Vrkoč, Ivo

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Institute of Mathematics AS CR, v. v. i.

EQUADIFF 3Ivo VrkočAn optimal control problem for Ito stochastic equationsIn: Miloš Ráb and Jaromír Vosmanský (eds.): Proceedings of Equadiff III, 3rd CzechoslovakConference on Differential Equations and Their Applications. Brno, Czechoslovakia, August 28 -September 1, 1972. Univ. J. E. Purkyně - Přírodovědecká fakulta, Brno, 1973. Folia FacultatisScientiarum Naturalium Universitatis Purkynianae Brunensis. Seria Monographia, Tomus I.pp. 205--207.Persistent URL: http://dml.cz/dmlcz/700059Terms of use:© Masaryk University, 1973Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czAN OPTIMAL CONTROL PROBLEM FOR ITO STOCHASTIC EQUATIONS by IVO VRKOC Let an Ito equation dx = a(t, x) dt + B(t, x) dw(t) (I) be given in a region Q = (0,L) x (x1, x2) where L is a positive number, (xx, x2) is an open interval and w(t) is a Wiener process (Ew(t) = 0, Ew2(t) = t, E is the mathe-matical expectation). Assume that i) the functions a(t, x), B(t, x) are defined in the closure Q, ii) a(f, x), B(t, x) are Lipschitz continuous in x and Holder continuous in t, iii) B(t, x) # 0 in Q, iv) an initial value x0 is a random variable which is stochastically independent of increments of w(t) and is situated in (x1, x2) (i.e. P(x0 e (xx, x2)) = 1). Denote S = {[l, x j ; 0 — t = L, i = 1, 2}. The solution x(t, x0) of (1) with adhesive barrier S is defined as usual i.e. x(t) = x*(t) for t = x, x(t) = x*(x) for t > x where x*(0 is the solution of (1) (provided that a(t, x), B(t, x) are extended on the whole plane) with initial value x0 and x is the moment of the first exit of x*(t) from (xx, x2). Under the assumptions i) to iv) the solution with adhesive barrier exists and is unique (in the sense of equivalent processes—see [1]). Denote P(B, a, x0, Q) the probability that x(t, x0) leaves the interval (xt, x2) at least once on the time-interval (0,L) i.e. P(B, a, x0, Q) = P{3{t : x(x, x0) $ (xx, x2), x = L}}. Definition 1. A function B(t, x) is called maximal with respect to a function a(t, x) and to a region Q (Q = (0, L) x (xt, x2)) if the functions a(t, x), B(t, x) fulfil i), ii), iii) and if P(B, a, x0,Q) = maxP(B\ a, x0, Q) B for all initial values x0 fulfilling iv) where B\t, x) can be every function fulfilling i), ii), iii) and \B\t,x)\ = \B(t,x)\. Let u(t, x) be a bounded solution of the parabolic equation du f . du 1 , d2u ,~. — = a(L-t,X)— + y B-(L - t, x ) — - (2) fulfilling the initial condition w(0, x) = 0 for xe (JC. , x2) (3) 205 and the boundary condition u(t9xd = 1 for t > 0, i = 1,2. (4) Under the conditions i), ii), iii) the solution u(t, x) exists and is unique in the class of bounded functions (see [2], [3]). The maximal functions B(t9 x) can be characterized by means of properties of the solutions u(t, x). Theorem 1. Let a(t, x), B(t, x) fulfil conditions i) to iii). The function B(f9 x) is maximal with respect to the function a(t9 x) and to the region Q if and only if the bounded solution u(t9 x) of (2) fulfilling (3) and (4) is convex as a function of x in Q. Since it is very difficult to verify the condition given by this Theorem more explicit conditions are formulated in the following theorems. The simplest case is when a(t, x) is a linear function in x. Theorem 2. Let functions a(t), fi(t) be defined and Holder continuous on <0, L>. Let B(t, x) fulfil conditions i) to iii). If a(f9 x) = a(t) + x/?(t) fulfils a(t9 xX) ^ 0 and a(t9 x2) ^ 0, then B(t9 x) is maximal with respect to the function a(t9 x) and to the region Q. Remark 1. The conditions a(t9 xx) = 0, a(t9 x2) ^ 0 are also necessary conditions. In the following it will be assumed that coefficients a, B do not depend on t. We shall say that the coefficients of (I) are symmetric if x1 = — x2, a(x) = — a(—x) and B(x) = B(-x). Theorem 3. Let coefficients of (1) fulfil i) to iii). If the coefficients are symmetric and fulfil a(t9 x) ?g 0 for x ^ 0, then B(x) is maximal with respect to the function a(x) and Q. Secondly, the case of nonnegative a(x) will be treated. Theorem 4. Let functions a(x)9 B(x) fulfil conditions i) to iii). If 0 < Kt ^ B2(x) ^ ^ K2 (K!, K2 are constants) and Ґ\ ^ / \ ^ -^2 — •* K2 I /K1 I / \ 0 й a(x) S ~ ~ 7 arcsin J-^-ì m (x^, x2), 2 (x2-Xl)2 l V л2J (5) then the function B(x) is maximal with respect to the function a(x) and to the region Q. NOW we shall deal with the case when a(x) can change the sign. Theorem 5. Let functions a(x)9 B(x) fulfil conditions i) lo iii). IfO < Kx fg B2(x) ^ ^ K2, a(x2) ^ 0 ^ a(Xj) and «,> _ „(„, 6 _ £ ^ - ^ [ a , ^ - j J I ) J 206 where X x - , JB2(0 "' x }B\O X2 X2 then B(x) is maximal with respect to the function a(x) and to the region Q. Remark 2. The conditions of the two last theorems are independent of L. It implies that if B is maximal with respect to a(x) and to a region Q* = (0,L*) x (xl, x2), then it is maximal with respect to a(x) and to a region Q = (0, L) x (xj, x2) for every L> 0. The next theorem states that there actually exists a couple of functions a(x), B(x) such that B(x) is not maximal with respect to a(x) and especially for B(x) == 1 there exists a function a(x) (fulfilling i) to iii)) such that B(x) = 1 is not maximal with respect to the function a(x). This theorem deals also with the necessity of condition (5). Due to the statement of the following theorem condition (5) cannot be omitted. Definition 2. A function cp(x, K±, K2) has property (M) if it is defined and continuous and positive in the domain 0 _ x = 1,0 < Kt = K2 and if a function B(x) is maximal with respect to a function a(x) and to Q = (0,1) x (0,1) under conditions that Kx = = B2(x) = K2, 0 = a(x) = <p(x, K!, K2). For example the function in the right-hand side of (5) has property (M). Theorem 6. Lel a function <p(x, Kl9 K2) have property (M), then (pyx,l,l)^~mm\y^(t,x)l~(t,x)\ for x > —, te(0,l), where v(t, x) is the bounded solution of dvjdt — \ d2vjdx2 fulfilling v(0, x) = 0 for 0 < x < 1 and v(t, 0) = v(t, 1) = 1 for t > 0. REFERENCES [1] I. VRКOČ: Some maximum principles for stochastic equations. Czеch. Math. J. V. 19 (94), 1969, 569-604. [2] I. VRКOČ: Some explicit conditions for maximal local diffusions in one-dimensional case. Czеch. Math. J. V. 21 (96), 1971, 236-256. [3] G. ЅCHLEINКOFER: Die erste Randwertaufgabe und das Cauchy-Problem für parabolische Differen-tialgleichungen mit unstetigen Anfangswerten. Mathеrnatiѕchе Zеitѕchrift 1969, B 111, 87—97. Authoґs address: Ivo Vrkoč Mathematical Institute, Czechoslovak Academy of Sciences Žitná 25, Praha 1 Czechoslovakia 207 

An optimal control problem for Ito stochastic equations

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An optimal control problem for Ito stochastic equations

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Institute of Mathematics AS CR, v. v. i.