention. It also allows one to characterize well-posed interconnections of analytic nonlinear input-output systems [13]. In most applications, a system’s inputs usually have noise components. In such circumstances, additional mathematical machinery is needed to properly describe an inputoutput map in the sense of Fliess. Several authors have formulated approaches under which Wiener processes are admissible inputs. One example is the series expansion of the solution of a stochastic differential equation, where iterated Itˆo and Stratonovich integrals play a central role [12], [16]. Sussmann gave a detailed description of the situation using Lie series and showed that a particularly suitable mathematical formulation involves the use of Stratonovich integrals because they obey the rules of ordinary differential calculus [22], [24]. On the other hand, Itˆo integrals are still useful for computing estimates of process moments [1], [19]–[21]. Similar approaches have been presented in [1], [6], [8], [11], [12], [18]. It is easily verified, however, that the corresponding output process of a nonlinear input-output system is in general not a Wiener process. Hence, these approaches are not suitable for modeling interconnected systems. In this paper, a broader class of stochastic processes

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FrB05.1 known as L2 -Itˆo processes are considered as inputs [4], [16]. It is argued that this input class is more appropriate for practical applications. Then stochastic versions of (1) and (2) are defined using Lebesgue and Stratonovich integrals. In this context, the paper has two specific goals. The first goal is to describe the theoretical framework under which a Fliess operator can be driven by a subset of L2 -Itˆo random processes. The second goal is to derive a sufficient condition for the stochastic convergence of the series which defines the corresponding output process. The paper is organized as follows. Section II will introduce the basic stochastic tools and definitions used throughout the paper. In Section III, bounds for stochastic iterated integrals are introduced. Section IV extends the definition of a Fliess operator to admit L2 -Itˆo random processes and provides the main convergence result for these operators. II. P RELIMINARIES Consider a Wiener process, W (t), defined over a probability space (Ω, F , P ). For a predictable function u : Ω × [t0 , t0 + T ] → Rm let kukp = max{kui kLp : 1 ≤ i ≤ m}, where k·kLp is the usual norm on Lp (Ω×[t0 , t0 +T ], P, P ⊗ λ), the set of all predictable functions defined on [t0 , t0 + T ] having finite k·kLp -norm, P is the predictable algebra, and λ is the Lebesgue measure. The Stratonovich integral is defined in terms of an Itˆo integral. Definition 1: For a stochastic process v ∈ L2 (Ω×[t0 , t0 + T ], P, P ⊗ λ), the Stratonovich integral of v is defined by Zt Zt 1 S v(s) dW (s) , v(s) dW (s) + hv, W i[t0 ,t] , 2 t0

t0

where the quadratic covariation is hv, W i[t0 ,t] , lim

n→∞

n−1 X k=0

(v(tk+1 ) + v(tk )) (W (tk+1 ) − W (tk )) .

A well known property of Stratonovich integrals is that they obey the usual integration by parts formula. Definition 2: [4], [16] Let T > 0 and t0 be fixed. An mdimensional stochastic process w over [t0 , t0 + T ] is called an L2 -Itˆo process if it can be written as w(t) =

Zt

t0

a(s) ds +

Zt

b(s) dW (s),

t0

Lm 2 (Ω

where a, b ∈ × [t0 , t0 + T ], P, P ⊗ λ). The set of all L2 -Itˆo processes will be denoted by I . It is known that I is a subset of Lm 2 (Ω × [t0 , t0 + T ], P, P ⊗ λ). Definition 3: Consider the set of all m-dimensional stochastic processes over [t0 , t0 + T ], denoted by m f [t0 , t0 + T ], which can be written as UV w(t) =

Zt

t0

Zt u(s) ds + S v(s) dW (s)

for some u, v ∈ I . The latter are called the drift and diffusion inputs, respectively. Moreover, the subset UV m [t0 , t0 + m f [t0 , t0 + T ] will refer to all processes satisfying: T ] ⊂ UV i. The integrands u and v are such that kuk1 , kvk2 , kvk4 ≤ R ∈ R+ .

ii. Each integrand consists of m components such that E[ui (t)] < ∞, E[vi (t)] < ∞, t ∈ [t0 , t0 + T ], and the random variables ui (t1 ), ui (t2 ), vi (t1 ) and vi (t2 ) are independent for 1 ≤ i ≤ m and t1 6= t2 . Observe that since u, v ∈ I , then by Definition 1 any w ∈ m f [t0 , t0 + T ] is also an L2 -Itˆo process. UV To describe an iterated integral over UV m [t0 , t0 + T ], consider the following alphabets: X = {x0 , x1 , . . . , xm }, Y = {y0 , y1 , . . . , ym } and XY = X ∪ Y . For each η ∈ XY ∗ , define recursively the mapping Eη : Lm 2 (Ω × [t0 , t0 + T ], P, P ⊗ λ) → Ca.s. [t0 , t0 + T ] by first setting E∅ = 1 and then letting Zt− Exi η′ [w](t) , ui (s)Eη′ [w](s) ds, xi ∈ X,

(3)

Zt− Eyi η′ [w](t) , S vi (s)Eη′ [w](s) dW (s), yi ∈ Y,

(4)

t0

t0

′

∗

where η ∈ XY , u0 = v0 = 1, and the notation t− (suppressed in subsequent sections) indicates that the integration is over [t0 , t). The following terminology is used throughout. Let Nm+1 be the set of all vectors with its m + 1 components in N = {0, 1, . . .}. For a fixed word η ∈ XY ∗ , define the vectors α = (α0 , · · · , αm ) ∈ Nm+1 and β = (β0P , · · · , βm ) ∈ m Nm+1 ,Pwhere αi = |η|xi , βi = |η|yi , k = i=0 αi and m β . Let ⊔⊔ : RhhXii × RhhXii → RhhXii n = i i=0 represent the shuffle product [2, p. 20]. As a consequence of the integration by parts formula, the shuffle product and the scalar product Eη1 [u](t)Eη2 [u](t) for η1 , η2 ∈ X ∗ are intimately related, i.e., Eη1 [u](t)Eη2 [u](t) = Fη1 ⊔⊔ η2 [u](t). ∗

(5) ∗

A straightforward extension of (5) from X to XY is possible since Stratonovich integration also obeys the same integration by parts rule. For any α, β ∈ Nm+1 deαm 0 fine the polynomials pα = xα 0 ⊔⊔ · · · ⊔⊔ xm and pβ = β0 βm y0 ⊔⊔ · · · ⊔⊔ ym , respectively. The summations over all possible αi ’s that sum to k and all ’s that sum P possible βiP to n are denoted, respectively, by kαk=k and kβk=n . Lemma 1: [5] Let X k Y n = {η ∈ XY ∗ , |η|X = k, |η|Y = n}. The characteristic series, X k Y n , of the language X k Y n can be written in terms of the shuffle product as X XkY n , η η∈X k Y n

=

X

kαk=k,kβk=n

t0

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pα ⊔⊔ pβ .

FrB05.1 Example 1: In the deterministic case, i.e., when the alphabet Y is empty and the drift inputs are deterministic, one can show that m Y Rk Uiαi (t) ≤ , (6) |Fpα [u](t)| ≤ αi ! α0 ! · · · αm ! i=0 when max{kuk1 , T } ≤ R on [0, T ]. Furthermore, it is easily ∗ verified that for any u ∈ Lm 1 [0, T ] and η ∈ X |Eη [u](t)| ≤ Eη [¯ u](t), 0 ≤ t ≤ T, where u ¯ ∈ Lm ¯j = |uj |, j = 1 [0, T ] has components u 0, 1 . . . , m. Now fix T > 0. Pick any u ∈ Lm 1 [0, T ] and let R = max{kuk1 , T }. From Lemma 1, observe that X

η∈X ∗

|(c, η)Eη [u](t)| ≤ ≤ ≤ =

∞ X X

|(c, η)| Eη [¯ u](t)

k=0 η∈X k ∞ X k

KM k!

k=0 ∞ X

k=0 ∞ X

X

u](t) Fpα [¯

kαk=k

K(M R)k

X

kαk=k

Using the above lemma, equation (6), and Definition 3, it follows that m ¯ 2αi Y R2k Ui (t) 2 ≤ , (8) kFpα [w](t)k2 ≤ (αi !)2 (α!)2 i=0 where α! , α0 ! · · · αm !. Now, given that Fpβ [w](t) is comprised exclusively of Stratonovich integrals, a different approach has to be em2 ployed to determine a bound for Fpβ [w](t) 2 . It is known that Stratonovich integrals lack certain important properties such as isometry [16]. But if a Stratonovich integral is written in terms of Itˆo integrals, then all the properties associated with Itˆo integrals are available. There exist several formulas for writing iterated Stratonovich integrals as sums of iterated Itˆo integrals [15], [17]. An analogous formula for Eξ [w], ξ ∈ Y ∗ , is obtained directly by succesive applications of Definition 1. Theorem 1: [5] Let ξ ∈ Y n and w ∈ UV m [0, T ] be arbitrary. Then

k! α0 ! · · · αm !

Eξ [w](t) =

n,⌊ n 2⌋

X

r1 =0,r2

K(M R(m + 1))k .

k=0

Therefore, the series defining Fc converges absolutely and uniformly on an open ball in L1 [0, T ] of radius R < 1/M (m + 1). In [14], the more conservative radius of convergence 1/M (m + 1)2 was proved.

1 r1 2 r2 2 =0

X

sr1 s ¯r2

Iξ

[w](t) ,

(9)

¯nr s ¯r2 ∈A 2 s ¯r

sr1 ∈Anr21

where sr2 , . . . , s¯1 ) ∈ Nr2 : s¯l2 + 1 < s¯l2 +1 , sr2 = (¯ A¯nr2 = {¯ 1 ≤ l2 ≤ r2 − 1, 1 ≤ s¯l2 ≤ n − 1}   for 1 ≤ r2 ≤ n2 , A¯n0 = ∅, s ¯

Anrr21 = {sr1 = (sr1 , . . . , s1 ) ∈ Nr1 : sl1 < sl1 +1 , 1 ≤ l1 ≤ r1 − 1, sl1 6= s¯l2 or s¯l2 + 1, s¯l2 ∈ s¯r2 , 1 ≤ sl1 ≤ n}

III. I TERATED S TOCHASTIC I NTEGRALS AND THEIR L2 U PPER B OUNDS For fixed α, β ∈ Nm+1 , w ∈ UV m [0, T ] and t ≥ 0, define the following sum of iterated integrals

s ¯

for 1 ≤ r1 ≤ n, An0r2 = ∅, and ⌊·⌋ is the floor function. In addition, if ξ = yin ξ ′ then

Sα,β [w](t) , Fpα ⊔⊔ pβ [w](t) = Fpα [w](t)Fpβ [w](t). The importance of Sα,β [w] comes from the fact that, using the commutativity of the shuffle product and relation (5), the Lebesgue integrals and Stratonovich integrals can be completely separated and thus, an L2 upper bound for Sα,β [w](t) can be obtained by calculating individual L2 upper bounds for the random vectors Fpα [w](t) and Fpβ [w](t). Then from the independence assumptions in Definition 3,

2 2 2 kSα,β [w](t)k2 = kFpα [w](t)k2 Fpβ [w](t) 2 . (7) The main goal of this section is to compute an upper bound 2 for the right handside of (7). A bound for kFpα [w](t)k2 can be easily calculated from the following lemma. Lemma 2: [5] Let u be the drift input of w ∈ UV m [0, T ]. Then for α = (α0 , . . . , αm ) ∈ Nm+1 and any real numbers t>s≥0 # "m m Y Y αi ¯ αi (t), U ≤ (Ui (t) − Ui (s)) E i

Rt 0

vin (tn )Iξ′ [w](tn ) dW (tn )

(10)

0

sr1 s ¯ Iξ r2 [w](t)

,

(11) R R Iξ [w](t) R vis¯l2 +1 vis¯l2 dW (t′ )dW (t)→ vis¯l2 +1 vis¯l2 dt

R

visl dW (t)→ 1

R

bisl dt 1

∈ L2 (Ω × [0, T ], P, P ⊗ λ), 1 ≤ l1 ≤ r1 , with bis l 1 ≤ l2 ≤ r2 , and isl1 , is¯l2 ∈ {0, . . . , m} are the indices of the sl1 -th and s¯l2 -th letters of ξ. The next two theorems present L2 upper bounds for the iterated Itˆo integrals (10) and (11). Theorem 2: [5] Let ξ ∈ Y n and w ∈ UV m [0, T ] be arbitrary. An L2 upper bound for the iterated Itˆo integral (10) at a fixed t ∈ [0, T ] is

i=0

i=0

¯i (t) , where U

Iξ [w](t) ,

Zt

E [|ui (s)|] ds.

7480

2

kIξ [w](t)k2 ≤

m Y V βi (t) i

i=0

βi !

,

(12)

FrB05.1  Rt  where Vi (t) = 0 E vi2 (s) ds. Theorem 3: Let ξ ∈ Y n and w ∈ UV m [0, T ] be arbitrary. An L2 upper bound for the iterated Itˆo integral (11) at a fixed t ∈ [0, T ] is

2

sr ¯ m

1 Y Viβi (t) V¯iγ¯i (t) Biγi (t)

s¯r2 √ , (13)

Iξ [w](t) ≤ 2r2 tr1 +r2

β¯i ! γ¯i ! γi !

Pr2 ¯l2 ∈ s¯r2 ; γi = where γ¯i = l2 =1 (δiis¯l2 + δi(is¯l2 +1) ), s P r1 ¯ δiis , sl1 ∈ sr1 ; βi = βi − γ¯i − γi ; V¯i2 (t) =  4 l1  R t l1 =1 Rt  E vi (s) ds and Bi (t) = 0 E b2i (s) ds. Here δij de0 notes the Kronecker delta function. Proof: This inequality is proved by induction over r = r1 + r2 . If r1 = 0 and r2 = 0 then inequality (13) reduces trivially to inequality (12). Now, assume that (13) holds up to r − 1 ≥ 0. Set ξ = yin · · · yisr ξ ′ , with ξ ′ ∈ Y sr −1 and sr = sr1 or s¯r2 + 1. For sr1 > s¯r2 + 1, applying the isometry property n − sr1 times and Theorem 2 gives  tsr1 +2

2

sr Zt Z

1



s¯r2 vin (tn ) · · · visr1 +1 ·

Iξ [w](t) = E 

2

0

tsr1 +1

sr1 −1 s ¯r2

bisr1 (tsr1 )Iξ′

0

0

[w](tsr1 )dtsr1 ·

2   dW (tsr1 +1 ) · · · dW (tn )  ≤

Zt

E

0





vi2n (tn )

tsr1 +2

···

Z 0

h E vi2sr

1 +1

≤

Zt 0

≤2 t

  E vi2n (tn ) · · ·

Z

h

E vi2sr

0

1

tsr1 +1

i=0

≤

Zt 0

E





vi2n (tn )

2 2r2 tsrr1 +r +1

···

Z 0

E

h

ts¯r2 +1

i (tsr1 +1 ) ·

0

0

h E vi2s¯r

2

i 2 ) 2r2 −1 tsr¯1r2+r2 −1 · (t )v (t s¯r2 +1 s¯r2 is¯r 2

i=0

≤

Zt

ts¯r2 +3

  E vi2n (tn ) · · · β¯i !

t

Z 0

t

Z

h E vi2s¯r

γi !

≤

·

0

i=0

l=0 l6=isr1

7481

2

m Y V¯lγ¯l (ts¯r2 +1 ) √ · γ¯l ! l=0

ts¯r2 +3

  E vi2n (tn ) · · ·

0 m Y

i 2 (ts¯r2 +1 ) 2r2 −1 tsr¯1r +r +1 ·

 21 2¯ γis¯r +1 2 i V¯is¯ +1 ) (t s¯r2 r2  dts¯r2  · +1 (ts¯r2 ) 2 γ¯is¯r2 !

dts¯r2 +1 · · · dtn

Zt

2 +2

 21 2¯ γis¯ i V¯is¯ r2 (ts¯r2 ) h  r2 E vi2s¯r (ts¯r2 ) dts¯r2  · 2 γ¯is¯r2 !

s ¯r2 +1

 

h E vi2s¯r

l6=is¯r2 ,is¯r2 +1

s ¯r2 +1

 

Z

0 β¯ γi Vi (ts¯r2 +1 ) Bi (ts¯r2 +1 )

0 m Y

i=0

i ) · (t +1 s r +1 1

i

′

¯ m Y Viβ (ts¯r2 ) V¯iγ¯i (ts¯r2 ) Biγi (ts¯r2 ) √ dts¯r2 dts¯r2 +1 · · · dtn γi ! β¯i ! γ¯i !

¯ m m Y Blγl (tsr1 +1 ) Viβ (tsr1 +1 ) V¯iγ¯i (tsr1 +1 ) Y √ · γl ! γ¯i ! β¯i !

i=0

Z 0

vi2sr +1 (tsr1 +1 ) 1

i (tsr1 +1 ) ·

i

2

V¯iγ¯i (tsr1 ) Biγi (tsr1 ) √ dtsr1 dtsr1 +1 · · · dtn γi ! γ¯i ! tsr1 +2

1 +1

ts¯r2 +3

2 Zt

sr Z

1 i h   2

s¯r2 E vi2s¯r +2 (ts¯r2 +1 ) ·

Iξ [w](t) ≤ E vin (tn ) · · · 2

1

β¯i !

0

h E vi2sr

Pm ′ where γi′ = γi + δiisr1 . Observe that i=0 γi = r1 . The remaining (n−sr1 ) nested integrals are evaluated in a similar manner except that the β¯i ’s increase instead of the γi ’s. Therefore,

2

sr ¯ m γ′

1 Y Viβ (t) V¯iγ¯i (t) Bl i (t)

s¯r2 r2 r1 +r2 √ ,

Iξ [w](t) ≤ 2 t ¯

γ¯i ! γi′ ! i=0 βi ! 2  Pn Pm  where i=0 β¯i + γ¯i + γi′ = n, β¯j = β¯j + l=sr +1 δjil 1 for il ∈ {0, . . . , m}, and βi = β¯i + γ¯i + γi′ . Similarly, for s¯r2 > sr1 , one applies instead the isometry property n − (¯ sr2 − 1) times. There are two situations. The first is when is¯r2 + 1 6= is¯r2 . It then follows by H¨older’s inequality that

i h E b2isr (tsr1 ) 2r2 trs1r −1+r2 ·

0 β¯ m Y Vi (tsr1 )

0

Z

i=0

tsr1 +1

Z

  E vi2n (tn ) · · ·

¯ γ m Y Viβ (tsr1 +1 ) V¯iγ¯i (tsr1 +1 ) Bl i (tsr1 +1 ) √ dtsr1 +1 · · · dtn , γ ′! β¯i ! γ¯i !

2

tsr1 +2

tsr1 +2

Zt

r2 r1 +r2

ts¯r2 +1

0

dtsr1 +1 · · · dtn

γis

i Bis r1 (tsr1 ) h r1 2 dtsr1 dtsr1 +1 · · · dtn E bisr (tsr1 ) 1 γisr1 !

0

2 Z

i sr1 −1 h s ¯

E b2isr (tsr1 ) Iξ′ r2 [w](tsr ) dtsr1 · 1

tsr1 +1

tsr1 +1

Z

i=0

2

Z

tsr1 +1

Z

h E vi2s¯r

2 +2

0 β¯ γi Vi (ts¯r2 +1 ) Bi (ts¯r2 +1 )

β¯i !

γi !

i 2 (ts¯r2 +1 ) 2r2 −1 tsr¯1r +r +1 ·

m Y V¯lγ¯l (ts¯r2 +1 ) √ · γ¯l ! l=0

l6=is¯r2 ,is¯r2 +1

2

FrB05.1 γ ¯i

+1

+1

γ ¯

is s ¯r2 +1 ¯ V¯is¯r +1 (ts¯r2 +1 ) V¯is¯r r2 (ts¯r2 +1 ) 2 q q2 dts¯r2 +1 · · · dtn (¯ γis¯r2 +1 + 1)! (¯ γis¯r2 + 1)!

≤

Zt

ts¯r2 +3

Z

  E vi2n (tn ) · · ·

h E vi2s¯r

2

0 ′ ¯ γ β m Y Vi (ts¯r +1 ) V¯i¯i (ts¯r +1 ) 2 p ′2 ¯i ! β γ¯i ! i=0 0

i 2 ) 2r2 −1 tsr¯1r +r (t +1 s ¯ r +1 · +2 2 2

Biγi (ts¯r2 +1 ) · γi !

where β! , β0 ! · · · βm ! and max{kvk2 , kv0 k2 , kvk4 } ≤ R. Proof: From (9) a Stratonovich iterated integral can be writ¯ ten in
terms of integrals. Note
thatn
#{Anr2 } =
Itˆo iterated s ¯r2 n−2r2 n n−r2 ≤ r1 . Using the ≤ r2 and #{Anr1 } = r1 r2 triangle inequality, Theorem 1, Theorem 3 and the binomial theorem

sr

n,⌊ n 2⌋

1 X X 1

s¯r2

[w](t) kEξ [w](t)k2 ≤ I

ξ

2 r1 2 r2 ¯ r1 =0,r2 =0

Pm

¯i′ i=0 γ

= where = γ¯i + δiis¯r2 + δi(is¯r +1) . Observe that 2 . Since (¯ γ +2) ≥ +1 = i 2r . In the second situation, i i s ¯ 2 s ¯ r2 r2 √ √ γ¯i + 1 γ¯i + 2, it follows that

≤

ts¯r2 +3

2 Zt

sr Z h

1 i   2

s¯r2

Iξ [w](t) ≤ E vin (tn ) · · · E vi2s¯r +2 (ts¯r2 +1 ) · 2

2 2r2 −1 trs¯1r +r 2 +1

0

m Y

¯ Viβ (ts¯r2 +1 )

β¯i !

i=0 ts¯r2 +1

Z 0

≤

Zt

0 m Y

i=0

0 γi m Bi (ts¯r2 +1 ) Y

γi !

E



V¯lγ¯l (ts¯r2 +1 ) √ · γ¯l !

l=0 l6=is¯r2 +1



ts¯r2 +3

···

Z

h E vi2s¯r

2

γ ¯i

γi !

≤

2

0 ¯ m Viβ (ts¯r2 +1 ) Biγi (ts¯r2 +1 ) Y

β¯i !

≤

V¯lγ¯l (ts¯r2 +1 ) √ · γ¯l !


0 there exist an N > 0 such that

X

j X

N2 X

< ǫ, (c, η)E [w](t) η

j=N1 k=0 η∈X k Y j−k

7482

2

FrB05.1 when N2 > N1 > N . Two m-dimensional random vectors x, y are called equivalent if P (ω ∈ Ω : x(ω) = y(ω)) = 1. It is well known that Lm 2 (Ω, F , P ) with its usual norm is a Hilbert space modulo this equivalence relation. The following theorem ensures that a Fliess operator converges in the mean square sense to produce a well defined output process when its coefficients satisfy a rational growth condition. Theorem 5: Suppose for a series c ∈ Rℓ hhXY ii there exists real numbers K > 0 and M > 0 such that

Finally, taking the indicated summations, j N2 X X

j=N1 k=0

≤K =K

|(c, η)| ≤ KM |η| , ∀η ∈ XY ∗ .

=K

ak,n (t) = KM n+k

Sα,β [w](t).

kαk=k,kβk=n

To show that (14) is mean square convergent, it is sufficient to show that it is a Cauchy series. Since |η| = |η|X + |η|Y = k + n = j, it follows immediately from Lemma 1 that

X

X j j N2 X X

N2 X

kak,j−k (t)k2 , (c, η)Eη [w](t) ≤

j=N1 k=0

j=N1 k=0 η∈X k Y j−k 2

kak,n (t)k2 ≤ KM

kαk=k,kβk=n

5 (m + 1)k X (n!) 4

k!(n!) kβk=n (β!)  2 k X n! (m + 1)   ≤ K(M R′ )k+n 1 β! k!(n!) 4 kβk=n

= K(M R′ )k+n ≤

(m + 1)k 1 4

k!(n!) K(M R′ (m + 1)2 )k+n 1

k!(n!) 4

2n

(m + 1)

.

k=0 ∞ X ′

(M R′ (m + 1)2 )n 1

(n!) 4

5 4

= K′

∞ X (M ′′ )n 1

n=0

(n!) 4 ′

, 2

(n!) 4

series. By the ratio test M ′′ sn+1 = lim 1 = 0. n→∞ (n + 1) 4 n→∞ sn lim

Hence, the series (14) is Cauchy. Note in (14) that there is an implied order to the summation over XY ∗ . This is to indicate that the current proof for the convergence of Fc is strictly speaking for conditional convergence. That is, the series converges when computed in the order indicated. The authors conjecture that the series is in P fact absolutely convergent, i.e., η∈XY ∗ |(c, η)| kEη [w]k2 < ∞, but this issue will be addressed elsewhere. An important observation applicable to the well-posedness of interconnected systems is made in the following corollary. Corollary 1: [5] In the context of Theorem 5, the operator f ℓ [0, T ] Fc : UV m [0, T ] → UV

Without loss of generality, it is assumed that R = max{kuk1 , kvk2 , kv0 k2 , kvk4 } ≥ 1. If R′ , 4R(R + 4), then

1 4

∞ (M R′ (m + 1)2 )k X (M R′ (m + 1)2 )n 1 k! (n!) 4 n=0

for any T > 0. Example 2: Consider an autonomous system modeled by a stochastic differential equation in integral form

√ √ X ( 2R( t + 2))2n Rk n! . 1 α!β! (β!) 4

kan,k (t)k2 ≤ K(M R′ )k+n

1

k!((j − k)!) 4

j=0 k=0 ∞ X

where M ′′ , (M R′ (m + 1)2 ) and K ′ , Ke(MR (m+1) ) . ′′ n Note that (M )1 is the n-th term of an absolutely convergent

where N2 > N1 > N , provided that N > 0 is sufficiently large. Applying the multinomial theorem, equations (7) and (8), and Theorem 4 n+k

j ∞ X X (M R′ (m + 1)2 )k (M R′ (m + 1)2 )j−k

n=0

Then for any random process w ∈ UV m [0, T ], T > 0, the series (14) converges in the mean square sense to a well defined random vector y(t) = Fc [w](t), t ∈ [0, T ]. Proof: Without loss of generality it is assumed that ℓ = 1. Pick a t ∈ [0, T ] and any w ∈ UV m [0, T ]. Let R = max{kuk1P , kvk2 , kv0 k2 , kvk4 }. For a word η ∈ XY ∗ , recall k = Pm i=0 αi is the number of Lebesgue integrals in η, m while n = i=0 βi is the number of stochastic integrals in η. Define X

kak,j−k (t)k2

zt = z0 +

Zt 0

f¯(zs ) ds +

Zt

g(zs ) dW (s),

(15)

0

where f¯(z) and g(z) are suitably defined functions and zs = z(s) [16]. For a C 2 function F , the Stratonovich chain rule is dF (zt ) = f (zt )

∂ ∂ F (zt ) dt + g(zt ) F (zt ) ds W (t), (16) ∂z ∂z

where ds W (t) refers to Stratonovich integration and f (z) = ∂ f¯(z) + g(z) 2 ∂z g(z). Using these equations one can identify ∂ and Lg = the Lie differentiation operators Lf = f (z) ∂z ∂ g(z) ∂z . Now, let F (z) in (16) be replaced by either f or g,

7483

FrB05.1 and substitute f (zt ) and g(zt ) into (15). This yields zt = z0 + f (z0 )

Zt 0

+

Zt Zs 0

0

R EFERENCES

Zt ds + g(z0 ) S dW (s) 0

Zt Zs Lf f (zr ) drds + S Lg f (zr ) dW (r)ds 0 0

Zt Zs Zt Zs Lf g(zr ) drdW (s) + S S Lg g(zr ) dW (r)dW (s) + S 0

0 0

0

= z0 + f (z0 )

Zt 0

Zt ds + g(z0 ) S dW (s) + R1 (zt ), 0

where R1 (zt ) contains all the integrals whose integrands do not depend on z0 . In light of (3)-(4), define X = {x0 }, Y = {y0 } and the iterated Lie derivatives Lgx0 η = Lgη Lgx0 and Lgy0 η = Lgη Lgy0 , where gx0 = f , gy0 = g, and η ∈ XY ∗ . Then repeating the procedure once more, zt = z0 + Lgx0 I(z0 )Ex0 [0](t) + Lgy0 I(z0 )Ey0 [0](t) +Lgx0 x0 I(z0 )Ex0 x0 [0](t) + Lgy0 x0 I(z0 )Ey0 x0 [0](t) +Lgx0 y0 I(z0 )Ex0 y0 [0](t) + Lgy0 y0 I(z0 )Ey0 y0 [0](t) +R2 (zt ), where I denotes the identity map. This produces the PeanoBaker formula for the solution of (15) zt =

j ∞ X X

X

Lgη I(z0 ) Eη [0](t).

j=0 k=0 η∈X k Y j−k

Thus (f, g, I, z0 ) realizes the operator Fc driven by only noise when (c, η) = Lgη I(z0 ), ∀η ∈ XY ∗ . This example is analogous to the known deterministic case described in [7]. Example 3: Suppose the alphabet X is empty and Y = {y0 }. The input-output operator associated with the series (c, η) = KM |η| , ∀η ∈ Y ∗ is y(t) = Fc [0](t) =

∞ X

n=0

Zt Zt2 KM S · · · S dW (t1 ) · · · dW (tk ) n

0

0

∞ X

W n (t) = KeMW (t) , KM n = n! n=0 which is a well defined output according to Theorem 5. If instead (c, η) = KM |η| |η|!, then Theorem 5 does not apply. However, if for a positive R < 1 there exist a stopping time τR , inf{t : |M W (t)| = R}, then y(t) will converge almost K surely to 1−MW (t) , ∀t ≤ τR . In which case, the convergence condition given in Theorem 5 is only sufficient.

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