SIAM J. CONTOTVol. 5, No. 4, 1967Printed in U.S.A.

ON THE ULTIMATE BOUNDEDNESS OF MOMENTS ASSOCIATEDWITH SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS*

MOSHE ZAKAI]

1. Introduction. Let x, be the solution to the stochastic differentialequation

(1) dx, m(t,x,) dt + G(t,x,) dw,, Xo a,

where x nd re(t, x) re vectors in the Euclid en r-spce E, G is n r X qmtrix nd w, is the standard q-dimensional Brownin motion. The follow-ing ssumptions re mde on m nd G"

() re(t, x) and G(t, x) re continuous in [0, o X E,(b) m(t, z) m(t, y) + G(t, x) G(t, y) <= c lx Y

where for vectors, ml ( 21]2’m and for mtrices,

The conditional expectation EaV(x) will be said to be ultimately bounded(see [1, p. 129]) if for all a in E,

(2) ]im lEaV(xt) <= lc < ,/c being independent of a. A Liapunov-type condition for the ultimateboundedness of certain functions V(x) will be derived in the next section.

Stability properties of x, in the sense of some convergence of x to thenull state have been discussed in the literature (i.e., [2], [3], [4]); however,the class of processes having such stability properties is, for many applica-tions, too restricted. In [5] Wonham suggested considering the propertythat x, admits a stationary probability measure as a weak stability propertyand derived Liapunov-type conditions for stability in the wek sense. Wemay also consider ultimate boundedness as a form of weak stability (withrespect to V(x)). For example, we may define x, to be nth order weaklystable if x, is ultimately bounded for V(x) Ix .

Recently Wonham [6] derived Liapunov-type sufficient conditions forthe ultimate boundedness of certain functions of x, under the assumptionthat the x process admits an invariant probability measure. Such an as-sumption has been avoided in 2 (at the expense of a stronger require-ment from the Liapunov function). Consequently, the results are appli-cable to a wider class of processes including nonsttionry processes and

* Received by the editors December 12, 1966, and in revised form March 11, 1967.] Faculty of Electrical Engineering, Technion-Israel Institute of Technology,

Haifa, Israel.

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STOCHASTIC DIFFERENTIAL EQUATIONS 589

processes for which the matrix GG’ is singular, and the proofs are based oastandard results on stochastic differential equations.Two examples will be considered in 3.2. A criterioa for ultimate bouadedaess. Let deote the differeatil

operator ssociated with (1)

(3) (t, x) 0 0,= + g,y( t, x)

where g is the i, jth entry of GG’, where the prime denotes the transpose.We will consider functions V(x) with the following properties:

(A) V(x) is real-valued, nonnegative and twice contuously differen-tiable in E.

(B) Let f(a, t) stand for any of the functions EV(x), E,gV(x) , orEa (OV(x)/Ox) G(t, x) . Then f(a, t) is, for each a, bounded

in any bounded intervM.It follows directly from [7] that if (1) satisfies conditions (a) and (b),

then Ex , p > O, is bounded ia any bounded intervM. Therefore,condition (B) is satisfied when V, 9V and (OV/Ox)G] are dominatedby polynoMs.TEonE 1. U x satisfies (1) with () and (b) and ff V(x) satisfies (A)

and B then

(4) v(x) v(x), > o, o,for all > 0 implies

(5) E V(x) V(a)e- + (1 e-).

If, in additi, m and G are independent of and for some e > O, h > O,

(6) Ei gV(x) + c <(c may depend a) for all 0 h, then (5) implies (4). (The last in-equality can be replaced by the more general condition that EagV(x) iscontinuous in at 0.)

Proof. By assumption (A), we may apply ItS’s formulu [8]; therefore,

V(xt) V(a) [Jo V(x,) + ’o OxG(xt) dwt.

By assumption (B), the expectation of the stochastic integral is zero and

(7) E V(xt) V(a) + ] E 9V(x,) ds.0

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590 MOSHE ZAKAI

It follows from (7) that EaV(xt) is absolutely continuous in (with respectto Lebesgue measure). Therefore, for almost all s, s ->- 0,

(8) dEa Y(xs) Eads

(9) <= ]1- 2Ea V(xs).

For all those s for which (9) holds, we havek2sd .Ea V xs ]1 e12so

dsk2 k2tSince e EaV(X) is also absolutdy continuous, ,V(x) is (everywhere)

the indefinite integrM of its Mmost-everywhere derivative. Integrating thelast inequMity we get

e E, V(x) V(a) 1),

from which (5) follows. Conversely, if (5) is true, then for > 0,

E V(xt) V(x) < k-(10)

since

li- E V(x) V(x) _<__ ] lc. V(.

(11) E V(x) V(x) 1 fo- E 9V(x,) ds.

Then if EgV(x,) is continuous in s at s 0, (4) follows from (10) and(11). In particular, since V(xs) is a.s. continuous, (6) implies the con-tinuity of EgV(xs) (see [9, Corollary 2, p. 164]).

3. Applications.Example 1. Consider the stochastic differential equation

(12) dxt Axt dt + f(xt) dt -t- G(xt) dwt,

where A is a constant r r matrix and f(x) a vector-valued function. Itis assumed that condition (b) is satisfied.TIEOREM 2. If all the characteristic values of A have negative real parts,

f(x) o(I x I) asx and Gj(x) o(I x I) asx---> , then Ea x ’is ultimately bounded for every positive p.

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STOCHASTIC DIFFERENTIAL EQUATIONS 591

Proof. Since E1/lx is nondecreasing in p, it suffices to prove thetheorem for even integers. Consider the matrix equation for B"

(13) A’B - BA -C,

where C is an arbitrary positive definite matrix. From the assumption onthe characteristic values of A it follows that (13) has a positive definitesolution B (see [1, p. 26]). Let km and k be the smallest and largest eigen-values of B, respectively. Setting V(x) x Bx, we have , x <= V(x)=< k Ix 2. For this choice of V(x), we have

cJV(x x’A’Bx -t- x’BAx + 2x’Bf(x) + trace {BG(x)G’(x)}-x’Cx -t- 2x’Bf(x) + trace IBG(x)G’(x)}-1/2x’Cx + [2x’Bf(x) + trace IBG(x)G’(x)} x Cx].

Since C is positive definite and f(x) and G(x) arethere exists an R > 0 such that the term in brackets will be negative forall xl >= R. Therefore, there exists a constant cl > 0 such that for allx in E,

V(x) <= 1/2 x’ Cx + c(14)

_-< -1/2where kc is the smallest eigenvalue of C. Since the last inequality satisfiesTheorem 1, Ealxt 12 < EaV(xt) is ultimately bounded. For p 2n,

3V(x) nV-l(x)CjV(x) + n(n 1)Y-2(x) trace -. G(x)G’(x)

and by (14),

x riVe(x)nV(x) + -vn(x)

o) } v_()+(- )-()tr(V (x)V’() +kOxSince V(x) is of the order of x ]:" and the other terms in the squarebracket are of lower order, there exist constants c, ca 0 such that forax inE,

V(x) -cV’(x) + c.

The ultimate boundedness of Eax follows from Theorem 1 andE. x (EV(x).Example 2. A generalization of Example 1 in [5] will be considered here;

in particular, G will not be required to be strictly positive definite. Let

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592 MOSHE ZAKAI

(15) dx Fx dt be(a) dt + f(x) dt - G(x) dw,,

o’--" CX

where F is a constant r X r matrix, b and c are constant r-vectors, isscalar-valued and f is vector-valued.THEOREM 3. Let the system (15) satisfy (b) and the following conditions"(i All the eigenvalues of F have negative real parts.(ii) a(a) > 0 for all 0, (a) is continuously differentiable and

(() d/d(r is bounded in , oo ).(iii) If(x)(iv) There exist two nonnegative constants and such that -[- > O,

Re (a - io)c’(ioI F)-lb > 0

for all real o, and if c’b O, then ac’Fb -[- flc’F2b O.Then Ea x, is ultimately bounded for every positive p.

Proof. For p 2, set

V() x’ Bx + f () d.ao

Then grad V(x) 2Bx + (crx)c and

Ox Ox#2B + fl()(c’x) cc’,

which is bounded. Therefore,

V(x) <- o(I x ) -[- (grd V(x))’(Fx bch(o’)).

By the results of Yacubovich-Kalman-Meyer [10, Lemm 4 and Theo-rem 1], there exist positive definite matrices B and Q such that

9V(x) <- o([ x ) x’Ox.Since Q is positive definite and !() =< h[z l, >- 0, there exists > 0such that #V(x) <= x’Qx. Therefore, there exist k and k2 such that Theorem1 applies for V(x), and since V(x) >= c x 12, E xt [ is ultimately bounded.For any integer n > 1, we have

grad V’(x) nV-(x) grad V(x),

Therefore,

nV’-(x)(2B + fl(c’x)cc’)

+ n(n- 1)V"-2(x)(grad V(x))(grad V(x))’.

9V(x) <= o(I x )and EV’(x), E. xt [" are ultimately bounded.

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STOCHASTIC DIFFE.RENTIAL EQUATIONS 59

Acknowledgment. The author wishes to thank a member of the editorialboard for pointing out an error in the previous version of this paper and forcalling his attention to the correction note of [10].

REFERENCES

[1] W. HAHN, Theory and Application of Liapunov’s Direct Method, Prentice Hall,Englewood Cliffs, New Jersey, 1963.

[2] R. Z. KHASMINSKII, On the stability of the trajectory of Markov processes, J. Appl.Math. Mech., 26 (1963), pp.. 1554-1565.

3] H. J. KUSHNER, On the theory of stochastic stability, Tech. Rep. 65-1, Center forDynamical Systems, Brown University, Providence, Rhode Island, 1965.Also in Advances in Control Systems, vol. 4, C. T. Leondes, ed., AcademicPress, New York, 1967.

[4] F. KozI, On almost sure asymptotic sample properties of diffusion processes de-

fined by stochastic differential equations, J. Math. Kyoto Univ., 4 (1965),pp. 515-528.

[5] W. M. WOHM, Liapunov criteria for weal stochastic stability, J. DifferentialEquations, 2 (1966), pp. 195-207.

[6] -------, A Liapunov method for the estimation of statistical averages, Ibid., to ap-pear.

[7] M. ZKAI, Some moment inequalities for stochastic integrals and solutions tostochastic differential equations, Israel J. Math.., to appear.

[8] K. ITS, On aformula concerning stochastic differentials, Nagoya Math. J., 3(1951),pp. 55-65.

[9] M. LovE, Probability Theory, Van Nostrand, Princeton, New Jersey, 1963..|10] K. R. MEYER, On the existence of Lyapunov functions for the problem of Lur$, this

Journal, 3(1966), pp. 373-383; note of correction, J. Differential Equations,to appear.

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