A Conditional Mean Square Estimate for the Solution of a SDE

Let F = (F (t), t ∈ R+) be a filtration on some probability space, and X be the strong solution of the equation X(t) = X̊ + ∫ t 0 Q(s, X(s))dι(s) + ∫ t 0 σ(s, X(s−))dY(s), where X̊ is an F (0)-measurable Rd-valued random variable, ι is a continuous increasing process with F (0)-measurable values at all times, Y is an Rm-valued locally square integrable martingale with respect to F subjected to some mild additional demands, Q and σ are continuous in x ∈ Rd random functions on R+×Rd (the former Rd-valued and F-progressive in (ω, t) ∈ Ω×R+, the latter (d×m)matrix-valued and F-predictable). Suppose also that there exists an F (0) ⊗ B+-measurable in (ω, t) nonnegative random process ψ such that, for all t, x, x Q(t, x) ≤ −ψ(t)|x|2 and ∫ t 0 ψ(s)dι(s) < ∞. Under these assumptions, E(|X(t)|2|F (0) is evaluated from above.


Introduction
The random processes under consideration are assumed given on a common probability space (Ω, F , P).Let F 0 be a sub-σ-algebra of F .We introduce the notation: E 0 = E • • • |F 0 (the definition of conditional expectation, in particular E 0 , adopted in this article does not demand finiteness of first absolute moment-see Section 2); V + 0 is the class of all increasing from zero numeral random processes whose values at all times are F 0 -measurable random variables, V +, c 0 is its subclass of continuous processes.If, besides, a filtration F = (F (t), t ∈ R + ) is given, then we identify F 0 with F (0).By M 2 we denote, following (Gikhman & Skorokhod, 1982, 2009), the class of all R m -valued (m will be determined by context, if matters) locally square integrable martingales w.r.t.F.
Let X be the strong solution of a stochastic differential equation of the kind where ι ∈ V +, c 0 and Y is chosen from some subclass of M 2 which is constructed and studied in Section 2 (and was introduced in (Yurachkivsky, 2013a(Yurachkivsky, , 2013b))).The goal of this article is to find an upper bound, much more exact than that provided by the Gronwall-Bellman lemma, for E 0 |X(t)| 2 .This is done in Section 3 containing the final result of the article together with its application to stability theory.In the case σ = const, an estimate for E 0 |X(t)| 2 was found in (Yurachkivsky, 2013a), and it is the starting point for our present research.Sections 1 and 2 contain preparatory technical results of which the Fubini-type theorem for conditional expectations (Proposition 2.3) may be of interest on its own right.
Stability of order p of the solution of a SDE is usually studied with the aid of Lyapunov's functions (see, e.g., Khasminsky, 2012;Shen & Sun, 2011).But this approach is fruitful only when Q, σ and ι are nonrandom and Y has independent increments, so that X is a Markov process.Our Theorem 3.1 yields, as a byproduct, sufficient conditions for mean square stability without these restrictive assumptions.The only specific condition of that theorem-inequality (26)-is of quadratic nature, which explains why the theorem concerns only the case p = 2.
All vectors are thought of as columns; with real entries, the class of all symmetric m × m matrices with real entries and its subclass of nonnegative (in the spectral sense) matrices, respectively.For A, B ∈ S, the inequality A ≤ B means that B − A ∈ S + (so that one may speak about increasing S-valued functions).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 I with two possible modes of writing the set: The reference books for the notions and results of stochastic analysis used in this paper are (Elliott, 1982;Gikhman & Skorokhod, 1982, 2009;Jacod & Shiryaev, 1987;Liptser & Shiryaev, 1989).

Deterministic Preliminaries
Lemma 1.1 For any S ∈ S and B ∈ Matr d×m , tr BS B ≤ B 2 tr S .
Proof.By a familiar property of symmetric matrices there exist real numbers λ 1 , . . ., λ m and an orthonormal basis In the subsequent three statements, H is a Borel Matr d×m -valued function and K is an increasing continuous Svalued function, both defined on ]a, b] ⊂ R + .The (i, j)th entry of the the matrix H(s) (respectively K(s)) will be denoted by h i j (s) (respectively κ i j (s)).For an arbitrary natural n, we denote n = {1, . . ., n}. (1) Then for all i ∈ d and λ, μ, ν ∈ m b a h iλ (s) 2 dκ μν (s) < ∞. (2) Proof.Obviously, the module of each entry of any matrix does not exceed the operator norm of the latter.So to deduce (2) from (1) it suffices to show that for all μ, ν ∈ m, s ∈]a, b] and t ∈]s, b] And this follows from the above-stated and the inequality S ≤ tr S for an arbitrary S ∈ S + (the norm of such a matrix equals to its greatest eigenvalue).
Corollary (4) Proof.For continuous H, the integrals on both sides of (4) are the limits of the Riemann-Stieltjes integral sums, so in this case the inequality is immediate from Lemma 1.1.
By the dominated convergence theorem the class of those functions H which satisfy (4) contains the limit of every pointwise convergent and uniformly bounded sequence of its elements.So it contains all bounded Borel functions, since each of them arises from continuous ones by virtue of at most countably many bounded pointwise passages to the limit.In case H is unbounded we introduce the functions .
By construction H n (s) ≤ n ∧ H(s) , so by what was proved Denote the (k, p)th entry of H n (s) by h nkp (s).From (3) and the inequality where k = (i, j, μ, ν), and all the more g nk ≤ 4 H 2 .Herein lim n→∞ g nk (s) = 0 (since, evidently, lim n→∞ H n (s) = H(s)), which together with the last inequality and (1) yields by the dominated convergence theorem And this jointly with (6) entails (5).

Extended Conditional Expectations
The definition of conditional expectation adopted in this article is due to Meyer (Shiryaev, 1996, Ch. II, § 7).It admits existence of the conditional expectation of a random variable with infinite first absolute moment.In this subsection, we recall some properties of thus generalized conditional expectation and prove several statements concerning this notion.
Let G be a sub-σ-algebra of F .The conditional given G expectation of an R + -valued random variable γ is defined, according to (Shiryaev, 1996), as the . Further E(γ|G) is defined in the obvious way for R d -valued (and even C d -valued if one needs) γ.Thus defined (on some extension of L 1 (Ω, F , P)) conditional expectation is called extended.
In this subsection, we consider R-valued random processes.The total variation on [a, b] of a function f will be denoted by var [a,b] f .
The following statement is immediate from Lemma 2.1, Proposition 2.1, Corollary 2.1 and the definition of total variation.Lemma 2.5 Let F be a random process on [a, b] such that var [a,b] F is a random variable (i.e., an F -measurable function of ω ∈ Ω) and Then: F holds.
Lemma 2.6 Let ξ be a càdlàg random process on [a, b].Assume that there exists a random variable Γ such that so that P{Ξ(t) > ε} = Eρ(t, ε).Right-continuity of ξ implies that, for any ε > 0, the left-hand side of the last equality tends to zero as t → a+.Hence, taking to account that ρ(•, ε) is an increasing process (since Ξ increases by construction and the operator E 0 is isotonic by Lemma 2.1), we get Denote Γ n = ΓI{Γ > n}.Writing, for arbitrary n ∈ N and ε ∈]0, 2n], the identity we obtain from (8), ( 9) and the definition of Ξ (with the use of Proposition 2.1 and Lemma 2.1, of course) E 0 Ξ(t) ≤ 2E 0 Γ n + 2nρ(t, ε) + ε, which together with (10) yields By construction the sequence (Γ n ) decreases to zero.By assumption E 0 Γ < ∞, whence by Lemma 2.1 Then Lemma 2.9 in (Yurachkivsky, 2013a) asserts that E 0 Γ n 0, which together with (11) where ε is arbitrary yields The proof of Lemma 2.6 will not change if we substitute a by an arbitrary inner point of [a, b], so, under its assumptions, lim for any t ∈ [a, b[.If, moreover, ξ increases, then by Lemma 2.1 E 0 ξ(t 1 ) ≤ E 0 ξ(t 2 ) as t 1 < t 2 .Hence, repeating, up to notation, the proof of Theorem II.7.4 (Shiryaev, 1996), we deduce the following statement (for classical conditional expectations, it is a particular case of that theorem).
Corollary 2.2 Let a càdlàg random process F on [a, b] satisfy condition (7).Then there exists a càdlàg random process G on [a, b] with property (12).
Otherwise speaking, Corollary 2.2 asserts existence, under the above stated assumptions, of the càdlàg version of E 0 F. In what follows, we consider namely it.
We denote by F 0 the filtration with F (t) = F (0), t ∈ R + (so that a random process is F 0 -adapted iff its values at all times are F (0)-measurable random variables).Then V + 0 can be defined equivalently as the class of all starting from zero F 0 -adapted increasing random processes.
Proposition 2.3 Let F and ϕ be random processes on [a, b], the former càdlàg and satisfying condition (7), the latter continuous and F 0 -adapted.Then (13) Note that the integral on the right-hand side of ( 13) is, due to continuity of ϕ, the limit of Riemann-Stieltjes sums and therefore an F -measurable function of ω.
Proof. 1 • .Denote Hence, recalling that ϕ is F 0 -adapted, we get by Theorem 1.17 in (Yurachkivsky, 2013b) As was shown above, var G < ∞ and G may be, without loss of generality, considered càdlàg, so the integrationby-parts formula yields , which together with ( 16) and ( 15) And this is none other than equality (13).
By construction all trajectories of ϕ n are continuously differentiable and therefore have finite variation on [a, b].
where χ n = b a ϕ n (s)dF(s) (existence of E 0 χ n is justified in the same manner as was done for E 0 χ in item 1 • ).Obviously, ϕ n ⇒ ϕ, which together with var G < ∞ yields ( By construction the random process ϕ n is F 0 -adapted, since so is ϕ.Continuity of both ϕ and ϕ n implies that Consequently, E 0 χ n → E 0 χ, which together with ( 18) and ( 19) entails ( 17).

A Subclass of the Class of Locally Square Integrable Martingales
Let our probability space (Ω, F , P) be endowed with a right-continuous flow of σ-algebras (or, in the terminology of (Jacod & Shiryaev, 1987;Liptser & Shiryaev, 1989), filtration) F = (F (t), t ∈ R + ).By K we denote the class of all F-adapted R m -valued (m will be determined by context, if matters) càdlàg random processes M satisfying the conditions: M3.The process E 0 |M| 2 is continuous.
Proposition 2.5 For any M ∈ K and stopping time τ, M τ ∈ K.
In (Yurachkivsky, 2013a), this statement was implicitly used in the proof of the main result.

The Main Result
In this section, we obtain, under appropriate assumptions, a conditional mean square estimate for the solution of the equation Along with this equation in its general form we will consider its particular case First of all we impose the following assumptions: S1.For every Z ∈ K Equation ( 24) has the unique strong solution on R + .
S2.For every N ∈ K Equation ( 25) has the unique strong solution on R + .
The Borel σ-algebra in R + will be denoted by B + .
The solution of (25) was evaluated in (Yurachkivsky, 2013a) by virtue of a special lemma cognate to the comparison theorems in (Ikeda & Watanabe, 1981, Ch. VI), without recourse to Lyapunov functions.That inequality underlies the derivation of our main result.
Theorem 3.1 Let X be an F (0)-measurable R d -valued random variable, ι be an F 0 -adapted increasing continuous random process, Y be an R m -valued random process of class K, Q and σ be continuous in for all x ∈ R d , t > 0; there exist random processes L ∈ V + 0 and R ∈ V +, c 0 such that Then the strong solution of the equation satisfies, for all t, the inequality By construction U n is a nonnegative increasing random process, so Denote Φ = e Ψ .The process ι, being F 0 -adapted and continuous, is F (0) ⊗ B + -measurable in (ω, t); for ψ this was assumed.Then it follows from (41) that Ψ is F (0) ⊗ B + -measurable in (ω, t) and all the more F 0 -adapted.Besides, it is continuous, since so is ι.These properties of Ψ together with relations ( 43), ( 38) and ( 30 Denoting V n = ΦE 0 |X n | 2 and noting that M n (0) = X because of (35), we get from ( 40) and (47) By assumption R ∈ V + 0 , so Theorem 2.19 in (Yurachkivsky, 2013a) asserts that By construction T increases and is continuous (since R possesses these properties and L is nonnegative).So (50) yields by the Gronwall-Bellman lemma V n (t) ≤ X 2 e T (t) .Multiplying both sides of this inequality by e −Ψ(t) , we get E 0 |X n (t)| 2 ≤ X 2 e T (t)−Ψ(t) . (51) b a means ]a,b] .We use the Euclidean norm | • | of vectors and the operator norm • of matrices A = sup |x|≤1 |Ax| .The symbols Matr d×m , S and S + signify the class of all d × m matrices us take an arbitrary orthonormal basis e 1 , . . ., e d in R d .Then for any symmetric d × d matrix A one has tr A = e 1 Ae 1 + . . .+ e d Ae d .In particular, Bh j )(Bh j ) e i , i.e., tr BS B = m j=1 λ j tr (Bh j )(Bh j ) .It remains to note that, firstly, for any x ∈ R d tr xx = |x| 2 and, secondly, |Bh j | ≤ B since |h j | = 1.
Proof.Denote Ξ(t) = sup a<s≤t |ξ(s) − ξ(a)|.Any càdlàg function is determined by its values on a dense subset of[a, b], so the supremum may be taken over s

Proposition 2. 2
Let F be a right-continuous increasing random process on [a, b] such that for any s ∈ [a, b], E 0 |F(s)| < ∞.Then there exists a right-continuous increasing random process G on [a, b] such that