The random processes under consideration are assumed, firstly, given on a common probability space (without any exception) and, secondly, càdlàg (the exceptions will be stipulated). Let be a sub-σ- algebra of. We introduce the notation:

—the class of all increasing from zero numeral random processes whose values at all times are -measurable random variables. If, besides, a filtration is given, then we identify with. By we denote, following [1], the class of all -valued (will be determined by the context, if matters) -martingales M such that for every; signifies (see ibid.) the class of all locally square integrable martingales w. r. t..

The definition of conditional expectation, in particular, adopted in this article is due to Meyer (see [2]). It admits existence of the conditional expectation of a random variable with infinite first absolute moment. Thus generalized conditional expectation inherits most of the familiar properties (listed, for example, in [2]) of the classical one, but in this case new proofs are required. They are gathered in Section 2.

Let be the solution of a stochastic differential equation of the kind

where is a continuous process from and is chosen from some subclass of which is constructed and studied in Section 3. The goal of this article is to find an upper bound, much more exact than that provided by the Gronwall—Bellman lemma, for. This is done in Section 4 containing the only final result of the article. The reader inclined to accept that result in less generality, when is a quasicontinuous process from and (so that), may skip all the preceding material. But for the approach underlying the derivations in Section 4 such a confinement is unnatural. That is a reason why 3/4 of the article’s volume are allocated to ancillary results. Another reason is that those results may prove useful beyond the context of this article.

Upper bounds for are usually obtained with the aid of Lyapunov’s functions (see, e.g., [3,4]). Our alternative approach is based on a “comparison theorem” (Corollary 4.2) allowing both to weaken the assumptions and to refine the conclusion (cf. our Theorem 4.3 with Theorem I.4.2 in [3]).

All vectors are thought of, unless otherwise stated, as columns; means. The space of all d-dimensional row vectors with real components is denoted. The words “almost surely” are tacitly implied in relations between random variables, including the convergence relation, unless it is explicitly written as the convergence in probability. Indicators are denoted by with two possible modes of writing the set: or.

The reference books for the notions and results of stochastic analysis used in this paper are [1,5,6].

Denote and, for , so that. In what follows, “nonnegative” means “-valued” (the value is not admitted). The Borel -algebra in will be denoted.

Let be a sub--algebra of. The conditional given expectation of an -valued random variable is defined, according to [2], as the -measurable -valued random variable such that

for every. For an valued random variable such that

we set by definition

. Further the conditional expectation of a -valued random variable is defined in the obvious way. Thus defined conditional expectation will be called extended. Unlike the classical conditional expectation (defined only for) it does not possess, generally speaking, the property

But for an -valued this property remains valid—with the same proof as for.

Obviously, the extended conditional expectation of coincides with the classical one and therefore

for every and. Equality (2) holds for -valued and, as well, which is immediate from the definition of extended conditional expectation. In particular,

for every -valued random variable.

The next two statements are immediate from the definition of extended conditional expectation.

Lemma 2.1. Let be an -valued random variable such that exists. Then Equality (3) holds for every.

Lemma 2.2. Let be an -valued random variable. Then for any

Lemma 2.3. Let and be nonnegative random variables such that. Then.

Proof. Denote . The assumption and the definition of extended conditional expectation yield for every. Consequently. □

Lemma 2.4. Let be an -valued random variable. Then for any and

Proof. By Formula (1)

By Lemma 2.2 By Lemmas 2.3 and 2.1 and therefore. It remains to write the evident inclusion

□

Corollary 2.5. Let and let be a nonnegative random variable such that. Then.

Proof. Lemma 2.4 and Formula (1) yield for arbitrary and

Passing in this inequality to the limit at first as and hereafter as, we get

.□

Lemma 2.6. Let be an increasing sequence of -valued random variables. Then.

Proof. In case the r.h.s is finite this is the Beppo Levi theorem. Having written, we obtain the same equality when.□

Lemma 2.7. Let be an increasing sequence of -valued random variables. Then

.

Proof. Denote. By construction is -measurable. Lemma 2.6 and the definition of conditional expectation yield, for arbitrary, So, which in view of -measurability of proves the lemma. □

Corollary 2.8. For every sequence of nonnegative random variables the inequality

is valid.

Proof. Denote. By Lemma 2.3

Herein, whence by Lemma 2.7. □

Lemma 2.9. Let be a decreasing sequence of nonnegative random variables such that. Then.

Proof. Retaining the notation of the proof of Lemma 2.7, we denote additionally . Then from the definition of conditional expectation we have

for arbitrary and. By condition, so whence, taking to account monotonicity of (and therefore of) we conclude by the Beppo Levi theorem that Juxtaposing these two equalities with (5), we see that

for any. Herein as, since by assumption. Then from (6) we get by Lemma 2.6.□

Theorem 2.10. Let be a sequence of - valued random variables almost surely converging to a random variable and such that

Then and.

Proof. Let first the’s be nonnegative. Denote

. Then

From the second relation we have by Corollary 2.8; the third relation together with (7) yields by Lemma 2.9

. Comparing these two conclusions with (8), we get. Thus we have proved the theorem for nonnegative random variables. The transition to the general case is trivial. □

Lemma 2.11. Let and be -valued random variables such that the conditional expectations and exist and are component-wise finite. Then exists and Equality (2) holds.

Proof. The assumptions of the lemma together with Equality (4) imply that

For nonnegative random variables Equality (2) ensues, as was pointed out above, directly from the definition of extended conditional expectation, so Inequalities (9) yield

Denote, for each,

. By construction and therefore. Consequently,

.

Obviously, Herein by construction, which together with (10) and (9) implies (7) and the same for and. Hence and from the above asymptotic relations we get by Theorem 2.10

□

Lemma 2.12. Let and be a - valued random variable such that. Then.

Proof. It suffices to consider the case. Then the last assumption of the lemma amounts to. Denote. By Formula (1) and therefore. Then by Lemma 2.11, which together with the previous inequality and the definition of extended conditional expectation yields . The inequalities imply, by Corollary 2.5, that, whence by the definitions of and extended conditional expectation we have.□

Lemma 2.13. Let and be nonnegative random variables, be -measurable. Then

Proof. Denote (due to -measurability of), ,

Formula (2) (for nonnegative random variables), Lemma 2.1 and the definition of yield

. Noting that

by Lemma 2.2, we convert this equality to Obviously, as. Then by Lemma 2.7 as, which together with the last equality yields It remains to let and again make use of Lemma 2.7. □

Lemma 2.14. Let and be random variables with values in and, respectively. Suppose that is -measurable and. Then

Proof. It suffices to consider the case. Writing, for arbitrary, the evident equalities, we get from Lemma 2.13

The assumption implies finiteness of the right-hand sides of both equalities. Consequently, the left-hand sides are finite, too. Then by the definition of extended conditional expectation, which together with the two preceding equalities completes the proof. □

Lemma 2.15. Let, and let and be random variables with values in and, respectively, such that: and is - measurable. Then (the null matrix).

Proof. From the last three assumptions we get by Lemma 2.14; the first assumption implies, according to Lemma 2.12, the equality .□

Lemma 2.16. Let be a converging in probability to zero sequence of nonnegative random variables such that for some increasing unbounded function

the sequence is stochastically bounded. Then.

Proof. From the first assumption we have for every, so it suffices to show that for any

Since increases to infinity, we shall have for sufficiently large (such that). Then by Lemma 2.3

for those N. Letting here, we deduce (11) from the last assumption of the lemma and unbounded growth of.□

Lemma 2.17. Let be an -valued measurable random process. Then for any -measurable random variable we have

Proof. Denote

. Let

. Thenwhence by the assumption about and by Lemma 2.2 we have

Thus contains all sets of the kind, where (“measurable rectangles”). Then it follows from (2) (for nonnegative random variables) that contains also all possible finite unions of pairwise disjoint measurable rectangles. According to Lemma 2.6 contains the union of every increasing sequence of its members. Consequently, it contains the -algebra generated by measurable rectangles, i.e. Equality (12) holds for.

Passing to the general case, we denote

(due to measurability of),. By construction for all and and therefore. From these relations we get by Lemma 2.7

As was shown (in another notation) in the proof of Lemma 2.13,. By what was proved, which together with the previous equality yields

. Juxtaposing this with (13), we arrive at (12). □

In the next two statements, the process need not be càdlàg.

Lemma 2.18. Let and be a bounded measurable random process on. Then

Proof. 1) Lemma 2.11 allows to consider, without loss of generality, that is. Then the boundedness assumption together with Lemma 2.1 allows to consider that.

Let at first, where and are a random variable and a Borel function, respectively. Then Equality (14) follows from Lemma 2.13.

2) Let for all, where is an increasing sequence of [0,1]-valued random processes such that for each

Then: for any s the sequence increases by Lemma 2.3 and by Lemma 2.13;

by the Beppo Levi theorem. By the same theorem we get from the first relation. The second relation jointly with Lemma 2.13 yields

. Comparing these two conclusions with (15), we obtain (14).

3) Let denote the class of all

such that Equality (14) holds for

. According to item 1) contains the algebra generated by measurable triangles. Then it follows from item 2) that.

4) Let us define the sequence by

, where the’s are the same as in the proof of Lemma 2.17. Item 3), Lemma 2.11 and Lemma 2.1 imply together (15) for each n. Herein by construction. It remains to refer to item 2). □

Theorem 2.19. Let and be a nonnegative measurable random process on. Then Equality (14) holds with possible value of both sides.

Proof. By Lemma 2.18 for any

By Lemma 2.6 for any

By the same argument as in the proof of that lemma,

and, in view of (17),

From (18) we have by Lemma 2.6

which together with (16) and (19) proves (14). □

The stochastic integral w.r.t. a local martingale will be written, following [5,6], as. The designation of this section is to find the least restrictive extra assumptions providing the properties

,

of underlying the derivations in Section 4. Herein we do not demand that, so the conditional expectations in these properties are not classical but extended.

The following statement differs from Doob’s optional theorem for nonnegative discrete-time submartingales only with the absence of the demand falling out of the proof if one uses the extended expectation instead of the ordinary one.

Lemma 3.1. Let be a sequence of nonnegative random variables adapted to a flow

and such that.

Then the inequality holds for any bounded stopping times (w.r.t. the same flow) and.

This result leads in the standard way to Doob’s inequality asserted by the following lemma.

Lemma 3.2. Under the assumptions of Lemma 3.1,

.

Let denote the class of all -adapted - valued (will be determined by the context, if matters) random processes satisfying the conditions:

M1. For all.

M2. For all

Lemma 3.3. Let. Then for every and -measurable -valued random variable such that.

Proof. Denote. Then: by condition M2 and the assumption that M is -adapted; by condition M1. It remains to refer to Lemma 2.15. □

Corollary 3.4. (from Lemmas 3.3 and 2.3) Let. Then for all

.

Hence and from the identity we get Corollary 3.5. Let. Then for all

.

Lemma 3.6. Let. Then for any

Proof. Denote

. By construction and condition M1

whence by Lemma 3.2

Herein is càdlàg (see the first sentence of the article), so. It remains to make use of Lemma 2.7. □

Henceforth “stopping time” means “stopping time w.r.t. the flow”.

Lemma 3.7. Let. Then the equality holds for every and bounded stopping time.

Proof. We consider, without loss of generality, -valued processes. Writing

and noting that the r.h.s. of the equality is -measurable, we get

. So it suffices, in view of Lemma 2.11, to show that

By assumption there exists a number such that. We will prove Equality (20) for (otherwise it is trivial). Denote

. By construction is a stopping time and for all. From the last relation and right-continuity of we have

. Herein, whence by Lemma 3.6, which in view of M1 proves stochastic boundedness of the sequence. Then by Lemma 2.16

Denote.

From M1 we have by Corollary 2.5

On the strength of M2

, which together with the previous relation results, by Lemma 2.14, in

By the same lemma and property M2 of

By the construction of

Herein and , which together with (24)-(22) and Lemma 2.11 yields This equality jointly with (21) proves (20). □

The class of all random processes such that the process is continuous will be denoted.

Lemma 3.8. contains the sum of every two its elements.

Proof. Let, where. Property M1 of ensues from Lemma 2.11. It follows from Lemma 2.3 that

for all and. Hence property M2 and, with account of Corollary 3.5, continuity of emerge. □

Theorem 3.9..

Proof. Let. Denote . By construction all’s are -measurable random variables (and therefore stopping times) and. The process is -adapted and right-continuous and therefore, by Theorem 2.1.1 [1], -progressive and all the more measurable. Then Lemma 2.17 applied to and yields

. By Corollary 3.5 is an increasing process and therefore. By the choice of the process is continuous, so. Consequently,

and therefore. Herein by Lemma 3.7

(as). Thus and

is a martingale. This means, since is an increasing to infinity sequence of stopping times, that. □

The quadratic variation of a semimartingale and the quadratic characteristic of a locally square integrable martingale M will be denoted and, respectively.

The following statement is immediate from Theorem 1.8.1 in [5] and the definition of quadratic characteristic.

Lemma 3.10. Let be an -valued locally square integrable martingale. Then for any stopping time .

Corollary 3.11. Let be an -valued locally square integrable martingale. Then for any stopping time.

Note that all the random variables in the above two statements are, generally speaking, -valued.

The Lebesgue - Stieltjes integral, where

is a random process of locally bounded variation, will be written shortly as.

In the next statement, the process need not be right-continuous and even may have second-kind discontinuities.

Lemma 3.12. Let be an -valued process of class and be an -valued -predictable random process such that

and the process is continuous. Then.

Proof. Lemma 3.8 allows us to confine ourselves to the case.

The assumptions of the lemma imply by Theorem I.4.40 [6] existence of the process. The same theorem asserts that and, From the last equality we also have by Corollary 3.11, which together with (25) proves property M1 of and continuity of.

The relation implies existence of an increasing to infinity sequence of stopping times such that for all

Setting in Lemma 3.10 at first and then and taking to account that is an increasing process, we get with account of Lemma 2.3 , which together with M1 entails stochastic boundedness of the sequences

and (in view of Lemma 2.4). So Lemma 2.16 asserts that. Thus, letting in (26), we obtain M2. □

Lemma 4.1. Let be a continuous increasing function, and be bounded in each interval Borel functions and be a function satisfying, for all, the equality

where is a Borel function with values in such that. Suppose also that

for all. Then, where is the solution of the equation

Proof. By condition (28) and the assumptions about the integral exists on and is a function of locally bounded variation. Equality (27) and the assumptions about and show that U is a Borel function. So. The assumptions of the lemma imply existence of the integral , as well (so that almost everywhere w.r.t. the measure with distribution function). This entitles us to define the function h by . It decreases, since, by assumption, and increases. Also, it is continuous, since so is.

Denoting and subtracting (27) from (29), we get the equation. Hence, taking to account that is continuous and starts from zero, we find

The function h being decreasing, the r.h.s. is nonpositive. □

Corollary 4.2. Let be a continuous increasing -adapted random process, be an -progressive random process with values in satisfying, for all, condition (28), be an -semimartingale and be a random process satisfying, for all, equality (27), where is a measurable random process such that. Then for all

where.

Proof. Denote. Noting that and taking to account continuity of, we write down the solution of (29):

By construction is a continuous process of locally bounded variation, so. By Proposition I.4.49d [6] the covariation of any such process and a semimartingale equals zero, so the integration-by-parts formula yields. Thus, which turns (31) into

Now, (30) follows from Lemma 4.1. □

The main result of this article concerns equations of the kind

and relies on the assumption S. For every -valued random process equation (32) has a unique strong solution.

As usually, signifies the continuous martingale constituent (see [1,5,6]) of a semimartingale.

Theorem 4.3. Let be an -valued process of class and be an -valued random function on, continuous in and - progressive in. Suppose also that condition S is fulfilled and there exists an - measurable in nonnegative random process such that and

Then the strong solution of the equation

satisfies, for all, the inequality

where.

Proof. Denote , so that is a stopping time, and

Let further denote the solution of the equation

(this definition of is correct due to condition S). Then as (because for these). Consequently,

By the choice of and by Corollary 3.11 and Theorem 3.9 for all and the process is continuous. Then because of (35)

for all. Obviously, the process is continuous, too. Thus Lemma 3.12 asserts thatwhence in view of (37)

Denote

From (36) we have by the assumptions about and

By Theorem 2.4.6 in [1] (or, the same, Theorem I.4.47 in [6])

Writing Itô’s formula for and putting

, so that (a twice covariant tensor), , we get with account of (36), (40) and (41), continuity of and the identity

By Theorem 3.9 and Corollary 3.11 for all since. Hence and from the evident inequality we have, which together with (38) yields, by Lemma 2.11,

By construction and the assumptions about and, whence by Formula (2) for nonnegative random variables The last three equalities together with Lemmas 2.11 and 2.13 imply that

By construction and the assumption on Q the process is càdlàg and non-positive. Then from (39) we have by the choice of and by Theorem 2.19

where. Then equality (42), whose l.h.s. is, evidently, an -valued process, together with established above finiteness of shows that for all (though may take the value with positive probability).

By the construction of, the assumption on and by Lemma 2.3 The process was assumed increasing and therefore increases, too; the process was assumed nonnegative, so by Lemma 2.3. Thus, which together with (42), (43) and finiteness of yields. Then from -measurability of we have by Lemma 2.15 and therefore

. From this inequality and (33), (42), (43) we get by Corollary 4.2

and all the more

Obviously, as . Then Corollary 2.8 asserts that

. It remains to note that

by Corollary 3.11. □