EXISTENCE OF STRONG SOLUTIONS OF A

STOCHASTIC DIFFERENTIAL INCLUSION.

1. THE LOCALLY LIPSCHITZEAN CASE

H. Fritsch

Institut für Optimierung und Stochastik, FB Mathematik und Informatik,

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

Abstract

In this paper we consider stochastic dynamic systems with a multivalued drift coefficient where the stochastic integral is defined in the sense of Ito. We show existence results for strong solutions under weakened classical conditions like locally Lipschitz condition. Key words: stochastic differential inclusion, locally Lipschitz condition, setvalued system, existence of solutions. AMS subject classifications: 60H10, 34A60.

1 INTRODUCTION

In the last 10 years various authors published papers on the field of differential

inclusions with random uncertainty which is splitted up into three main directions.

The first one is the consideration of random inclusions like in [13], [17] and others.

The second direction is determined by questions of stochastic evolution inclusions

([5], [1], [3], [2]) and in the third we have stochastic differential inclusions like in this

article or with a setvalued diffusion term ([19], [4], [14], [15], [12], [11]). The aim of

this paper is a translation of the classical weakening of conditions for existence of solutions of stochastic differential equations in the sense of Ito to stochastic inclusions of the third direction.

On a complete probability space (Ω, 𝔉, P) we consider the following stochastic inclusion

x(t) ∈ x^{0} + ∫_{0}^{t} F (s, 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 understand 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]}.

By (𝔉_{t})_{t∈[0,T]} we describe a family of σ-fields generated by (w(s))_{s∈[0,t]}. We denote the set of all nonempty, compact and convex subsets of ℝ^{n} by P_{kc}(ℝ^{n}) and the ε-ball at x by B_{ε}(x). Let F : Ω × [0, T] × ℝ^{n} → P_{kc}(ℝ^{n}) be a 𝔉_{t}-measurable multifunction with E {∫_{0}^{T} | F (t, x(t)) | dt} < ∞, where | A |= sup_{a∈A} ∥a∥ for a subset A of a linear space 𝕏 with a corresponding norm ∥.∥. 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} < ∞, where we write ∥.∥_{F} for the Frobenius norm of a real matrix. Following reflections of [16] for the family of nonempty, compact valued and convex valued multifunctions defined on Ω×[0, T]×ℝ^{n} we use the embedding theorem of [18] to view the elements of this as single valued functions taking values in a real Banach space. In this way we by-pass the lack of a difference of sets in the sense of Minkowski. Combining this idea with the definition of Hausdorff continuity (e.g. in [9]) we define the ”strong” Hausdorff continuity.

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).

Remark 1 Because of h(A, B) ≤| A − B | it is obviously that from strong hcontinuity the h-continuity in [9] follows, where h(., .) stands for the well-known Hausdorff distance.

We say that for any N > 0 the processes F and b fulfil 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

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

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

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

where K is a positive constant.

If X(.) is a set of trajectories x(.) we have for a multifunction F, which fulfils condition (3), obviously

| F (., X(.)) |^{2} ≤ sup_{x∈X} | F (., X(.)) |^{2} ≤ K (1+ | X(.) |^{2})  (4)

with F (., X(.)) = ⋃_{x∈X} F (., x(.)). From Theorem 4.1. of [8] we derive

| ∫_{0}^{t} F (s, X(s))ds |≤ ∫_{0}^{t} | F (s, X(s)) | ds  (5)

and from [10] we know that for a setvalued stochastic integral (in the sense of Ito) the following equation is derivable

E {| ∫_{0}^{t} b(s, X(s))dw(s) |^{2}} = E {∫_{0}^{t} | b(s, X(s)) |^{2} ds} ,  (6)

where in the right hand side | . | is applied to the Frobenius norm. From [7] we have the following version of the well-known Doob maximal martingale inequality for setvalued stochastic integrals in the sense of Kisielewicz ([10] , [11] )

E { sup_{t∈[0,T]} | (K)∫_{0}^{t} B(s) dw(s) |^{2}} ≤ 4 ∫_{0}^{T} E {| B(t) |^{2}} dt  (7)

where B(.) is a well-defined setvalued stochastic process like in the simplest way obtained by parametrization B(.) = b(X(.)) = {b(x(.)) : x(.) ∈ X(.)}.

2 EXISTENCE OF SOLUTIONS OF DIFFERENTIAL

INCLUSIONS WITH LOCALLY LIPSCHITZEAN

COEFFICIENTS

Now we want to show the existence of solutions of inclusion (1) for given initial value under weaker conditions. If uniqueness of solution is important for the consideration we have to assume that the multivalued drift F is dissipative too (see e.g. [6], [15]). To weaken the assumptions for existence of solutions of (1) we need the following statement about locally dependence of the solution set of (1) from the coefficients F and b of the inclusion. At first for some N > 0 we define a stopping time T^{N} as the earliest moment that any solution of (1) is leaving clB_{N}(\underline{0}), i.e. T^{N} = T

if ∥x(t)∥ ≤ N, ∀t ∈ [0, T], ∀x(.) ∈ X(.) or T^{N} = inf_{x(.)∈X(.)} inf_{t∈[0,T]} {t : ∥x(t)∥ > N}

otherwise. Using Hausdorff distance we have

T^{N} = \{ \begin{matrix}T & : & | X(t) |≤ N, ∀t ∈ [0, T] \\ inf_{t∈[0,T]} \{t : | X(t) |> N\} & : & otherwise \end{matrix}  (8)

where X(.) is defined as set of solutions of (1).

Lemma 1 Assuming that the processes F_{i}(t, x) and b_{i}(t, x) with i = 1, 2 of the stochastic inclusions

x^{i}(t) ∈ x^{0} + ∫_{0}^{t} F_{i}(s, x^{i}(s))ds + ∫_{0}^{t} b_{i}(s, x^{i}(s))dw(s)  (9)

satisfy the conditions of section 1, a global Lipschitz condition with a constant L > 0 and (3). For some N > 0 the equations

F_{1}(t, x) = F_{2}(t, x) , b_{1}(t, x) = b_{2}(t, x)  (10)

hold for ∥x∥ ≤ N. Let X^{1}(.) and X^{2}(.) be the sets of solutions of inclusions (9) and let T_{i}^{N} be stopping times for X^{i}(.) (i = 1, 2) defined by (8).

Then we have P ( sup_{t∈[0,T_{1}^{N}]} | X^{1}(t) − X^{2}(t) |= 0) and P (T_{1}^{N} = T_{2}^{N}) = 1.

Proof: We define the characteristic function of the interval [0, T_{1}^{N}]

χ_{[0,T_{1}^{N}]}(t) = \{ \begin{matrix}1 & : & sup_{s∈[0,t]} | X^{1}(s) |≤ N \\ 0 & : & sup_{s∈[0,t]} | X^{1}(s) |> N \end{matrix}

and derive the following estimation

E {χ_{[0,T_{1}^{N}]}(t) | X^{1}(t) − X^{2}(t) |^{2}}
≤ E {χ_{[0,T_{1}^{N}]}(t) | ∫_{0}^{t} F_{1}(s, X^{1}(s)) − F_{2}(s, X^{2}(s))ds+
+ ∫_{0}^{t} b_{1}(s, X^{1}(s)) − b_{2}(s, X^{2}(s))dw(s) |^{2}}
≤ 2E {χ_{[0,T_{1}^{N}]}(t) | ∫_{0}^{t} F_{1}(s, X^{1}(s)) − F_{2}(s, X^{2}(s))ds |^{2} +
+χ_{[0,T_{1}^{N}]}(t) | ∫_{0}^{t} b_{1}(s, X^{1}(s)) − b_{2}(s, X^{2}(s))dw(s) |^{2}}
≤ 4E {χ_{[0,T_{1}^{N}]}(t) | ∫_{0}^{t} F_{1}(s, X^{1}(s)) − F_{2}(s, X^{1}(s))ds |^{2} +

+χ_{[0,T_{1}^{N}]}(t) | ∫_{0}^{t} F_{2}(s, X^{1}(s)) − F_{2}(s, X^{2}(s))ds |^{2} +
+χ_{[0,T_{1}^{N}]}(t) | ∫_{0}^{t} b_{1}(s, X^{1}(s)) − b_{2}(s, X^{1}(s))dw(s) |^{2} +
+χ_{[0,T_{1}^{N}]}(t) | ∫_{0}^{t} b_{2}(s, X^{1}(s)) − b_{2}(s, X^{2}(s))dw(s) |^{2}}
{(10) \atop =} 4E \{| ∫_{0}^{t} χ_{[0,T_{1}^{N}]}(s) (F_{2}(s, X^{1}(s)) − F_{2}(s, X^{2}(s))) ds |^{2} +
+ | ∫_{0}^{t} χ_{[0,T_{1}^{N}]}(s) (b_{2}(s, X^{1}(s)) − b_{2}(s, X^{2}(s))) dw(s) |^{2}}
{(5),(6) \atop ≤} 4T E \{∫_{0}^{t} χ_{[0,T_{1}^{N}]}(s) | F_{2}(s, X^{1}(s)) − F_{2}(s, X^{2}(s)) |^{2} ds\}
+4E {∫_{0}^{t} χ_{[0,T_{1}^{N}]}(s) | b_{2}(s, X^{1}(s)) − b_{2}(s, X^{2}(s)) |^{2} ds}
{(4) \atop ≤} 4(T + 1)L^{2} ∫_{0}^{t} E \{χ_{[0,T_{1}^{N}]}(s) | X^{1}(s) − X^{2}(s) |^{2}\} ds .  (11)

Applying the Gronwall lemma to (11) we receive E {χ_{[0,T_{1}^{N}]}(t) | X^{1}(t) − X^{2}(t) |^{2}} = 0 and therefore P (X^{1}(t) = X^{2}(t)) = 1 for 0 ≤ t ≤ T_{1}^{N}. Because of the (strong) classical conditions to the coefficients of (9) the elements of X^{1}(.) and X^{2}(.) are continuous with probability 1 and so we receive P ( sup_{t∈[0,T_{1}^{N}]} | X^{1}(t) − X^{2}(t) |= 0) = 1. In this way we have T_{1}^{N} ≤ T_{2}^{N} with probability 1. By changing indexes above in the proof we have also P (T_{2}^{N} ≤ T_{1}^{N}) = 1 and so the proof is complete.

□

Now we are able to weaken the global Lipschitz condition.

Theorem 1 Let the coefficient processes F and b fulfil the usual conditions and the conditions (2) and (3). Then the inclusion (1) has a nonempty solution set with probability 1.

Proof: At first we define an initial value y_{N}^{0} with y_{N}^{0} =

\{ \begin{matrix}x^{0} & : & ∥x^{0}∥ ≤ N \\ \frac{N}{∥x^{0}∥}x^{0} & : & ∥x^{0}∥ > N \end{matrix} and using x^{N} = \{ \begin{matrix}x & : & ∥x∥ ≤ N \\ \frac{N}{∥x∥}x & : & ∥x∥ > N \end{matrix} the new coef-

ficient processes F_{N}(t, x) = F (t, x^{N}) and b_{N}(t, x) = b(t, x^{N}). Let (y_{N}(t)) be a solution of the inclusion

y_{N}(t) ∈ y_{N}^{0} + ∫_{0}^{t} F_{N}(s, y_{N}(s))ds + ∫_{0}^{t} b_{N}(s, y_{N}(s))dw(s) .  (12)

For the inclusion (12) all assumptions from the known theorems about existence of solutions ([5], [11] a.o.) are fulfilled. So there is at least a solution (y_{N}(t)) of (12) which is continuous with probability 1 and for which we have E { sup_{t∈[0,T^{N}]} ∥y_{N}(t)∥^{2}} < ∞, where T^{N} is defined by (8).

Now let N′ > N. For x ∈ B_{N}(\underline{0}) we have F_{N}(t, x) = F_{N′}(t, x) and b_{N}(t, x) = b_{N′}(t, x). By Lemma 1 both solution sets Y_{N}(t) and Y_{N′}(t) correspond with probability 1 on the interval [0, T^{N}]. Then for N′ > N we have

P ( sup_{t∈[0,T]} | Y_{N}(t) − Y_{N′}(t) |> 0) = P (T^{N} < T )
=  P (Y_{N}(T^{N})\B_{N}(\underline{0}) ≠ ∅) = P ( sup_{t∈[0,T]} | Y_{N}(t) |> N) .

If we can show now that P ( sup_{t∈[0,T]} | Y_{N}(t) |> N) tends to zero for N → ∞, then we get the uniform convergence with probability 1 in the Hausdorff distance of the set Y_{N}(t) to a limit set X(t). If we apply the strong h-limit for N → ∞ to the inclusion (12) we see that the limit set X(t) contains the continuous solutions of inclusion (1). Let g(α) = \frac{1}{1+α^{2}} for a real number α. So we estimate

E {g(∥x^{0}∥) | Y_{N}(t) |^{2}}

≤ 4E {g(∥x^{0}∥)∥y_{N}^{0}∥^{2} + g(∥x^{0}∥) | ∫_{0}^{t} F_{N}(s, Y_{N}(s))ds |^{2} +

+g(∥x^{0}∥) | ∫_{0}^{t} b_{N}(s, Y_{N}(s))dw(s) |^{2}}

{(6) \atop ≤} 4E \{1 + T g(∥x^{0}∥) ∫_{0}^{t} | F_{N}(s, Y_{N}) |^{2} ds+

+g(∥x^{0}∥) ∫_{0}^{t} | b_{N}(s, Y_{N}(s)) |^{2} ds}

{(4) \atop ≤} 4 + 4(T + 1)K ∫_{0}^{t} E \{g(∥x^{0}∥) (1+ | Y_{N}(s) |^{2})\} ds
≤ 4 (1 + (T + 1)T KE {g(∥x^{0}∥)}) +

+4(T + 1)K ∫_{0}^{t} E {g(∥x^{0}∥) | Y_{N}(s) |^{2}} ds

and using Gronwall inequality

E {g(∥x^{0}∥) | Y_{N}(t) |^{2}} ≤ C̃_{0}e^{4(T +1)Kt} ≤ C_{0}  (13)

where the constant C_{0} > 0 is independent from N. Furthermore we receive

E {g(∥x^{0}∥) sup_{t∈[0,T]} | Y_{N}(t) |^{2}}

≤ 4E {g(∥x^{0}∥)∥y_{N}^{0}∥^{2} + g(∥x^{0}∥) sup_{t∈[0,T]} | ∫_{0}^{t} F_{N}(s, Y_{N}(s))ds |^{2} +

+g(∥x^{0}∥) sup_{t∈[0,T]} | ∫_{0}^{t} b_{N}(s, Y_{N}(s))dw(s) |^{2}}

{(7) \atop ≤} 4E \{1 + T g(∥x^{0}∥) ∫_{0}^{T} | F_{N}(s, Y_{N}) |^{2} ds+  (14)

+4g(∥x^{0}∥) ∫_{0}^{T} | b_{N}(s, Y_{N}(s)) |^{2} ds}

{(4) \atop ≤} 4 + 4(T + 4)K ∫_{0}^{T} E \{g(∥x^{0}∥) (1+ | Y_{N}(s) |^{2})\} ds
{(13) \atop ≤} C_{1}  (15)

with a constant C_{1}, independent from N too. For an arbitrary δ > 0 we get, in the well-known way,

P ( sup_{t∈[0,T]} | Y_{N}(t) |> N)

= P (g(∥x^{0}∥) sup_{t∈[0,T]} | Y_{N}(t) |^{2}> N^{2}g(∥x^{0}∥))
≤ P (g(∥x^{0}∥) sup_{t∈[0,T]} | Y_{N}(t) |^{2}> δN^{2}) + P (g(∥x^{0}∥) ≤ δ)
≤ \frac{C_{1}}{δN^{2}} + P (g(∥x^{0}∥) ≤ δ) .

So it holds

lim sup_{N→∞} P ( sup_{t∈[0,T]} | Y_{N}(t) |> N) ≤ P (g(∥x^{0}∥) ≤ δ) .

Because of P (g(∥x^{0}∥) = 0) = 0 and δ is any positive number we derive that P ( sup_{t∈[0,T]} | Y_{N}(t) |> N) tends to zero for N → ∞. By this we have the existence of solutions of inclusion (1).

□

Following the traditional way of proving existence theorems for stochastic differential equations we change now the condition (3) by

| x^{T} F (t, x) | +∥b(t, x)∥_{F}^{2} ≤ K (1 + ∥x∥^{2})  (16)

for all (ω, t, x) ∈ Ω × [0, T] × ℝ^{n} with a constant K > 0, where x^{T} F (., x) = {x^{T} f(., x) : f(., x) ∈ F (., x)}.

Theorem 2 Let the processes F and B fulfil the usual assumptions and the conditions (2) and (16). Then the solution set of inclusion (1) is not empty with probability 1.

Proof: Analogously to the proof of Theorem 1 we have to point out that for a constant C_{2} > 0, independent from N, the inequality E {g(∥x^{0}∥) | Y_{N}(t) |^{2}} ≤ C_{2} is realized. Using Ito formula (see [6]) to (12) we obtain

∥y_{N}(t)∥^{2} ∈ ∥y_{N}^{0}∥^{2} + ∫_{0}^{t} (2y_{N}(s)^{T} F_{N}(s, y_{N}(s)) + ∥b_{N}(s, y_{N}(s))∥_{F}^{2} ) ds +
+2 ∫_{0}^{t} y_{N}(s)^{T} b_{N}(s, y_{N}(s))dw(s)

and E {g(∥x^{0}∥) | Y_{N}(t) |^{2}}

≤ E {g(∥x^{0}∥) sup_{y_{N} ∈Y_{N}} | ∥y_{N}^{0}∥^{2} + ∫_{0}^{t} (2y_{N}(s)^{T} F_{N}(s, y_{N}(s))+
+∥b_{N}(s, y_{N}(s))∥_{F}^{2} ) ds |}
≤ 1 + 2E {g(∥x^{0}∥) sup_{y_{N} ∈Y_{N}} ∫_{0}^{t} | y_{N}(s)^{T} F_{N}(s, y_{N}(s)) | ds} +

+E {g(∥x^{0}∥) sup_{y_{N} ∈Y_{N}} ∫_{0}^{t} ∥b_{N}(s, y_{N}(s))∥_{F}^{2} ds}

{(16) \atop ≤} 1 + 3KE \{g(∥x^{0}∥) sup_{y_{N} ∈Y_{N}} ∫_{0}^{t} (1 + ∥y_{N}(s)∥^{2}) ds\}
≤ 1 + 3KT E {g(∥x^{0}∥)} + 3K ∫_{0}^{t} E {g(∥x^{0}∥) | Y_{N}(s) |^{2}} ds .

By applying the Gronwall inequality we get E {g(∥x^{0}∥) | Y_{N}(t) |^{2}} ≤ C_{2} and therefore the statement of Theorem 2.

□

References

[1] N.U. Ahmed. Nonlinear stochastic differential inclusions on Banach space. Stochastic Analysis and Applications, 12(1):1–10, 1994.

[2] N.U. Ahmed. Optimal relaxed controls for nonlinear infinite dimensional stochastic differential inclusions. In N.H. Pavel, editor, Optimal Control of Differential Equations, pages 1–19. Springer-Verlag New York, 1994.

[3] N.U. Ahmed. Existence of solutions of nonlinear stochastic differential inclusions on Banach space. In V. Lakshmikantham, editor, World Congress of Nonlinear Analysts ´ 92, pages 1699–1712. de Gruyter Berlin, 1996.

[4] J.-P. Aubin and G. Da Prato. The viability theorem for stochastic differential inclusions. Stochastic Analysis and Applications, 16(1):1–15, 1998.

[5] G. Da Prato and H. Frankowska. A stochastic Filippov theorem. Stochastic Analysis and Applications, 12(4):409–426, 1994.

[6] H. Fritsch. Optimal control of a setvalued stochastic dynamic system. Optimization, 35(3):251–263, 1995.

[7] H. Fritsch. A Note to Multivalued Martingales. Report No. 28-99, FB Mathematik und Informatik, Martin-Luther-Universität Halle-Wittenberg, 1999.

[8] F. Hiai and H. Umegaki. Integrals, conditional expectations and martingales of multivalued functions. Journal of Multivariate Analysis, 7:149–182, 1977.

[9] S. Hu and N.S. Papageorgiou. Handbook of Multivalued Analysis. Vol. I: Theory. Kluwer Dordrecht, 1997.

[10] B.K. Kim and J.H. Kim. Stochastic integrals of set-valued processes and fuzzy processes. Journal of Mathematical Analysis and Applications, 236:480–502, 1999.

[11] M. Kisielewicz. Properties of solution set of stochastic inclusions. Journal of Applied Mathematics and Stochastic Analysis, 6(3):217–236, 1993.

[12] M. Kisielewicz. Set-valued stochastic integrals and stochastic inclusions. Stochastic Analysis and Applications, 15(5):783–800, 1997.

[13] M. Michta. On weak solutions of random differential inclusions. Journal of Applied Mathematics and Stochastic Analysis, 8(4):393–396, 1995.

[14] J. Motyl. Note on strong solutions of a stochastic inclusion. Journal of Applied Mathematics and Stochastic Analysis, 8(3):291–297, 1995.

[15] J. Motyl. On the solution of stochastic differential inclusion. Journal of Mathematical Analysis and Applications, 192:117–132, 1995.

[16] N.S. Papageorgiou. On the theory of Banach space valued multifunctions. 1. Integration and conditional expectation. Journal of Multivariate Analysis, 17:185–206, 1985.

[17] N.S. Papageorgiou and F. Papalini. Random evolution inclusions of the subdifferential type. Stochastic Analysis and Applications, 15(5):801–821, 1997.

[18] H. Radström. An embedding theorem for spaces of convex sets. Proceedings of the American Mathematical Society, 3:165–169, 1952.

[19] B. Truong-Van and Truong Xuan Duc Ha. Existence results for viability problem associated to nonconvex stochastic differentiable inclusions. Stochastic Analysis and Applications, 17(4):667–685, 1999.