4.2.1
A General Large Deviation Result.
Recall Condition 2.3.2 in Chapter 2. The first condition requires continuity in the control for deterministic controlled systems. The second condition is a law of large numbers result for small noise controlled stochastic systems. In both cases we are allowed to assume the controls take values in a compact set.When applied to the SDE (4.1.1),G will be the mapping that takes the PRM into X.
Recall the rate function defined in (2.3.4). We will state it here again for easy reference. For φ ∈ U, define Sφ = {g ∈S:φ =G0(νg
defined by
I(φ) = inf
g∈Sφ
{LT(g)}, φ∈U. (4.2.1)
By convention, I(φ) =∞ if Sφ=∅.
For applications, the following strengthened form of Theorem 2.3.2 is useful. The proof follows by straightforward modifications; for completeness we include a sketch in the appendix of this chapter.
Let {Kn⊂X, n= 1,2, . . .} be an increasing sequence of compact sets such that ∪∞
n=1Kn=X. For eachn let
¯
Ab,n
.
= ϕ∈A¯: for all (t, ω)∈[0, T]×M¯, n ≥ϕ(t, x, ω)≥1/n if x∈Kn
and ϕ(t, x, ω) = 1 if x∈Knc},
and let ¯Ab =∪∞n=1A¯b,n. Define ˜UN =UN ∩A¯b.
Theorem 4.2.1. Suppose Condition 2.3.2 holds with UN replaced by U˜N. Then the
conclusions of Theorem 2.3.2 continue to hold.
4.2.2
A family of SPDEs driven by Poisson Random Mea-
sures.
In this section we introduce the basic SPDE model that will be studied in this work. We begin by giving a precise meaning to a solution for such a SPDE and then recall a result from [52] which gives sufficient conditions on the coefficients ensuring the strong existence and pathwise uniqueness of solutions.
If Φ is a countable Hilbertian nuclear space (recall the definition of CHNS in 3.2.2), and {Φn}n∈N0 are the completions of Φ with respect to {|| · ||n}n∈N0, respectively,
then {Φn}n∈N0 is a sequence of decreasing Hilbert spaces and Φ = ∩
∞
we identify Φ00 with Φ0 using Riesz’s representation theorem, and denote the space
of bounded linear functionals on Φn by Φ−n. This space has a natural inner product
(and norm) which we denote by h·,·i−n (resp. || · ||−n), n ∈N0 such that {Φ−n}n∈N0
is a sequence of increasing Hilbert spaces and the topological dual of Φ, denoted as Φ0 equals ∪∞
n=0Φ−n (see Theorem 1.3.1 of [52]). Elements of Φ0 need not have much
regularity. Solutions of the SPDE considered in this chapter will have sample paths in Φ0. In fact under the conditions imposed here the solutions will take values in D([0, T] : Φ−n) for some finite value of n.
We will assume that there is a sequence {φj} ⊂ Φ such that {φj} is a complete
orthonormal system (CONS) in Φ0 and is a complete orthogonal system (COS) in
each Φn, n ∈ Z. Then {φnj} ={φj||φj||−n1} is a CONS in Φn for each n ∈ Z. Define
the map θp : Φ−p → Φp by θp(φ
−p j ) = φ
p
j. It is easy to check that for all p ∈ N,
θp(Φ) ⊆Φ (see Remark 6.1.1 of [52]). Also, for eachr >0,η∈Φ−r and φ∈Φr,η[φ]
is defined by the formula
η[φ] =
∞
X
j=1
hη, φji−rhφ, φjir. (4.2.2)
We refer the reader to Example 1.3.2 of [52] for a canonical example of such a Count- able Hilbertian Nuclear Space (CHNS) defined using a closed densely defined self- adjoint operator on Φ0. A similar example was considered in Section 3.2.
Following [51], we introduce the following conditions on the coefficients A and G in equation (4.1.1). Let A : [0, T]×Φ0 → Φ0, G : [0, T]×Φ0 ×X → Φ0 be maps satisfying the following condition.
Condition 4.2.1. There exists p0 ∈N such that, for everyp≥p0, there exists q≥p
and a constant K =K(p, q) such that the following hold.
1. (Continuity) For all t ∈ [0, T] and u ∈ Φ−p, A(t, u) ∈ Φ−q and G(t, u,·) ∈
L2(
2. (Coercivity) For all t∈[0, T], and φ∈Φ,
2A(t, φ)[θpφ]≤K(1 +||φ||2−p).
3. (Growth) For all t∈[0, T], and u∈Φ−p,
||A(t, u)||2 −q ≤K(1 +||u|| 2 −p) and Z X ||G(t, u, v)||2 −pν(dv)≤K(1 +||u|| 2 −p).
4. (Monotonicity) For all t∈[0, T], and u1, u2 ∈Φ−p,
2hA(t, u1)−A(t, u2), u1 −u2i−q
+ Z
X
||G(t, u1, v)−G(t, u2, v)||2−qν(dv)≤K||u1−u2||2−q.
We now give a precise definition of a solution to the SDE (4.1.1).
Definition 4.2.1. Let ( ¯M,B( ¯M),P¯,{F¯t}) be the filtered probability space from Sec-
tion 2.2.2. Fix p ∈ N0, suppose that X0 is a ¯F0-measurable Φ−p-valued random
variable such that E||X0||2−p < ∞. A stochastic process {Xt}t∈[0,T] defined on ¯M is
said to be a Φ−p-valued strong solution to the SDE (4.1.1) with initial valueX0, if
(a) Xt is a Φ−p-valued ¯Ft-measurable random variable for all t∈[0, T];
(b) X∈D([0, T] : Φ
−p) a.s.;
(c) there is a q ≥ p such that for all t ∈ [0, T] and u ∈ Φ−p, A(t, u) ∈ Φ−q and
G(t, u,·)∈L2(
X, ν; Φ−q), and there exists a sequence{σn}n≥1 of{F¯t}-stopping times
increasing to infinity such that for each n≥1, ¯ E Z T∧σn 0 Z X ||G(s, Xs, v)||2 −qν(dv)ds <∞
and ¯ E Z T∧σn 0 ||A(s, Xs)||2 −qds <∞;
(d) for all t∈[0, T], almost all ω ∈M¯, and all φ∈Φ Xt[φ] =X0[φ] + Z t 0 A(s, Xs)[φ]ds+ Z t 0 Z X G(s, Xs−, v)[φ] ˜N−1(dsdv). (4.2.3) In Definition 4.2.1, ˜N−1
is the compensated version of N−1
as defined below (4.1.1), with N−1
having jump rates that are scaled by 1/and is constructed from ¯
N, as below (2.2.1).
One can similarly define a Φ−p-valued strong solution on an arbitrary filtered
probability space supporting a suitable PRM.
Definition 4.2.2(pathwise uniqueness). We say that the Φ−p-valued solution for the
SDE (4.1.1) has thepathwise uniquenessproperty if the following is true. Suppose that X and X0 are two Φ−p-valued solutions defined on the same filtered probability
space with respect to the same Poisson random measure and starting from the same initial condition X0. Then the paths of X and X0 coincide for almost all ω.
The following theorem is taken from [52] (see Theorem 6.2.2, Lemma 6.3.1 and Theorem 6.3.1 therein).
Theorem 4.2.2. Suppose that Condition 4.2.1 holds. LetX0 be aΦ−p-valued random
variable satisfying E||X0||2−p <∞. Then for sufficiently large p1 ≥ p, the canonical
injection from Φ−p to Φ−p1 is Hilbert-Schmidt, and for all such p1 the SDE (4.1.1)
with initial value X0 has a pathwise unique Φ−p1-valued strong solution.