Dynamic Systems and Applications 15 (2006) 301-314

SECOND ORDER STOCHASTIC EQUATIONS AND INCLUSIONS

MARIUSZ MICHTA AND JERZY MOTYL

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland

ABSTRACT.The purpose of the paper is to consider some issues of the stochastic inclusions theory applicable in solving stochastic control problems. We discuss the existence of weak solutions to a stochastic equation and more generally, stochastic inclusion, driven by two general semimartingales, under weaker concepts of continuity than lower semicontinuity, imposed on a set-valued integrand. AMS (MOS) Subject Classification. 60H20, 93B05, 93C30.

1. PRELIMINARIES

Looking at deterministic optimal control problems one can find the theory which is extensively developed in connection with set-valued analysis and differential inclusions (see e.g. [4, 5]). Similarly, stochastic inclusions appear as a theoretical description of stochastic control problems in a natural way (see e.g. [3, 16, 19, 20] and references therein). The major contribution in this field has been connected with strong solutions to stochastic inclusions ([1, 2, 6, 16, 22, 25, 26]). The case of weak solutions to stochastic inclusions, as far as we know, has been studied only in few papers [7, 18, 21, 23]. Moreover, in these papers only the Itô’s type stochastic inclusions have been considered. Similarly as in the theory of stochastic equations, in the set-valued case the continuity assumption imposed on set-valued integrands play the crucial role obtaining the existence of weak solutions. In the setvalued case one can choose between different concepts of semicontinuity, which in the single valued case coincide with continuity of single valued mapping (see [11], [8]. In [7] and [21] the authors have proved the existence of weak solutions to Ito‘s type stochastic inclusions, under upper semicontinuity assumptions imposed on set-valued integrands and describing only convex-valued set-valued functions. But the assumption of the convexity of values is rather inconvenient in practical problems. Similar results without the convexity assumption have been considered in [17] for inclusions with lower semicontinuous set-valued functions. The lower semicontinuous (l.s.c.) case has been recently studied in [24] for the second order stochastic inclusion:

Received April 30, 2004           1056-2176 $15.00 \begin{matrix}c \\ ◯ \end{matrix}Dynamic Publishers, Inc.

(SI) X_{t} ∈ H_{t} + ∫_{0}^{t}[∫_{0}^{s−} F (τ, X)dY_{τ}]dZ_{s}, t ∈ IR^{+},

driven by general semimartingales Y and Z. In the present paper we discuss the existence of weak solutions to inclusion (SI) under weaker assumptions then lower semicontinuity imposed on a set-valued integrand than those presented in [24].

2. AUXILIARY RESULTS ON INTEGRATORS

In this section, we recall some properties of semimartingales and limiting properties of stochastic integrals, needed in the sequel. For more details we refer the reader to [14, 13, 30, 31, 32]. Let D(IR^{+}, IR^{d}) denote a Polish metric space of all cádlág (i.e. right continuous, having left limits) functions and provided with a Skorohod topology. The required definitions can be found in [9, 27]. Let (Ω, F, (F_{t})_{t≥0}, P) denote a complete filtered probability space with filtration (F_{t})_{t≥0} satisfying usual conditions. Let Z = (Z_{t})_{t≥0} be a semimartingale adapted to the filtration (F_{t})_{t≥0}, Z_{0} = 0. Let h: IR^{d} → IR^{d} be a bounded function with a compact support and such that h(x) = x in some neighbourhood of 0. Then, process Z can be decomposed as Z = J^{h} + M^{h} + B^{h}, where M^{h} is an (F_{t}) adapted local martingale, B^{h} is a predictable process with bounded variation and J_{t}^{h} = ∑_{0<s≤t}(ΔZ_{s} − h(ΔZ_{s})) (see e.g. [14]). Following Stricker [32] and Jakubowski, Memin and Pages [13] we recall the (UT) condition for a sequence of semimartingales. Let {Z^{n}} be a sequence of semimartingales defined possibly on different probability spaces (Ω^{n}, F^{n}, (F_{t}^{n}), P^{n}).

Definition 1. ([32]). The sequence {Z^{n}} of semimartingales satisfies the condition (UT) if for every q ∈ IR^{+} the family of random variables

(1)  {∫_{0}^{q} U_{s}^{n}dZ_{s}^{n} : U^{n} ∈ U_{q}^{n}, n ∈ IN} is tight in IR,

where U_{q}^{n} denotes the family of predictable processes of the form

U_{s}^{n} = U_{0}^{n} + ∑_{i=0}^{k} U_{i}^{n}I_{{t_{i}<s≤t_{i+1}}}, for 0 = t_{0} < t_{1} < ⋯ < t_{k} = q

and every U_{i}^{n} being an F_{t_{i}}^{n} measurable random variable such that |U_{i}^{n}| ≤ 1, for every i ∈ IN ∪ {0}, k, n ∈ IN.

It is clear that a sequence {Y^{n}} = {{Y_{k,j}^{n}}}_{k=1,...,d, j=1,...,d} of d × d dimensional semimartingales satisfies (UT) if and only if (UT) holds for every sequence {Y_{k,j}^{n}}_{n}, k = 1, …, d, j = 1, …, d.

For every n ∈ IN let us take the decomposition of the semimartingale Z^{n}, Z^{n} = J^{n,h} +M^{n,h} +B^{n,h}, where h is a function defined above. We have the following result.

Theorem 1. The following conditions are equivalent:

(i) {Z^{n}} satisfies the condition (UT),

(ii) there exists a bounded function h : IR^{d} → IR^{d} with a compact support, such that h(x) = x in some neighbourhood of 0 and such that for every q ∈ IR^{+} the sequences of random variables {V ar(J^{n,h})_{q}}_{n∈IN}, {V ar(B^{n,h})_{q}}_{n∈IN}, {[M^{n,h}]_{q}}_{n∈IN} are tight. [X] denotes here the square variation (bracket) process associated to semimartingale X.

(iii) for every q ∈ IR^{+} the family of random variables {∫_{0}^{q} U_{s}^{n}dZ_{s}^{n} : U^{n} ∈ U^{n}, n ∈ IN} is tight, where U^{n} denotes a class of predictable processes U^{n} such that sup_{t≤q} |U_{t}^{n}| ≤ 1, n ∈ IN,

(iv) for every q ∈ IR^{+} and ϵ > 0 there exists α > 0 such that for every n ∈ IN and predictable process U^{n}

(2)  P^{n} (sup_{t≤q} |U_{t}^{n}| > α) < α ⇒ P^{n} (sup_{t≤q} | ∫_{0}^{t} U_{s}^{n}dZ_{s}^{n}| > ϵ) < ϵ.

Proof. We omit the proof, because the result follows in the same way as Theorem 1.1 in [31], where the function h was originally taken as h(x) = I_{{|x|≤a}} for a > 0.

Corollary 1. If {Z^{n}} is a sequence of semimartingales satisfying (UT) and {U^{n}} is a sequence of predictable processes such that {sup_{t≤q} |U_{t}^{n}|} is tight in IR, then the sequence of stochastic integrals {∫_{0}^{·} U_{s}^{n}dZ_{s}^{n}} also satisfies (UT).

The below Lemma 2.1 was proved in [24] however it plays the key role in the main result, so for the convenience of the reader we present its proof.

Lemma 1. Let {R^{n}}, {Y^{n}}, and {Z^{n}} be sequences of (F_{t}^{n}) adapted semimartingales satisfying (UT) and suppose that {H^{n}} is a sequence of (F_{t}^{n}) adapted processes such that the sequence {(H^{n}, Y^{n}, Z^{n})} is tight in D(IR^{+}, IR^{d^{2}+2d}). Then

(i) the sequence {(H^{n}, ∫_{0}^{·} R_{τ−}^{n}dY_{τ}^{n}, Y^{n}, Z^{n})} is tight in D(IR^{+}, IR^{2d^{2}+2d}), (ii) if for every q ∈ IR^{+} the sequence {sup_{t≤q} |R_{t−}^{n}|} is tight in IR, then

the sequence

{(H^{n}, ∫_{0}^{·} (∫_{0}^{s−} R_{τ−}^{n}dY_{τ}^{n}) dZ_{s}^{n}, Y^{n}, Z^{n})}

is tight in D(IR^{+}, IR^{d^{2}+3d}) also.

Proof. By Prohorov‘s theorem [9] we may assume that

(3)  (H^{n}, Y^{n}, Z^{n}) →^{D} (H, Y, Z).

Then, by Corollary 2.8 of [31], there exists a family {Y^{n,i}} of (F_{t}^{n}) adapted processes satisfying

(4) {(H^{n}, ∫_{0}^{·} R_{τ−}^{n}dY_{τ}^{n,i}, Y^{n,i}, Y^{n}, Z^{n})} is tight in D(IR^{+}, IR^{3d^{2}+2d}),

(5) lim_{i→∞} lim sup_{n→∞} P {sup_{t≤q} |Y_{t}^{n} − Y_{t}^{n,i}| ≥ ϵ} = 0, for every ϵ > 0, q ∈ IR^{+},

and such that the sequence {Y^{n} − Y^{n,i}} satisfies (UT) for every i ∈ IN. Using integration by parts we get the equality

(6)  ∫_{0}^{t} R_{τ−}^{n}d(Y^{n} − Y^{n,i})_{τ}

= R_{t}^{n}(Y_{t}^{n} − Y_{t}^{n,i}) + ∫_{0}^{t}(Y_{τ−}^{n} − Y_{τ−}^{n,i})dR_{τ}^{n} + [Y^{n} − Y^{n,i}, R^{n}]_{t}.

We claim that

(7)  lim_{i→∞} lim sup_{n→∞} P {sup_{t≤q} | ∫_{0}^{t} R_{τ−}^{n}d(Y^{n} − Y^{n,i})_{τ} | ≥ ϵ} = 0

for every ϵ > 0, q ∈ IR^{+}.

Indeed, by equality (5) and Theorem 2.1 we get

(8)  lim_{i→∞} lim sup_{n→∞} P {sup_{t≤q} | ∫_{0}^{t}(Y_{τ−}^{n} − Y_{τ−}^{n,i})dR_{τ}^{n}| ≥ ϵ} = 0

for every ϵ > 0, q ∈ IR+. Since {R^{n}} satisfies (UT), then by [32], sequences {sup_{t≤q} |R_{t}^{n}|} and {[R^{n}]_{q}} are tight for every q ∈ IR^{+}. Consequently, applying the Kunita-Watanabe inequality to the last term of the right side of (6) and using again Theorem 2.1 we obtain

(9)  lim_{i→∞} lim sup_{n→∞} P {sup_{t≤q} |[Y^{n} − Y^{n,i}, R^{n}]_{t}| ≥ ϵ} = 0

for every ϵ > 0, q ∈ IR^{+}.

Finally, by (6), (8) and (9), it follows (7). Hence, by formulas (4) and (7) we deduce the tightness of the sequence

{(H^{n}, ∫_{0}^{·} ∫_{0}^{s−} R_{τ−}^{n}dY_{τ}^{n}dZ_{s}^{n}, Y^{n}, Z^{n})} .

For proving (ii) we observe that the sequence (∫_{0}^{·} R_{τ−}^{n}dY_{τ}^{n}) satisfies (UT) by Corollary 2.1. Therefore, the proof follows immediately from part (i) applied to sequences (∫_{0}^{·} R_{τ−}^{n}dY_{τ}^{n}) and (H^{n}, Y^{n}, Z^{n}).

3. CONTINUITY CONCEPTS OF SET-VALUED INTEGRANDS

As it was mentioned in the Introduction, there are some other concepts of lower semicontinuity, weaker than (l.s.c.). We present them briefly, in order to use them at the end of this section. For details we refer to [28], [29]. Let F : X → 2^{Y} be a set-valued mapping defined on a topological space X and taking on its values in the family of all nonempty subsets of a real linear normed space (Y, || · ||). For x ∈ X let U(x) denote the family of all open neighbourhoods of x. For a set D ⊂ Y we define a set-valued mapping F_{D} : X → 2^{Y}:

F_{D}(x) = ⋃_{U∈U(x)} ⋂_{z∈U} F (z) − D.

Let ϵ > 0 and let B_{0} = {y : ||y|| < 1}. Then for D = ϵB_{0} we write F_{ϵ} instead of F_{ϵB_{0}}.

Definition 2. ([29]). A set-valued mapping F : X → 2^{Y} is said to be: lower semicontinuous (l.s.c.) if F ⊆ F_{0}, where F_{0}(x) := ⋂_{ϵ>0} F_{ϵ}(x). weak lower semicontinuous (w.l.s.c.) if for every ϵ > 0, x ∈ X, and

U ∈ U(x) there exists z ∈ U such that F (z) ⊆ F_{ϵ}(z).

K-ball-Lipschitz lower semicontinuous (K-b-L. l.s.c.) if F_{ϵ}(x) is nonempty and F_{ϵ}(x) ⊆ F_{μ}(x) − KϵB_{0}, for some K ≥ 1 and every x ∈ X, ϵ > 0, μ > 0. ball uniformly lower semicontinuous (b.u.l.s.c.) if: F_{ϵ} is nonempty and for every ϵ > 0 there exists δ > 0 such that F_{δ}(x) ⊆ F_{μ}(x) − ϵB_{0}, for all μ > 0 and x ∈ X.

ball uniformly lower semicontinuous (b.u.l.s.c.) if: F_{ϵ} is nonempty and for every ϵ > 0 there exists δ > 0 such that F_{δ}(x) ⊆ F_{μ}(x) − ϵB_{0}, for all μ > 0 and x ∈ X.

Remark 1. Let us note that the condition imposed in the Definition 2a is equivalent to the classical concept of lower semicontinuity of a set-valued function.

Proposition 1. ([29] Theorem 2.4.) The following chain of implications for set-valued mapping F : X → 2^{Y} holds true:

(l.s.c) ⇒ (w.l.s.c) ⇒ (K − b − L. l.s.c) ⇒ (b.u.l.s.c) ⇒ (a.l.s.c).

There exist set-valued functions with compact values for which the above implications are noninverticable.

Example 1. Let X = Y = IR^{1}. Set F (x) = [0, n^{−1}] for x = q2^{−n}, with q odd integer and n positive integer and F (x) = [−1, 0] for remaining x. Since for every x ∈ IR^{1} there exists a sequence (x_{k}) convergent to x, x_{k} = q_{k}2^{−n_{k}}, n_{k} → ∞, it follows that F_{ϵ}(x) = (−ϵ, ϵ) for all x and that F is weak l.s.c., but not l.s.c. at every x ∈ IR^{1}. Set F (x) = [0, 1] for rational x and F (x) = [−1, 0] for x irrational. Then F is (K-b-L. l.s.c), but not weak l.s.c. Let Y = IR^{2}. Set F (x) = {(t, 0) : 0 ≤ t ≤ 1} for x irrational and F (x) = {(t, s) : 0 ≤ t ≤ 1, t^{2} ≤ s ≤ 1} for x rational. F is not (K-b-L. l.s.c) but it is ball uniformly l.s.c.

Proposition 2. ([29] Prop. 2.5.) If F : X → IR^{1} and takes on convex values then (K-b-L. l.s.c) continuity is equivalent to an (a.l.s.c.) one.

Proposition 3. ([29] Corol. 4.7) Let X be a paracompact Hausdorff topological space, Y a Banach space. Assume that F : X → 2^{Y} is a set-valued mapping with closed convex values and ball uniformly l.s.c. Then it admits a continuous selection.

We define

S(F (y), ϵ) = {v : inf_{u∈F (y)} ||v − u|| < ϵ} ,

It follows from Definition 2, that a set-valued function F is (a.l.s.c.) if and only if

∀ϵ ∀y ∃δ ⋂_{z∈y+δB} S(F (z), ϵ) ≠ ∅

Let F : IR^{+}×IR^{d} → Comp (IR^{n}) be a set-valued function if two variables taking its values in the family of nonempty and compact subsets of IR^{n}. Similarly as above for each t ∈ [k, k + 1) we define

S_{k}(F (t, y), ϵ) = {v : inf_{u∈F (t,y)} ||v − u|| < ϵ} ,

k = 0, 1, …

Definition 3. We call a set-valued function F : IR^{+} × IR^{d} →CompIR^{n} an almost lower semicontinuous (a.l.s.c.) with respect to its second variable if

∀ϵ ∀k ∀y ∃δ ∀t ∈ [k, k + 1) ⋂_{z∈y+δB} S_{k}(F (t, z), ϵ) ≠ ∅

Lemma 2. If a set-valued function F : IR^{+} × IR^{d} →CompIR^{n} is (a.l.s.c.) with respect to its second variable then F^{∗}(t, y) = ∫_{0}^{t} F (s, y)ds is also (a.l.s.c.) with respect to its second variable.

Proof. Let

(10)  ∀ϵ ∀k ∀y ∃δ_{k} ∀t ∈ [k, k + 1) ⋂_{z∈y+δB} S_{k}(F (t, z), ϵ) ≠ ∅

and assume that ∫_{0}^{t} F (s, y)ds is not almost l.s.c. It means that

(11)  ∃ϵ̄ ∃y ∃k̄ ∀δ ∃t̄∈ [k̄, k̄ + 1) ∃z̄_{1}, z̄_{2} ∈ y + δB
Sk̄(F^{∗}(t̄, z̄_{1}), ϵ̄) ∩ Sk̄(F^{∗}(t̄, z̄_{2}), ϵ̄) = ∅,

where t̄ is such that k̄ ≤ t̄ < k̄ + 1. For fixed ϵ̄, y let k̄ be the smallest integer satisfying (11). Then, for fixed ϵ̄, y, k̄, z̄_{1}, z̄_{2} and all measurable selectors f(·) ∈ F (·, z̄_{1}) and g(·) ∈ F (·, z̄_{2}) we have

(12)  | ∫ t̄ ; 0f(s)ds − ∫ t̄ ; 0g(s)ds| > 2ϵ̄,

while for every z_{1}, z_{2} ∈ y + δB, there exist measurable selectors f_{1}(·) ∈ F (·, z_{1}) and f_{2}(·) ∈ F (·, z_{2}) such that

(13)  | ∫ k̄ ; 0 f_{1}(s)ds − ∫ k̄ ; 0 f_{2}(s)ds| = α < 2ϵ̄.

Now, let us take ϵ = ϵ̄ − α/2 in the formula (10). Then there exists δ̄ such that

(14)  ⋂_{z∈y+δ̄B} Sk̄(F (t, z), ϵ) ≠ ∅

for every t ∈ [k̄, k̄ + 1). For this δ̄ we find t̄, z̄_{1}, z̄_{2} satisfying (11). From the property (14) we get

(15)  Sk̄(F (t, z̄_{1}), ϵ) ∩ Sk̄F (t, z̄_{2}), ϵ) ≠ ∅

for each t ∈ [k̄, k̄ + 1).

Let us define set-valued function

H(t) = Cl(Sk̄(F (t, z̄_{1}), ϵ) ∩ Sk̄F (t, z̄_{2}), ϵ).

It is measurable (see e.g. [15]), so by the Kuratowski and Ryll-Nardzewski measurable selection theorem it admits a measurable selector h(t) ∈ H(t) such that

dist(h(t), F (t, z̄_{1})) < ϵ and dist(h(t), F (t, z̄_{2})) < ϵ for all t ∈ [k̄, k̄ + 1). By Filippov’s Selection Theorem (e.g. [8]) we deduce that there exist selectors f̄_{1} ∈ F (t, z̄_{1}) and f̄_{2} ∈ F (t, z̄_{2}) such that |h(t) − f̄_{1}| < ϵ and |h(t) − f̄_{2}| < ϵ for each t ∈ [k̄, k̄ + 1). Therefore,

(16)  |f̄_{1} − f̄_{2}| < 2ϵ = 2ϵ̄− α.

Now, we define measurable selectors f(·) of F (·, z̄_{1}) and g(·) of F (·, z̄_{2}) by formulas:

f(t) = 1l_{[0,k̄)}f_{1}(t) + 1l_{[k̄,t̄)}f̄_{1}(t)

and

g(t) = 1l_{[0,k̄)}f_{2}(t) + 1l_{[k̄,t̄)}f̄_{2}(t).

Finally, using (13) and (16) we obtain for this selectors

| ∫ t̄ ; 0f(s)ds − ∫ t̄ ; 0g(s)ds| < α + (2ϵ̄− α)(t̄− k̄) = 2ϵ̄.

Contradiction with the inequality (12).

Below we describe the class of set-valued integrands for stochastic inclusion (SI). By IR^{2d} we denote the space of d × d real valued matrices, with the norm || · || defined by:

||a|| = (∑_{i≤d} ∑_{j≤d}(a^{ij})^{2})^{1/2} for a = (a^{ij})_{d×d}.

Let F̃ : IR^{+} × D(IR^{+}, IR^{d}) → Comp(IR^{2d}) be a set-valued function with values in the space of all nonempty and compact subsets of IR^{2d}. For every x ∈ D(IR^{+}, IR^{d}) let

SF̃(x) := {v ∈ D(IR^{+}, IR^{2d}) : v(t) ∈ F̃(t, x), t ∈ IR^{+}}

denote the set (possibly empty) of all cádlág selections of F̃(·, x).

Definition 4. (Condition C, [24]). A set-valued function F̃ : IR^{+}×D(IR^{+}, IR^{d}) → Comp(IR^{2d}) satisfies Condition C if there exists a function V : IR^{+}×D(IR^{+}, IR^{d}) → IR^{2d} such that V (·, x) ∈ SF̃(x) for every x ∈ D(IR^{+}, IR^{d}) and V (·, x_{n}) → V (·, x) in D(IR^{+}, IR^{2d}) for every x_{n} → x in D(IR^{+}, IR^{d}), i.e. if a set-valued function SF̃(·) : D(IR^{+}, IR^{d}) → 2^{D(IR^{+},IR^{2d})} admits a continuous selection.

Example 2. (a) If g : IR^{+} × IR^{d} → IR^{2d} is a continuous single-valued function, then F̃ : D(IR^{+}, IR^{d}) → D(IR^{+}, IR^{2d}) defined by F̃(t, x) := {f(t+, x)}, where f(t, x) = g(t, x(t−)), satisfies Condition C (see Lemma 4.4 of [31]).

(b) Let a set-valued function F̃(·, ·) be (l.s.c.) jointly with respect to all its variables and having nonempty, closed, and convex values. Then by Michael‘s Selection Theorem (see e.g. [8]) it satisfies Condition C and for every x ∈ D(IR^{+}, IR^{d}) the set SF̃(x) is a nonempty closed and convex subset of the space D(IR^{+}, IR^{2d}).

Remark 2. It is easy to see that if a set-valued function F̃ satisfies Condition C, then a set-valued mapping D(IR^{+}, IR^{2d}) ∋ (x, z) → SF̃(x) × {z} ∈ 2^{D(IR^{+},IR^{2d})}×D(IR^{+}, IR^{d}) admits a continuous selection. Hence, if f and g are functions from Example 2a, then the function V : IR^{+} × D(IR^{+}, IR^{d}) → IR^{2d} defined by V (t, x) = f(t+, x) satisfies the following condition:

for every (x, z) ∈ D(IR^{+}, IR^{2d}) and every sequence {(x^{n}, z^{n})} ⊂ D(IR^{+}, IR^{2d}), if (x^{n}, z^{n}) → (x, z) in D(IR^{+}, IR^{2d}), then (V (·, x^{n}) , z^{n}) → (V (·, x), z) in D(IR^{+}, IR^{2d}) × D(IR^{+}, IR^{d}).

Theorem 2. Let F : IR^{+} × IR^{d} → Comp(IR^{1}) be a set-valued function satisfying:

(i) F is integrably bounded, i.e. there exists a function m(·) ∈ L^{1}(IR^{+}, IR) such that H(F (t, y), 0) ≤ m(t) a.e. t ∈ IR^{+}, and all y ∈ IR^{d},

(ii) F is (IR^{+} × IR^{d})-Borel measurable, (iii) F (t, y) is (a.l.s.c.) with respect to y. If F^{∗}(t, y) = ∫_{0}^{t} F (s, y)ds, then a set-valued function F̃ : IR^{+}×D(IR^{+}, IR^{d}) →

Comp(IR^{1}) defined by F̃(t, x) = F^{∗}(t, x(t−)) satisfies Condition C.

Proof. First we show, that F^{∗}(t, y) is Hausdorff continuous in t uniformly with respect to y. Indeed, let us take an arbitrary t_{n} → t. Then for every measurable selector f(·) ∈ F (·, y) we get

dist (∫_{0}^{t_{n}} f(s)ds, ∫_{0}^{t} F (s, y)ds) ≤ inf_{a∈∫_{0}^{t} F (s,y)ds}∥ ∫_{0}^{t_{n}} f(s)ds − a∥

≤ ∥ ∫_{0}^{t_{n}} f(s)ds − ∫_{0}^{t} f(s)ds∥ ≤ ∫_{t}^{t_{n}} m(s)ds → 0.

Therefore, for every y

sup_{f}dist (∫_{0}^{t_{n}} f(s)ds, ∫_{0}^{t} F (s, y)ds) ≤ ∫_{t}^{t_{n}} m(s)ds → 0

where the supremum is taken over all measurable selectors f(·) of F (·, y). Then, by the definition of the Hausdorff metric ([15]), we deduce

h (∫_{0}^{t_{n}} F (s, y)ds, ∫_{0}^{t} F (s, y)ds) → 0

as t_{n} → t, uniformly in y. Using this we are able to deduce that F^{∗} is (a.l.s.c.) jointly with respect to its all variables i.e.

∀ϵ ∀(t, y) ∃δ ⋂_{(s,z)∈(t,y)+δB} S(F^{∗}(s, z), ϵ) ≠ ∅.

Contradicting, let us assume similarly as in Lemma 2 that

(17)  ∃ϵ̄ ∃(t, y) ∀δ ∃(t̄_{1}, z̄_{1}), (t̄_{2}, z̄_{2}) ∈ (t, y) + δB
S(∫_{0}^{t_{1}} F (s, z̄_{1})ds, ϵ̄) ∩ S(∫_{0}^{t_{2}} F (s, z̄_{2})ds, ϵ̄) = ∅.

Without any loss of generality we assume t_{1} < t_{2}. Let δ be such small that taking |t_{2} − t_{1}| < δ we get

h (∫_{0}^{t_{1}} F (s, z)ds, ∫_{0}^{t_{2}} F (s, z)ds) < ϵ̄/2

for each z. Such a δ exists by the uniform continuity of F^{∗}. Taking δ̄ in the formula (17) and t_{1} = t̄_{1}, t_{2} = t̄_{2} we get

∫ t̄_{2} ; 0 F (s, z̄_{1})ds ⊂ S(∫ t̄_{1} ; 0 F (s, z̄_{1})ds, ϵ̄/2),

and therefore,

S (∫ t̄_{2} ; 0 F (s, z̄_{1})ds, ϵ̄/2) ⊂ S(∫ t̄_{1} ; 0 F (s, z̄_{1})ds, ϵ̄).

Therefore by (17) we deduce that

S (∫ t̄_{2} ; 0 F (s, z̄_{1})ds, ϵ̄/2) ∩ S(∫ t̄_{2} ; 0 F (s, z̄_{2})ds, ϵ̄) = ∅

what is the same as the condition (11) in Lemma 2 and therefore, leads as before to the contradiction with the property (12) there. By the above F^{∗}(t, y) is jointly (a.l.s.c.). Moreover, it admits convex values (see e.g. [8, 15]). Using Propositions 2 and 3 we deduce that there exists a selection g(t, y) of F (t, y) jointly continuous in IR^{+} × IR^{d}.

Finally, we define the function V : IR^{+}×D(IR^{+}, IR^{d}) → IR^{2d} by V (t, x) = f̃(t+, x) where f̃(t, x) = g(t, x(t−)). By Remark 2, it is continuous and therefore, V (·, x) is a continuous selection of a set-valued mapping x → SF̃(x), where F̃(t, x) = F^{∗}(t, x(t−)) with F^{∗}(t, y) = ∫_{0}^{t} F (s, y)ds. Thus F̃ satisfies Condition C.

Since the property of almost l.s.c. is the weakest one of all described in the Definition 2 it is clear that we have:

Corollary 2. Suppose F : IR^{+} × IR^{d} → Comp(IR^{1}) is a set-valued function satisfying assumptions (i) and (ii) of Theorem 3.2. Let F (t, ·) satisfy any type of lower semicontinuity from Definition 3.2, with respect to its second variable and for every fixed t ∈ IR^{+}. If F^{∗}(t, y) = ∫_{0}^{t} F (s, y)ds, then a setvalued function F̃ : IR^{+} × D(IR^{+}, IR^{d}) → Comp(IR^{1}) defined by F̃(t, x) = F^{∗}(t, x(t−)) satisfies Condition C.

4. SECOND ORDER STOCHASTIC EQUATIONS AND INCLUSIONS

In this section, we present the main result of the paper (Theorem 4.2). Let F : IR^{+} ×D(IR^{+}, IR^{1}) → Comp(IR^{1}) be a set-valued function taking

its values in the family of all nonempty and compact subsets of IR^{1}. Let Y = (Y_{t})_{t≥0} and Z = (Z_{t})_{t≥0} be one dimensional {F_{t}} semimartingales. We discuss below the existence of solutions to a second order stochastic inclusion of the form

(SI) X_{t} ∈ H_{t} + ∫_{0}^{t}[∫_{0}^{s−} F (τ, X)dY_{τ}]dZ_{s}, t ∈ IR^{+}.

We say that inclusion (SI) has a weak solution if there exist a probability space (Ω̂, F̂, (F̂_{t}), P̂) and F̂_{t}-adapted processes X̂, Ĥ, Ŷ, Ẑ such that D(Ĥ, Ŷ, Ẑ) = D(H, Y, Z) and such that the inclusion

X̂_{t} ∈ Ĥ_{t} + ∫_{0}^{t}[∫_{0}^{s−} F (τ, X̂)dŶ_{τ}]dẐ_{s}

holds for every t ∈ IR^{+}.

In order to obtain the existence of a weak solution to a stochastic inclusion with a set-valued function satisfying Condition C, one has to describe the second order single-valued stochastic equation

(SE) X_{t} = H_{t} + ∫_{0}^{t}[∫_{0}^{s−} V (τ−, X)dY_{τ}]dZ_{s}, t ∈ IR^{+}.

Let us assume that V : IR^{+} × D(IR^{+}, IR^{1}) → IR^{1} is a bounded function such that: D(IR^{+}, IR^{1}) ∋ x → V (·, x) ∈ D(IR^{+}, IR^{1}) is a continuous mapping. Let {Y^{n}}, {Z^{n}} be sequences of (F_{t}^{n}) semimartingales and let {H^{n}} be a sequence of (F_{t}^{n}) adapted cádlág processes. Let us consider the sequence of stochastic equations

(SE(n)) X_{t}^{n} = H_{t}^{n} + ∫_{0}^{t}[∫_{0}^{s−} V (τ−, X^{n})dY_{τ}^{n}]dZ_{s}^{n}, t ∈ IR^{+}. Then the following limiting result can be stated.

Theorem 3. ([24]) Let {X^{n}} be a sequence of solutions to equations (SE(n)). We assume that there exists a family of semimartingales {{R^{n,i}}} such that for every i ∈ IN a sequence {R^{ni}}_{n} satisfies (UT) and such that

(18)  lim_{i→∞} lim sup_{n→∞} P^{n} {sup_{t≤q} ||R_{t−}^{n,i} − V (t−, X^{n})|| ≥ ϵ} = 0

for every ϵ > 0, q ∈ IR^{+}. If {Y^{n}} and {Z^{n}} satisfy (UT) and (H^{n}, Y^{n}, Z^{n}) →^{D} (H, Y, Z), then the sequence {(X^{n}, H^{n}, Y^{n}, Z^{n})} is tight and each of its limit points states a weak solution to the stochastic equation

(19)  X_{t} = H_{t} + ∫_{0}^{t}[∫_{0}^{s−} V (τ−, X)dY_{τ}]dZ_{s}, t ∈ IR^{+}.

Now, we are ready to state some results of the existence of weak solutions to stochastic inclusions as consequences of Theorems 3.2 and 4.1.

Corollary 3. Let us suppose that a set-valued mapping F : IR^{+}×D(IR^{+}, IR^{1}) → Comp(IR^{1}) is bounded and satisfies Condition C. Assume that the selection V defined in Condition C satisfies the assumptions of Theorem 4.1 for some sequence {X_{n}} such that:

X_{t}^{n} = H_{t}^{n} + ∫_{0}^{t}[∫_{0}^{s−} V (τ−, X^{n})dY_{τ}^{n}]dZ_{s}^{n}, t ∈ IR^{+}, n ∈ IN

and for some sequences {Y^{n}}, {Z^{n}} satisfying (UT). If (H^{n}, Y^{n}, Z^{n}) →^{D} (H, Y, Z), then, there exists a weak solution to the stochastic inclusion

X_{t} ∈ H_{t} + ∫_{0}^{t}[∫_{0}^{s−} F (τ, X)dY_{τ}]dZ_{s}, t ∈ IR^{+}.

Theorem 4. Let Y = (Y_{t})_{t≥0} and Z = (Z_{t})_{t≥0} be one dimensional {F_{t}} semimartingales and let F : IR^{+} ×IR^{1} → Comp(IR^{1}) be a set-valued function satisfying:

(i) F is integrably bounded by a function m(·),

(ii) F is (IR^{+} × IR^{1})-Borel measurable,

(iii) F (t, ·) is almost lower semicontinuous with respect to its second variable.

If F̃ : IR^{+}×D(IR^{+}, IR^{1}) → Comp(IR^{1}) is defined by F̃(s, y) = ∫_{0}^{t} F (s, y(s−))ds, then there exists a weak solution to the stochastic inclusion:

X_{t} ∈ H_{t} + ∫_{0}^{t}[∫_{0}^{s−} F̃(τ, X)dY_{τ}]dZ_{s}, t ∈ IR^{+}.

Proof. Let us take a partition (t_{n,k})_{k≥0,n≥1} of IR^{+}, i.e. let (t_{n,k}) be a sequence such that:

(a) 0 = t_{n,0} < t_{n,1} < ⋯, lim_{k→∞} t_{n,k} = ∞, for every n ∈ IN, (b) max_{k≥1}(t_{n,k} − t_{n,k−1}) → 0, as n → ∞. Let α_{n}(t) = max{t_{n,k} : t_{n,k} ≤ t}. For x ∈ D(IR^{+}, IR^{1}) we define a sequence of piecewise constant functions

x^{α_{n}}(t) := x(t_{n,k})1l_{[t_{n,k},t_{n,k+1})}(t), n ∈ IN, k ≥ 0.

Let us define sequences of piecewise constant processes:

H^{n} = H^{α_{n}}, Y^{n} = Y^{α_{n}}, Z^{n} = Z^{α_{n}}, for n ∈ IN.

Consequently, (H^{n}, Y^{n}, Z^{n}) is convergent to (H, Y, Z) with P.1, and hence the convergence in distribution also holds. By the definition, both Y^{n} and Z^{n} are semimartingales with locally bounded variation, and {Y^{n}}, {Z^{n}} satisfy (UT). In the same way as in the proof of Theorem 3.2, we find the measurable selection f of F, and the continuous function f^{∗}(t, y) = ∫_{0}^{t} f(s, y)ds.

Similarly as in the proof of Lemma 4.4 (i) of [31], we can construct a sequence {g^{i}} of functions such that for every i ∈ IN, g^{i} ∈ C^{1,2}(IR^{+}×IR^{1}, IR^{1}) and

(20)  sup_{x∈K} sup_{t≤q} |g^{i}(t, x) − f^{∗}(t, x)| → 0

for all compact subsets K of IR^{1} and q ∈ IR^{+}.

Since H^{n} →^{D} H, then similarly as in the proof of Lemma 2.1 (formula (5)), we can find a family {{H^{n,i}}} of processes such that

(21) lim_{i→∞} lim sup_{n→∞} P {sup_{t≤q} |H_{t}^{n} − H_{t}^{n,i}| ≥ ϵ} = 0, for every ϵ > 0, q ∈ IR^{+},

and such that the sequence {H^{n,i}}_{n} satisfies condition (UT) for every i ∈ IN. Define for every n, k ∈ IN,

(22)  X_{t}^{n,i} = H^{n,i} + ∫_{0}^{·} f^{∗}(s, X_{s−}^{n})dZ_{s}^{n},

and

(23)  R_{t}^{n,i} = g^{i}(t, X^{n,i}).

Using the same argumentation as in Lemma 4.4 (i) of [31] we deduce that the function V (t, x) = f^{∗}(t, x(t−)) satisfies (20) and therefore,

(24)  lim_{i→∞} lim sup_{n→∞} P^{n} {sup_{t≤q} ||R_{t−}^{n,i} − V (t−, X^{n})|| ≥ ϵ} = 0

for every ϵ > 0, q ∈ IR^{+}.

For every n ∈ IN let X^{n} be a strong solution to the equation

X_{t}^{n} = H_{t}^{n} + ∫_{0}^{t}[∫_{0}^{s−} f^{∗}(τ−, X_{τ−}^{n})dY_{τ}^{n}]dZ_{s}^{n}, t ∈ IR^{+}.

Now, using Theorems 3.2, 4.1 and Corollary 4.1 the result follows.

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