Page 112 - IJOCTA-15-1

D. Kasinathan et.al. / IJOCTA, Vol.15, No.1, pp.103-122 (2025)

2. Preliminaries
ς
Z
In this section, we briefly recollect some basic C D + f(ς) = 1 (ς − ϑ) n−q−1 (n) (ϑ)dϑ,
q
f
definitions, namely Riemann-Liouville (R-L) frac- 0 Γ(n − q)
tional derivative and integral, Caputo fractional 0
n − 1 < q < n,
derivative, existing lemmas, and semigroup the-
ory of bounded linear operators which are used th
where (n) denotes the n derivative, provided that
in a sequel. Let (Ω, F ς , P) be a complete filtered
the R.H.S is pointwise defined on J.
probability space furnished with complete fam-
ily of right continuous increasing sub σ-algebras Definition 4. 39–42 A one parameter family of
{F ς , ς ∈ J} hold with F ς ⊂ F. A H-valued ran- bounded linear operators (C(ς)) ς∈R on X is called
dom variable is a F ς -measurable function x(ς) : a strongly continuous cosine family if
Ω → H, and a collection of random variable
S = {x(ς, ω) : Ω → H : ς ∈ J} is called a stochas- (1) C(ϑ + ς) + C(ϑ − ς) = 2C(ϑ)C(ς) ∀ ϑ, ς ∈
tic process. Let γ n (ς)(n = 1, 2, . . . ) be a sequence R; C(0) = I
of real valued one-dimensional standard Brownian (2) ς 7−→ C(ς)x is continuous on R ∀ x ∈ X
motions independent of (Ω, F ς , P). Set w(ς) = ;
√
∞ λ n γ n (ς)ζ n (ς), ς ≥ 0, where, λ n ≥ 0 are
P
n=1 (3) The associated sine family (S(ς)) ς∈R to
non-negative real numbers and {ζ n }(n = 1, 2, . . . ) (C(ς)) ς∈R is defined by
is complete orthonormal basis in K. Let Q ∈ ς
Z
0
L (K, H) be an operator defined by Qζ n = λ n ζ n S(ς)x = C(ϑ)xdϑ, x ∈ X, ς ∈ R.
Q
P ∞
with finite Tr(Q) = n=1 λ n < ∞. Then the
above K-valued stochastic process w(ς) is called a 0
0
Q-Wiener process. Let Ψ ∈ L (K, H) and define, Lemma 1. 9,10 (Burkholder-Davis-Gundy
Q
0
Inequality) For any p ≥ 2 and for an L (K, H)-
Q
∞ valued predictable process G(ϑ), we have
2
2
∗
∥Ψ∥ = Tr(ΨQΨ ) = X p λ n Ψζ n ∥ .
∥
Q
n=1  p

2

ς Z
p ς Z 2
If ∥Ψ∥ Q < ∞, then Ψ is known as Q-Hilbert

˜  (E∥G(ϑ)∥ p 0 ) dϑ  ,
p
sup E
G(ϑ)dw(ϑ)
≤ C p L
Schmidt operator. For more details on con- ϑ∈J 0 0 Q
cepts and theory on SDEs, one can refer the p

˜
monographs 13–15 and references therein. where C p = p(p−1) 2 , and E denotes the mathe-
2
Definition 1. 36 The R-L fractional integral of matical expectation.
0
order n − 1 < q < n, for a continuous function Lemma 2. 22 For any σ : [0, T] → L (K, H)
Q
f : J → R defined as P ∞ 1
such that ∥σQ 2 e n ∥ 1 < ∞ hold.
n=1
L H ([0,∞);H)
ς Also, for a given ϑ and for any a, b ≥ 0,
Z
1
q q−1
I + f(ς) = (ς−ϑ) f(ϑ)dϑ, ς > 0, q > 0,
0 Γ(q) Z b
H
0 E∥ σ(ϑ)dB (ϑ)∥ p
Q
provided that the R.H.S is pointwise defined on J. a
∞ Z b 1
X
p
Definition 2. 37 The R-L fractional derivative of ≤ CHς pH−1 ∥σ(ϑ)Q 2 e n ∥ dϑ.
a
order q, of a function f : J → R defined as, n=1
P ∞ 1
Moreover, if ∥σQ 2 e n ∥ is uniformly conver-
ς n=1
Z gent then
1 d n
q n−q−1
D + f(ς) = (ς − ϑ) f(ϑ)dϑ,
0 Γ(n − q) dς
Z b Z b
0 H p pH−1 p
ς > 0, n − 1 < q < n, E∥ σ(ϑ)dB (ϑ)∥ X ≤ CHς ∥σ(ϑ)∥ dϑ.
Q
0
L p
a a
where n = [q] + 1, [q] denotes the integral part For details on basic preliminaries of fBm, one can
of number q, provided that the R.H.S is pointwise refer to. 13,22
defined on J, Γ is the Gamma function. 43
Lemma 3. (Banach Fixed Point Theo-
Definition 3. 38 The Caputo fractional deriva- rem) Let X be a Banach space. If f : X → X is
tive of order q for a function f : J → R defined a continuous contracting map on X, then f has a
as unique fixed point. Moreover,
106