EXISTENCE OF STRONG SOLUTIONS OF A

STOCHASTIC DIFFERENTIAL INCLUSION.

2. THE UPPER SEMICONTINUOUS CASE

H. Fritsch

Institut für Optimierung und Stochastik, FB Mathematik und Informatik,

Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle, Germany

Abstract

In this second part of a paper about a stochastic differential inclusion with setvalued drift we consider the question of existence of solutions for the case of a kind of upper semicontinuous multifunctions. The conditions to the dispersion matrix are weakened in the classical way. To show existence results for such inclusions we use a locally Lipschitzean approximation of upper semicontinuous multifunctions. Key words: stochastic differential inclusion, setvalued system, existence of solutions, upper semicontinuous multifunction, locally Lipschitzean approximation. AMS subject classifications: 60H10, 34A60.

1 INTRODUCTION

This work is a continuation of [4]. Here we consider the existence of a nonempty set

of solutions of the stochastic inclusion

x(t) ∈ x^{0} + ∫_{0}^{t} F (x(s))ds + ∫_{0}^{t} b(s, x(s))dw(s)  (1)

where the first integral is of Aumann type and the stochastic integral is to un-

derstand in the sense of Ito with a ℝ^{d}-valued Brownian motion (w(t))_{t∈[0,T]} with independent components. For the initial value x^{0} of the setvalued system (1) we assume E {∥x^{0}∥^{2}} < ∞ and the independence from (w(t))_{t∈[0,T]}. We demand for the second integral of inclusion (1) the same weakened conditions like in the first paper [4] and that the multivalued drift is upper semicontinuous in a way described below. Before coming to existence results for (1) we introduce a locally Lipschitzean approximation of upper semicontinuous multifunctions in Banach spaces like in [2].

We define a family of σ-fields (𝔉_{t})_{t∈[0,T]} generated by (w(s))_{s∈[0,t]}. The set of all nonempty, compact, convex subsets of ℝ^{n} is denoted by P_{kc}(ℝ^{n}) and the ε-ball at x is denoted by B_{ε}(x). Let F : Ω × ℝ^{n} → P_{kc}(ℝ^{n}) be a measurable multifunction with E {∫_{0}^{T} | F (x(t)) | dt} < ∞. Further let b : Ω × [0, T] × ℝ^{n} → ℝ^{n,d} be a 𝔉_{t}- measurable process with E {∫_{0}^{T} ∥b(t, x(t))∥_{F}^{2} dt} < ∞. We write | A | = sup_{a∈A} ∥a∥ for a subset A of a linear space 𝕏 with a corresponding norm ∥.∥. Assumptions and background for using differences of sets are the same like in [4]. We define a ”strong” Hausdorff continuity in the following way.

Definition 1 Let 𝕏 be a Hausdorff topological space and let 𝕐 be a metric space. A multifunction F : 𝕏 → P_{kc}(𝕐) is said to be

(a) strong h-upper semicontinuous at x^{0} ∈ 𝕏 if for every ε > 0, there is δ = δ(ε) > 0 such that for all x ∈ B_{δ}(x^{0}) we have (F (x) − F (x^{0})) ⊂ B_{ε}(\underline{0}),

(b) strong h-lower semicontinuous at x^{0} ∈ 𝕏 if for every ε > 0, there is δ = δ(ε) > 0 such that for all x ∈ B_{δ}(x^{0}) we have \underline{0} ∈ (F (x) − F (x^{0})) + B_{ε}(\underline{0}),

(c) strong h-continuous at x^{0} ∈ 𝕏 if (a) and (b) are fulfilled. If (a) (resp. (b), (c)) are valid at every x^{0} ∈ 𝕏, then we call F strong h-upper semicontinuous (resp. strong h-lower semicontinuous, strong h-continuous).

We say that for any N > 0 the process b fulfils a locally Lipschitz condition with a constant L_{N} > 0 for every (ω, t, x), (ω, t, y) ∈ Ω × [0, T] × clB_{N}(\underline{0}) if

∥b(t, x) − b(t, y)∥_{F} ≤ L_{N}∥x − y∥  (2)

with ∥b∥_{F}^{2} = tr(b^{T} b) and that F and b are bounded in growth for all (ω, t, x) ∈ Ω × [0, T] × ℝ^{n} if

| F (x) |^{2} +∥b(t, x)∥_{F}^{2} ≤ K (1 + ∥x∥^{2}) ,  (3)

where K is a positive constant.

Remark 1 From Remark 1.19 of [5] we derive for a nonempty subset A of a normed space 𝕏 the equation | cl convA | = | A |.

Remark 2 If the multifunction F fulfils relation (3) we have | F (Y) |^{2} ≤

K (1+ | Y |^{2}) with F (Y) = ⋃_{y∈Y} F (y).

The following notion of a continuous partition of unity is well-known and can be looked up for instance in [1].

2 LOCALLY LIPSCHITZEAN APPROXIMATION

OF UPPER SEMICONTINUOUS MULTIFUNC-

TIONS

We know from [7] that a continuous function f : Y → 𝕏, with Y ⊂ 𝕐 open, can be approximated by locally Lipschitzean maps. For h-upper semicontinuous multifunctions F (for definition see [5]) there is a transformed approximation result in [2]. If the multifunction F is strong h-upper semicontinuous in the sense of Definition 1 this result can be modified to the following.

Lemma 1 Let Y be a nonempty subset of a Banach space 𝕐 and 𝕏 be a Banach space. Further let F : Y → P_{kc}(𝕏) be a multifunction. For r_{k} = 3^{−k} let (A_{λ})_{λ∈Λ} be a locally finite refinement of the open covering Y = ⋃_{y∈Y} [B_{r_{k}}(y) ∩ Y] and (ψ_{λ})_{λ∈Λ} be a locally Lipschitzean partition of unity subordinated to (A_{λ})_{λ∈Λ}. For choosed y_{λ} ∈

A_{λ} ⊆ [B_{r_{k}}(y_{λ}) ∩ Y] let F_{k}(y) = ∑_{λ∈Λ} ψ_{λ}(y)M_{λ} with M_{λ} = cl convF (B_{2r_{k}}(y_{λ}) ∩ Y).

Then it holds

(a) F (y) ⊆ F_{k+1}(y) ⊆ F_{k}(y) ⊆ cl convF (Y) on Y.

(b) If F is uniformly locally bounded then F_{k} is locally (strong) Lipschitzean for k ≥ 1.

(c) If F is strong h-upper semicontinuous then | F (y) − F_{k}(y) |→ 0 with k → ∞ on Y.

Remark 3 If Y is a compact subset of 𝕐 we have a finite index set Λ.

Proof: (a) If ψ_{λ}(y) > 0 then by definition y ∈ supp(ψ_{λ}(y)) ⊆ A_{λ} ⊆ [B_{r_{k}}(y_{λ}) ∩ Y], because (A_{λ})_{λ∈Λ} is a locally finite refinement of [B_{r_{k}}(y_{λ}) ∩ Y]. By this F (y) ⊆ M_{λ} and consequently F (y) ⊆ F_{k}(y) = ∑_{λ∈Λ} ψ_{λ}(y)M_{λ}. Obviously it holds M_{λ} ⊆ cl convF (Y). Therefore we have also F_{k}(y) ⊆ cl convF (Y) . It follows for arbitrary k the relation F (y) ⊆ F_{k}(y) ⊆ cl convF (Y). We have to show now that

F_{k+1}(y) ⊆ F_{k}(y). Let F_{k+1}(y) = ∑_{μ∈Λ} ψ̃_{μ}(y)M̃_{μ} with M̃_{μ} = cl convF (B_{2r_{k}}(ỹ_{μ}) ∩ Y).

If ψ_{λ}(y) > 0 and ψ̃_{μ}(y) > 0 then ∥y_{λ} −ỹ_{μ}∥ ≤ ∥y_{λ} −y∥+∥y−ỹ_{μ}∥ ≤ r_{k} +r_{k+1}, because y ∈ supp(ψ_{λ}(y)) and y ∈ supp(ψ̃_{μ}(y)) respectively. Since r_{k} + r_{k+1} = \frac{4r_{k}}{3} < 2r_{k} it follows that B_{2r_{k+1}}(ỹ_{μ}) ⊆ B_{2r_{k}}(y_{λ}) and so M̃_{μ} ⊆ M_{λ}. By definition we receive F_{k+1}(y) ⊆ F_{k}(y) for all y ∈ Y.

(b) Remember we call F_{k} locally (strong) Lipschitzean with constant l_{N} > 0 if for any N > 0 there is a l_{N} with | F_{k}(y^{1}) − F_{k}(y^{2}) |≤ l_{N}∥y^{1} − y^{2}∥ for all y^{1}, y^{2} ∈ Y with ∥y^{1}∥ ≤ N, ∥y^{2}∥ ≤ N and F is called uniformly locally bounded if there exist N > 0 and k_{N} > 0 so that | F (y) |≤ k_{N} for all y ∈ Y with ∥y∥ ≤ N. If F is uniformly locally bounded then we have also | M_{λ} |= sup {∥f∥_{𝕏} : f ∈ M_{λ}} ≤ k_{N} for corresponding λ with N ≥ 4r_{k}. So we receive the locally Lipschitz condition for F_{k} by the locally Lipschitz condition of the partition of unity (ψ_{λ})_{λ∈Λ} subordinated to the locally finite refinement (A_{λ})_{λ∈Λ} of the open covering of Y from above. Then

| F_{k}(y^{1}) − F_{k}(y^{2}) | ≤ ∑_{λ∈Λ} ∥ψ_{λ}(y^{1}) − ψ_{λ}(y^{2})∥ | M_{λ} | ≤ l_{N}∥y^{1} − y^{2}∥ .

(c) Recalling Definition 1 of strong h-upper semicontinuity of F we see that it is equivalent to | F (y) − F (y^{0}) |< ε for all y ∈ [B_{δ}(y^{0}) ∩ Y ] with δ = δ(y^{0}, ε) > 0 and all y^{0} ∈ Y. Assume now statement (c) is false for instance in y^{0} ∈ Y. Then at least one ε > 0 exists so that | F (y^{0}) − F_{k}(y^{0}) |≥ ε for all k ≥ 1, i.e. (F (y^{0}) − F_{k}(y^{0})) ⊄ B_{ε}(\underline{0}) for k ≥ 1. By this (F (y^{0}) − M_{λ}) ⊄ B_{ε}(\underline{0}). Therefore it exists at least one y_{k} ∈ [B_{2r_{k}}(y_{λ}) ∩ Y] with

(F (y^{0}) − F (y_{k})) ⊄ B_{ε}(\underline{0})  (4)

for all k ≥ 1. By consideration of y^{0} ∈ supp(ψ_{λ}(y^{0}) ⊆ A_{λ} ⊆ [B_{r_{k}}(y_{λ}) ∩ Y] we obtain

∥y^{0} − y_{k}∥ ≤ ∥y^{0} − y_{λ}∥ + ∥y_{λ} − y_{k}∥ ≤ r_{k} + 2r_{k} = r_{k−1} ⟶ 0

for k → ∞. By assumption F is strong h-upper semicontinuous and consequently a number k_{0} ≥ 1 exists so that for all k ≥ k_{0} it holds (F (y^{0}) − F (y_{k})) ⊂ B_{ε}(\underline{0}). This is a contradiction to (4) and so we have | F (y) − F_{k}(y) |→ 0 with k → ∞ for all y ∈ Y.

□

Remark 4 If the multifunction F fulfils the estimation (3) then for all y ∈ Y we receive for the locally Lipschitzean approximation F_{k} with k ≥ 1 the following relation

| F_{k}(y) |^{2}≤ K (1 + r_{k−1} + ∥y∥^{2})  (5)

which can be proved using Remarks 1 and 2.

3 EXISTENCE OF STRONG SOLUTIONS OF IN-

CLUSIONS WITH UPPER SEMICONTINUOUS

DRIFT

Using the locally Lipschitzean approximation of a strong h-upper semicontinuous multifunction described in Lemma 1 we show now the existence of a solution of the stochastic inclusion (1). Theorem 1 For fixed ω ∈ Ω let F : Ω × ℝ^{n} → P_{kc}(ℝ^{n}) be a strong h-upper semicontinuous multifunction satisfying ( 3). Furthermore let b : Ω × [0, T] × ℝ^{n} → ℝ^{n,d} fulfil the conditions (2) and (3). Then the inclusion

x(t) ∈ x^{0} + ∫_{0}^{t} F (x(s))ds + ∫_{0}^{t} b(s, x(s))dw(s) has a nonempty set X(.) of solutions x(.).

Remark 5 If F satisfies the boundedness condition (3) for all x ∈ ℝ^{n} then the multifunction is uniformly locally bounded too .

Proof: For N > 0 and fixed ω ∈ Ω let (F_{N}^{k})_{k≥1} be a sequence of locally Lipschitzean (with constant l_{N}) multifunctions on Ω × clB_{N}(\underline{0}) ⊂ Ω × ℝ^{n} defined by Lemma 1 and approximating F. So we construct sequences of inclusions

y_{N}(t) ∈ y_{N}^{0} + ∫_{0}^{t} F_{N}(y_{N}(s))ds + ∫_{0}^{t} b_{N}(s, y_{N}(s))dw(s)  (6)
y_{N}^{k}(t) ∈ y_{N}^{0} + ∫_{0}^{t} F_{N}^{k}(y_{N}^{k}(s))ds + ∫_{0}^{t} b_{N}(s, y_{N}^{k}(s))dw(s)  (7)

with an initial condition y_{N}^{0} = \{ \begin{matrix}x^{0} & : & ∥x^{0}∥ ≤ N \\ \frac{N}{∥x^{0}∥}x^{0} & : & ∥x^{0}∥ > N \end{matrix} and coeffi-

cient processes b_{N}(t, x) = b(t, x^{N}), F_{N}(x) = F (x^{N}) and F_{N}^{k}(x) =

F^{k}(x^{N}), where x^{N} = \{ \begin{matrix}x & : & ∥x∥ ≤ N \\ \frac{N}{∥x∥}x & : & ∥x∥ > N \end{matrix} . We define stopping

times T_{N} = \{ \begin{matrix}T & : & | Y_{N}(t) |≤ N, ∀t ∈ [0, T] \\ inf_{t∈[0,T]} \{t : | Y_{N}(t) |> N\} & : & otherwise \end{matrix} and

T_{N}^{k} = \{ \begin{matrix}T & : & | Y_{N}^{k}(t) |≤ N, ∀t ∈ [0, T] \\ inf_{t∈[0,T]} \{t : | Y_{N}^{k}(t) |> N\} & : & otherwise \end{matrix} for the sets

Y_{N}(.) and Y_{N}^{k}(.) of solutions of (6) and of (7). Set T̃_{N} = T_{N} ∧ T_{N}^{k}. We have to show now that for t ∈ [0, T̃_{N}] the set Y_{N}^{k}(t) converges in the strong-h-sense to the set Y_{N}(t) with k → ∞. From the proof of Theorem 1 of [4] we know that Y_{N}^{k}(.) ≠ ∅. So we estimate using Theorem 3.14 of [6]

E {| Y_{N}(t ∧ T̃_{N}) − Y_{N}^{k}(t ∧ T̃_{N}) |^{2}}

≤ 4T E {∫_{0}^{t∧T̃_{N}} | F_{N}(Y_{N}(s)) − F_{N}^{k}(Y_{N}(s)) |^{2} ds} +

+4T E {∫_{0}^{t∧T̃_{N}} | F_{N}^{k}(Y_{N}(s)) − F_{N}^{k}(Y_{N}^{k}(s)) |^{2} ds} +

+2E {∫_{0}^{t∧T̃_{N}} | b_{N}(s, Y_{N}(s)) − b_{N}(s, Y_{N}^{k}(s)) |^{2} ds}

≤ 4T E {∫_{0}^{t∧T̃_{N}} | F_{N}(Y_{N}(s)) − F_{N}^{k}(Y_{N}(s)) |^{2} ds} +

+2 (2T l_{N}^{2} + L_{N}^{2}) ∫_{0}^{t} E {| Y_{N}(s ∧ T̃_{N}) − Y_{N}^{k}(s ∧ T̃_{N}) |^{2} ds}

and with Gronwall lemma

E {| Y_{N}(t ∧ T̃_{N}) − Y_{N}^{k}(t ∧ T̃_{N}) |^{2}}

≤ 4T E {∫_{0}^{t∧T̃_{N}}| F_{N}(Y_{N}(s)) − F_{N}^{k}(Y_{N}(s)) |^{2} ds} +
+4T C_{0} ∫_{0}^{t} [e^{C_{0}(t−s)}E {∫_{0}^{s∧T̃_{N}}| F_{N}(Y_{N}(τ)) − F_{N}^{k}(Y_{N}(τ)) |^{2} dτ}] ds . (8)

Going to the limit for k → ∞ in (8) and using Fatou lemma and Lemma 1 we obtain

lim_{k→∞} E {| Y_{N}(t ∧ T̃_{N}) − Y_{N}^{k}(t ∧ T̃_{N}) |^{2}}

≤ 4T lim sup_{k→∞} E {∫_{0}^{t∧T̃_{N}} | F_{N}(Y_{N}(s)) − F_{N}^{k}(Y_{N}(s)) |^{2} ds} +

+4T C_{0} lim sup_{k→∞}∫_{0}^{t}[e^{C_{0}(t−s)}E {∫_{0}^{s∧T̃_{N}} | F_{N}(Y_{N}(τ)) − F_{N}^{k}(Y_{N}(τ)) |^{2} dτ}] ds

≤ 4T E {∫_{0}^{t∧T̃_{N}} lim_{k→∞} | F_{N}(Y_{N}(s)) − F_{N}^{k}(Y_{N}(s)) |^{2} ds} +

+4T C_{0} ∫_{0}^{t}[e^{C_{0}(t−s)}E {∫_{0}^{s∧T̃_{N}}lim_{k→∞} | F_{N}(Y_{N}(τ)) − F_{N}^{k}(Y_{N}(τ)) |^{2} dτ}] ds

= 0 .  (9)

Now we estimate uniformly with a generalized Doob inequality (see [3]) and Hoelder inequality (for p = ∞, q = 1)

E { sup_{t∈[0,T̃_{N}]} | Y_{N}(t) − Y_{N}^{k}(t) |^{2}}

≤ 2T E {∫ T̃_{N} ; 0 | F_{N}(Y_{N}(s)) − F_{N}^{k}(Y_{N}^{k}(s)) |^{2} ds} +

+8E {∫ T̃_{N} ; 0 | b_{N}(s, Y_{N}(s)) − b_{N}(s, Y_{N}^{k}(s)) |^{2} ds}

≤ 4T E {∫ T̃_{N} ; 0 | F_{N}(Y_{N}(s)) − F_{N}^{k}(Y_{N}(s)) |^{2}} +

+4 (T l_{N}^{2} + 2L_{N}^{2}) E {∫_{0}^{T} χ_{[0,T̃_{N}]}(s) | Y_{N}(s) − Y_{N}^{k}(s) |^{2} ds}

≤ 4T E {∫ T̃_{N} ; 0 | F_{N}(Y_{N}(s)) − F_{N}^{k}(Y_{N}(s)) |^{2} ds} +

+C_{1} ∫_{0}^{T} ess sup_{ω} χ_{[0,T̃_{N}]}(s)E {| Y_{N}(s ∧ T̃_{N}) − Y_{N}^{k}(s ∧ T̃_{N}) |^{2}} ds

≤ 4T E {∫ T̃_{N} ; 0 | F_{N}(Y_{N}(s)) − F_{N}^{k}(Y_{N}(s)) |^{2} ds} +

+C_{1} ∫_{0}^{T} E {| Y_{N}(s ∧ T̃_{N}) − Y_{N}^{k}(s ∧ T̃_{N}) |^{2}} ds

{(8) \atop ≤} 4T E \{∫ T̃_{N} ; 0 | F_{N}(Y_{N}(s)) − F_{N}^{k}(Y_{N}(s)) |^{2} ds\} +

+C_{2} ∫_{0}^{T} E {∫_{0}^{s∧T̃_{N}} | F_{N}(Y_{N}(t)) − F_{N}^{k}(Y_{N}(t)) |^{2} dt} ds +

+C_{2} ∫_{0}^{T} ∫_{0}^{s} e^{C_{0}(s−t)}E {∫_{0}^{t∧T̃_{N}} | F_{N}(Y_{N}(τ)) − F_{N}^{k}(Y_{N}(τ)) |^{2} dτ} dtds .

Going to the limit for k → ∞ we receive lim_{k→∞} E { sup_{t∈[0,T̃_{N}]} | Y_{N}(t) − Y_{N}^{k}(t) |^{2}} = 0 . Furthermore we have

lim_{k→∞} P ( sup_{t∈[0,T̃_{N}]} | Y_{N}(t) − Y_{N}^{k}(t) |> 0) = lim_{k→∞} P (∥T_{N} − T_{N}^{k}∥ > 0)  (10)

and with g(a) = (1 + a^{2})^{−1}

lim_{k→∞} P ( sup_{t∈[0,T̃_{N}]} | Y_{N}(t) − Y_{N}^{k}(t) |> 0)

≤ lim_{k→∞} P ( sup_{s∈[0,T]} | Y_{N}^{k}(s) |> N) + P ( sup_{s∈[0,T]} | Y_{N}(s) |> N)
= lim_{k→∞} P (g(∥x^{0}∥) sup_{s∈[0,T]} | Y_{N}^{k}(s) |^{2}> N^{2}g(∥x^{0}∥)) +

+P (g(∥x^{0}∥) sup_{s∈[0,T]} | Y_{N}(s) |^{2}> N^{2}g(∥x^{0}∥))

≤ lim_{k→∞} P (g(∥x^{0}∥) sup_{s∈[0,T]} | Y_{N}^{k}(s) |^{2}> N^{2}δ) + 2P (g(∥x^{0}∥) ≤ δ) +

+P (g(∥x^{0}∥) sup_{s∈[0,T]} | Y_{N}(s) |^{2}> N^{2}δ)

≤ lim_{k→∞} E \{g(∥x^{0}∥) sup_{s∈[0,T]} | Y_{N}^{k}(s) |^{2}\} \frac{1}{N^{2}δ} + 2P (g(∥x^{0}∥) ≤ δ) +
+E \{g(∥x^{0}∥) sup_{s∈[0,T]} | Y_{N}(s) |^{2}\} \frac{1}{N^{2}δ} .  (11)

Analogously to the proof of Theorem 1 of [4] we obtain

E {g(∥x^{0}∥) sup_{s∈[0,T]} | Y_{N}(s) |^{2}} ≤ C_{3}
E {g(∥x^{0}∥) sup_{s∈[0,T]} | Y_{N}^{k}(s) |^{2}} ≤ C_{4}(1 + r_{k−1})

(12)

where the constants are independent from N. In this way we continue estimation (11)

lim_{k→∞} P ( sup_{t∈[0,T̃_{N}]} | Y_{N}(t) − Y_{N}^{k}(t) |> 0)

{(12) \atop ≤} lim_{k→∞} C_{4} \frac{(1 + r_{k−1})}{N^{2}δ} + 2P (g(∥x^{0}∥) ≤ δ) + \frac{C_{3}}{N^{2}δ}
= \frac{(C_{4} + C_{3})}{N^{2}δ} + 2P (g(∥x^{0}∥) ≤ δ) .

And so we get

lim sup_{N→∞} lim_{k→∞} P ( sup_{t∈[0,T̃_{N}]} | Y_{N}(t) − Y_{N}^{k}(t) |> 0) ≤ 2P (g(∥x^{0}∥) ≤ δ)

and because of δ is any positive number and P (g(∥x^{0}∥) = 0) = 0 the existence of the limit for N → ∞ follows. By this lim_{N→∞} lim_{k→∞} P ( sup_{t∈[0,T̃_{N}]} | Y_{N}(t) − Y_{N}^{k}(t) |> 0) = 0

and lim_{k→∞} P (∥T_{N} − T_{N}^{k}∥ > 0) = 0.

□

Remark 6 Stochastic differential inclusions involving a multifunction F which depends explicitly from time t ∈ [0, T] can considered in the same way if we can find a pair [U, ϕ] so that for all (t, x) ∈ [0, T] × ℝ^{n} and fixed ω ∈ Ω we have F (t, x) = ϕ(x, U(t)). Here ϕ : ℝ^{n} × ℝ^{m} → ℝ^{n} is a continuous map and U(.) is a strong h-upper semicontinuous multifunction with nonempty, compact and convex values in ℝ^{m}. Such a representation of F is called parametrization.

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