3. On Stochastic Functional Dierential Equations 29
3.3. Representations for Linear RFDEs with Additive Noise
As linear SRFDEs we refer to systems where the drift coecientH(t, ψ)is ane linear inψ for eacht, which meansH(t, ψ) =L(t, ψ) +h(t)for some operatorL:R× C(J,Rn)→Rn that is linear with respect to the second argument, and an inhomogeneity map h: [t0,∞)→ Rn. As it is common practice we will use the notationsL(t, ψ) =L(t)(ψ) =L(t)ψ, and we will occasionally refer toLas a family of operators, e.g.(L(t))t∈[t0,∞). For later referencing we put this special case of (3.1.1) in display:
dx(t) =L(t)xtdt+h(t)dt+σ(t)dW(t) fort≥t0,
xt0 = Υ. (3.3.1)
Remember that, by the Riesz representation theorem, the linear operators (L(t))t∈[t0,∞) may uniquely be extended from C(J,Rn) to Bb(J,Rn), where Bb denotes the measurable and bounded mappings. This unique extension will be tacitly applied when needed, and the extended family of linear operators will also be denoted by the same symbols(L(t))t∈[t0,∞). Regarding the deterministic version, the conditions for existence and uniqueness are carried over from the account of Hale and Lunel, [HVL93, chapter 6], thereby xing the related notations to have them at hand later on.
Assumption 3.1 (HaleLunel conditions for global existence and uniqueness). There is an m∈Lloc1 [t0,∞)×R,Rn×n
, which means locally Lebesgue-integrable, n×nmatrix-valued function η(t, u), measurable in (t, u)∈R×R, so that
η(t, u) =
0 foru≥0,
η(t,−r) foru≤ −r, (3.3.2) continuous from the left in allu∈(−r,0)and has bounded variation inuon[−r,0]for each t. And the variation with respect touis bounded through
Var[−r,0]η(t,·)≤m(t) for all t≥t0, (3.3.3) and the linear mapping L(t) :C(J,Rn)→Rn is given by
L(t)ψ= Z 0
−r
ψ(u)duη(t, u) for allt∈(−∞,∞), ψ∈ C(J,Rn),
where du indicates that the LebesgueStieltjes integration is carried out with respect to the u-argument of the integrator, andt is xed. In particular,|L(t)ψ| ≤m(t)kψk.
Together with the Lloc1 -assumption on hAssumption 3.1 ensures existence and uniqueness of global solutions in the deterministic case. These HaleLunel conditions are satised if we, for instance, assume the family L to be continuous with repect to the sup-norm on [t0, T]× C(J,Rn), given by
k(t, ψ)k[t0,T]×C(J,Rn):= max{|t|,kψk} for allt≥t0, ψ∈ C(J,Rn). (3.3.4) Fundamental Solutions. The concept of fundamental solutions, which is a generalization from classical theory of ordinary dierential equations, will be of vital importance for this work due to its crucial role in the variation-of-constants formula. This extract from the book [HVL93] outlines a formal denition of the fundamental matrix solution, and reviews the solution representation through the variation-of-constants formula in the nonautonomous deterministic case; we will not present every detail, but mainly follow the main ideas from the introduction of an appropriate resolvent kernel in order to rigorously dene fundamen-tal solutions to solution representations. All details can be found in [HVL93, chapter 6].
Informally speaking the variation-of-constants formula originates from the linear dierential law and does not get in the way of the retarded feedback mechanism.
First, we rewrite the solution of the deterministic version of (3.3.1) with an application of the integration by parts formula, which is applicable due to absolute continuity of the solu-tionx, and where we write the formal weak derivative ofxas x˙. As we mentioned before,
related dierential formulas have to be understood as integrated equations. We obtain that
˙ x(t) =
Z t t0
x(u)duη(t, u−t) + Z t0−t
−r
Υ(t−t0+u)duη(t, u) +h(t)
=−η(t, t0−t)x(t0)− Z t
t0
η(t, u−t) ˙x(u)du+ Z t0−t
−r
Υ(t−t0+u)duη(t, u) +h(t) for allt∈[t0,∞).
(3.3.5) We dene k(t, s) := η(t, s−t), s, t∈ [t0,∞), a kernel of type L1loc on [t0,∞), in order to reformulate (3.3.5) withy(t) = ˙x(t)as a Volterra equation of the second kind,
y(t) = Z t
t0
k(t, u)y(u)du+g(t) for Lebesgue-a.e. t∈[t0,∞), (3.3.6) where g ∈ Lloc1 ([t0,∞),Rn) is given by the collection of terms from inhomogeneity and initial-segment inuence, namely
g(t) :=−η(t, t0−t)Υ(0) + Z t0−t
−r
Υ(t−t0+u)duη(t, u) +h(t) for allt≥t0. From the corresponding theory of Volterra equations, we conclude that there is a Volterra resolvent Rsatisfying
R(t, s) =−η(t, s−t) + Z t
s
R(t, u)η(u, s−u)du for allt≥s, s∈[t0,∞), (3.3.7) and it is unique in theL1-sense on every nite time horizon. By means of a Gronwall-type argument, the variation condition (3.3.3) implies
|R(t, s)| ≤m(t) exp Z t
s
m(u)du
for allt≥s, s∈[t0,∞). (3.3.8) We dene the fundamental matrix solutionxˇ as
ˇ
x(t, s) :=In− Z t
s
R(u, s)du for alls∈[t0,∞), t≥s, (3.3.9) whereIndenotes then-dimensional unit matrix. We may interpret the fundamental solution (ˇx(t, u) : u ∈ [t0,∞), t ≥ u−r) as the family of matrix solutions of the homogeneous deterministic systems
dx(t) =L(t)xtdt fort≥u,
x(t) =1{u}(t)In fort∈[u−r, u], (3.3.10) where the dierential law L(t)is taken as separately acting on the column vectors. As we have pointed out before, the existence of solutions of the deterministic version of (3.3.1) follows from an application of the Schauder xed-point theorem, and crucially relies on the continuity of the initial segment Υ, which means that (3.3.10) is not covered through that approach due to its discontinuous initial segment. The slight detour to the Volterra resolvent provides a rigorous denition of the fundamental solution. In the rst argument the
fundamental solution is absolutely continuous, solves the integral equation and its dierential law applies almost everywhere with respect to the Lebesgue measure. Continuing from (3.3.8), we can conclude that
|ˇx(t, s)| ≤exp Z t
s
m(u)du
for alls∈[t0,∞), t≥s, (3.3.11) and for any nite time horizon T >0, due to boundedness of the resolvent in (3.3.8), there iscR=cR(T)>0 such that for all∆∈R witht+|∆| ≤T andt− |∆| ≥u
|ˇx(t+ ∆, u)−x(t, u)| ≤ˇ cR|∆| for allu∈[0, T], t∈[u, T]. (3.3.12) That means the fundamental solution is locally uniformly Lipschitz in the rst argument with respect to compacts of the second argument. The general existence and uniqueness result for solutions of the deterministic version of (3.3.1) also covers the corresponding homogeneous system started at any intermediate time points∈[t0, T]initiated with some ψ∈ C(J,Rn), formally given by
dx(t) =L(t)xtdt fort≥s,
xs=ψ. (3.3.13)
That means that there is a solution semi group (Tt,sdet : s ∈ [t0,∞), t ≥ s) that shoves segments fromC(J,Rn)along the solution path intoC(J,Rn)according to the deterministic dierential law. In other words, if we denote (z(t) : t ≥ t0) the solution of (3.3.13) for s=t0, thenzt=Tt,tdet
0ψfor allt≥t0. Due to [HVL93, Chapter 6.1, 6.2] the unique solution of the inhomogeneous system is then given by
x(t) =Tt,tdet0Υ(0) + Z t
t0
ˇ
x(t, u)h(u)du for allt≥t0. (3.3.14)
Example 3.2. a) This special case is taken from [HVL93]. For arbitary N∈N andr >0 let Ak ∈ Rn×n, k ∈ {1, . . . , N} be a family of constant matrices, and rk ∈ (0, r), k ∈ {1. . . N}, a collection of delay lengths. Assume further someA:R×R→Rn×n,(t, u)7→
A(t, u), that is integrable inufor everyt, and that there is some functiona∈Lloc1 (R,R) such that
Z 0
−r
A(t, u)ψ(u)du
≤a(t)kψk for allt∈R, ψ∈ C(J,Rn).
If we moreover assume thath∈Lloc1 , and lett0∈R, and Υ∈ C(J,Rn) arbitrary, then the system
dx(t) =PN
i=1Aix(t−ri)dt+R0
−rA(t, u)x(t+u)du dt+h(t)dt fort≥t0,
xt0 = Υ, (3.3.15)
satises Assumption 3.1 and therefore, there is a unique solution and it may be repre-sented in the form (3.3.14). The reason for bringing up this particular example is that J. Hale and S. Verduyn Lunel refer to it as the most common type of linear systems with nite lag which is known to be useful in applications, see [HVL93, Chapter 6.1].
b) This one is a modication of the above example. It is an instance of a continuous family of continuous linear operators, which is to say that (L(t))t∈[t0,T], as a mapping from [t0, T]×C(J,Rn), is continuous with respect tok·k[t0,T]×C(J,Rn), see (3.3.4). This example keeps jump positions xed, but allows time dependence for the height of jumps. For arbitary N ∈ N and r > 0 let Ak : [t0,∞) → Rn×n, k ∈ {1, . . . , N}, be a family of continuously dierentiable Rn×n-valued functions, and rk ∈ (0, r), k ∈ {1. . . N}, a collection of delay lengths. Assume further someA: [t0,∞)×R→Rn×n,(t, u)7→A(t, u) that is integrable in u for everyt and that there is some functiona ∈Lloc1 (R,R) such that
Z 0
−r
A(t, u)ψ(u)du
≤a(t)kψk for all t∈[t0,∞), ψ∈ C(J,Rn).
We additionally assume that A(t, u) is continuously dierentiable in t. Then, for h ∈ Lloc1 , andΥ∈ C, the system
dx(t) =
N
X
i=1
Ai(t)x(t−ri)dt+ Z 0
−r
A(t, u)x(t+u)du dt+h(t)dt fort≥t0, xt0 = Υ,
(3.3.16)
satises condition (3.1) from above with
η(t, u) =− Z 0
u
A(t, v)dv−
N
X
i=1
Ai(t)1{u≤−ri} fort∈R, u∈J. (3.3.17)
It is generally true that systems of this form admit fundamental solutions that are Lipschitz-continuous in both arguments, see Lemma A.3 in the appendix. Further, this special case contains systems of the form
dx(t) =−a(t)x(t)dt+b(t)x(t−r)dt+h(t)dt fort∈[t0, T],
xt0= Υ, (3.3.18)
if we assume the coecients a, b∈ C1([t0, T],R), i.e. to be continuously dierentiable.
Those systems play a crucial role in the second part of this work.
In case of an autonomous drift coecient L(·) = L, the local Lipschitz property simplies to ordinary continuity of L. In case of additive noise the stochastically perturbed system can also be described with the help of the fundamental solution by means of a stochastic variation-of-constants formula. Especially, for systems of the form
dx(t) =Lxtdt+σ(t)dW(t) fort≥t0,
xt0 = Υ, (3.3.19)
we cite a representation result from the book of S.-E. A. Mohammed, [Moh84]. For the deterministic version of (3.3.19), the solution semi group (Tt,udet : u≥ t0, t ∈[u,∞))from C(J,Rn)toC(J,Rn)does only depend ont−uwhich motivates us to write
Tt−udet :=Tt,udet for allu∈[t0,∞), t≥u.
And analogously for the fundamental solutionx(tˇ −u) := ˇx(t, u)foru∈[t0,∞), t≥u−r. Proposition 3.3 ( [Moh84], Chapter 4, Theorem (4.1), Remark (4.2)). Suppose that(Tsdet)s≥0 denotes the solution semi group of the deterministic version of (3.3.19) whereL:C(J,Rn)→ Rn is continuous linear, σ: [t0,∞)→Rn×m is locally square integrable, Υ∈ C(J,Rn)and (W(u))u∈[t0,∞)is anm-dimensional Brownian motion. Then there is a unique strong solu-tionx= (x(t))t∈[t0,∞)of the SRFDE (3.3.19) and it admits the representation
x(t) =Tt−tdet0Υ(0) + Z t
t0
ˇ
x(t−u)σ(u)dW(u) for allt≥t0 P-a.s. (3.3.20) The proof that is presented in [Moh84, Lemmas 4.3, 4.4, Theorem 4.1] uses relatively strong assumptions due to ensure a formula for the dierential of a stochastic integral. We will generalize the result by closely related ideas using absolute continuity of the fundamental solution in the rst argument and the stochastic Fubini theorem, which one can nd in [Jac79, Théorème 5.44] for the nite-dimensional case in french language, or in a rather general Hilbert-space setting in [DPZ14, Theorem 4.33]. Our rst objective is to show that our candidate solution has a (Hölder)-continuous modication.
Lemma 3.4. If we denote the fundamental solution of (3.3.1) by (ˇx(t, u) :u∈[t0, T], t∈ [u−r, T]) and assume thatσ∈ Bb([t0, T],Rn×m), i.e. bounded and Borel-measurable, with supu∈[t0,T]|σ(u)|=:σ+, in case of the HaleLunel conditions 3.1 the processz, dened by
z(t) :=
Z t t0
ˇ
x(t, u)σ(u)dW(u) for allt∈[t0, T] has a Hölder-continuous version of order γ∈(0,1/2).
Proof. This can be seen by an application of the Kolmogorov continuity criterion applied to Z t
t0
σ(u)dW(u)− Z t
t0
ˇ
x(t, u)σ(u)dW(u) fort∈[t0, T].
Due to the local Lipschitz continuity of xˇ in the rst argument, see (3.3.12), we nd that for∆>0
E
Z t+∆
t0
In−x(tˇ + ∆, u)
σ(u)dW(u)− Z t
t0
In−x(t, u)ˇ
σ(u)dW(u)
2
=E
"
Z t t0
x(tˇ + ∆, u)−x(t, u)ˇ
σ(u)dW(u)
2#
+E
Z t+∆
t
In−x(tˇ + ∆, u)
σ(u)dW(u)
2
≤ Z t
t0
ˇx(t+ ∆, u)−x(t, u)ˇ
2σ2+du+ Z t+∆
t
In−x(tˇ + ∆, u)
2σ2+du
≤σ2+(t−t0)c2R∆2+σ+2 Z t+∆
t
c2R∆2du≤const ∆2,
where in the second to the last inequality we have used Itô isometry and that In = ˇx(u, u) and therefore,
ˇx(t+ ∆, u)−In
≤cR∆for allu∈[t, t+ ∆]for appropriatecR, see (3.3.12).
From the general theory, we know thatRt
t0σ(u)dW(u),t∈[t0, T], admits Hölder-continuous sample paths of order γ ∈ (0,1/2) almost surely, and therefore there is an almost surely Hölder-continuous version ofRt
t0x(t, u)σ(u)dWˇ (u), t∈[t0, T]of orderγ∈(0,1/2).
In the following, when considering the stochastic integral process, dened in Lemma 3.4, we will refer to its continuous version. The next objective is to give a generalization of the solution representation that is presented in [Moh84], stated above as Proposition 3.3.
Theorem 3.5 (General Representation Theorem). Consider the situation of the Lemma 3.4 and let (Tt,udet:u≥t0, t≥u)denote the solution semi group from C(J,Rn) toC(J,Rn) of the deterministic version of the homogeneous SRFDE
dx(t) =L(t)xtdt+σ(t)dW(t) fort∈[t0, T),
xt0 = Υ. (3.3.21)
Then, for arbitrary nite time horizon T > t0, the unique solution of (3.3.21) is P-almost surely given by
y(t) :=Tt,tdet0Υ(0) + Z t
t0
ˇ
x(t, u)σ(u)dW(u) for allt∈[t0, T],
where the stochastic integral term is understood as the continuous version ensured by the previous Lemma 3.4.
Proof. We go over the arguments deliberately in small steps. Due to its denition(Tt,tdet0Υ(0) : t≥t0)solves the deterministic version of (3.3.21) in t, which is to say that
Tt,tdet0Υ(0) = Υ(0) + Z t
t0
∂
∂sTs,tdet0Υ(0)ds and
∂
∂tTt,tdet
0Υ(0) =L(t)(Tt,tdet
0Υ) = Z 0
−r
Tt,tdet
0Υ(θ)dθη(t, θ) for allt∈[t0, T]. (3.3.22) Further, we know that the fundamental solution solves the respective integral equation of the deterministic system in the rst argument, which means
ˇ
x(t, u) = ˇx(u, u) + Z t
u
Z 0
−r
ˇ
x(s+θ, u)dθη(s, θ)ds for allu∈[t0, T], t≥u. (3.3.23) And due to the fact thatx(sˇ +θ, u) = 0for alls∈[t0, u)andθ∈[−r,0], we may exchange theufort0in the lower boundary of the right-hand side integral above. We obtain that
ˇ
x(t, u) =In+ Z t
t0
Z 0
−r
ˇ
x(s+θ, u)dθη(s, θ)ds for allt≥u.
Using (3.3.22) and (3.3.23) to rewrite(y(t))t∈[t0,T] leads to
y(t) = Υ(0) + Z t
t0
∂
∂sTs,tdet0Υ(0)ds +
Z t t0
ˇ
x(u, u)σ(u) + Z t
u
Z 0
−r
ˇ
x(s+θ, u)σ(u)dθη(s, θ)ds dW(u) for allt∈[t0, T].
As before, we may replace theuin the lower integral boundary on the right byt0, because
the integrand is zero for alls∈[t0, u).
y(t) = Υ(0) + Z t
t0
Z 0
−r
Ts,tdet
0Υ(θ)dθη(s, θ)ds+ Z t
t0
Z t t0
Z 0
−r
ˇ
x(s+θ, t0)σ(u)dθη(s, θ)ds dW(u) +
Z t t0
ˇ
x(u, u)σ(u)dW(u) for allt∈[t0, T].
For the triple-integral term, if we understandRt
t0x(sˇ +θ, u)dW(u),u∈[t0, t]as a stochastic integral parametrized by (s, θ), we may apply the stochastic Fubini theorem (see [Jac79, Théorème 5.44] (or [DPZ14, Theorem 4.33]) to interchange the order of integration. As an intermediate step η(s,·)⊗ds must formally be split into a dierence of two positive, nite measures. We have to check that both appearing stochastic integrals are well-dened, which means predictability, i.e. measurability with respect to the ltration that is generated by the left-continuous and adapted processes, of the integrand as well as L2-integrability.
But concerning the rst stochastic integralRt
t0x(sˇ +θ, u)σ(u)dW(u)intricate measurability issues do not arise, because the integrand x(sˇ +θ, u)σ(u)is deterministic and bounded, in particular predictable. And therefore, the stochastic integral Rt
t0x(sˇ +θ, u)σ(u)dW(u) is predictable int, see e.g. [Jac79]. And alsoL2-integrability is ensured by boundedness of the integrand. Regarding the second stochastic integral
Z t t0
Z t u
Z 0
−r
ˇ
x(s+θ, u)σ(u)dθη(s, θ)ds dW(u),
the same reasoning holds true and is not aected by a decomposition of the dθ(η(t, θ))dt -measure in positive and negative part. We obtain that
Z t t0
Z t t0
Z 0
−r
ˇ
x(s+θ, u)σ(u)dθη(s, θ)ds dW(u) = Z t
t0
Z 0
−r
Z t t0
ˇ
x(s+θ, u)σ(u)dW(u)dθη(s, θ)ds P-almost surely for a dense subset in tfrom[t0, T].
(3.3.24) Note further that, simply because xˇ solves the integrated equation for the homogeneous deterministic system,
Z t t0
Z t u
Z 0
−r
ˇ
x(s+θ, u)σ(u)dθη(s, θ)ds dW(u) (3.3.25)
= Z t
t0
Z t u
Z 0
−r
ˇ
x(s+θ, u)σ(u)dθη(s, θ)ds dW(u)
= Z t
t0
Z t u
L(s) ˇx(s+θ, u) :θ∈J
σ(u)ds dW(u)
= Z t
t0
(ˇx(t, u)−x(u, u))σ(u)dWˇ (u)
= Z t
t0
(ˇx(t, u)−In)σ(u)dW(u) for allt∈[t0, T]. (3.3.26) That provides continuous paths intalmost surely with respect toPdue to Lemma 3.4 and the choice of the continuous version. Continuing from (3.3.24) we may decline the right-hand side inner integral to an upper boundary ofs+θ, which ensures continuity of the stochastic integral term on the right due to construction. Further, with regard to (3.3.26), we know
that the term in line (3.3.25) isP-almost surely continuous in t, again due to construction.
Therefore, we can understand the two sides of (3.3.24) as two continuous processes in t that match P-almost surely on a dense subset of [t0, T]. So, they must be the same up to indistinguishability, i.e.
Z t t0
Z t t0
Z 0
−r
ˇ
x(s+θ, u)σ(u)dθη(s, θ)ds dW(u)
= Z t
t0
Z 0
−r
Z s+θ t0
ˇ
x(s+θ, u)σ(u)dW(u)dθη(s, θ)ds for allt∈[t0, T] P-almost surely.
Applying that to the termy we nd thatP-almost surely
y(t) = Υ(0) + Z t
t0
Z 0
−r
Ts,tdet
0Υ(θ) + Z s+θ
t0
ˇ
x(s+θ, u)σ(u)dW(u)dθη(s, θ)ds+ Z t
t0
σ(u)dW(u)
= Υ(0) + Z t
t0
Z 0
−r
y(s+θ)dθη(s, θ)ds+ Z t
t0
σ(u)dW(u) for allt∈[t0, T].
Or, in other words and short-hand dierential notation respectively, we nd that
y(t) = Υ(0) + Z t
t0
L(s)ysds+ Z t
t0
σ(u)dW(u), or dy(t) =L(t)ytdt+σ(t)dW(t)
for allt∈[t0, T]P-a.s. (3.3.27) By uniqueness of solutions, which is covered in Section 3.2 of this work, this settles the proof.
Of course, due to the linearity of the inhomogeneous nonautonomous system (3.3.1), we nd that the solution of (3.3.1) may now be given explicitly. To put a label to it, we stow that fact in the following corollary.
Corollary 3.6. Under the assumptions of Lemma 3.4, the solution of (3.3.1) is P-almost surely given by
x(t) =Tt,tdet0Υ(0) + Z t
t0
ˇ
x(t, u)h(u)du+ Z t
t0
ˇ
x(t, u)σ(u)dW(u) for allt≥t0. (3.3.28) In particular, the solution is a continuous Gaussian process in Rn.
Remark 3.7. We will refer to the solution formulas of the form (3.3.14), (3.3.20) and (3.3.28) as variation-of-constants formulas. Their kind has approved as a helpful tool in the study of stochastic retarded functional dierential equations. And they will do so in the second part of this work.