Teor Imov r. ta Matem. Statist. Theor. Probability and Math. Statist. Vip. 88, 2013 No. 88, 2014, Pages 139–149 S 0094-9000(2014)00924-2 Article electronically published on July 24, 2014
CONVERGENCE OF EXIT TIMES FOR DIFFUSION PROCESSES UDC 519.21
YU. S. MISHURA AND V. V. TOMASHYK
Abstract. We establish the convergence in probability of exit times from a strip for stochastic processes that are solutions of stochastic differential equations with diffusion coefficients satisfying the Yamada condition under the assumption that the coefficients converge. As an auxiliary result, the uniform convergence in probability of pre-limit processes to the limit process is proved.
1. Introduction The importance of the problem of convergence of exit times from a strip for a stochastic process is explained by a possible application in studying the limit behavior of optimal stopping times. The classical approach to solve the optimal stopping time problem is based on the excessive functions used to describe the so-called reference set that, in turn, defines explicitly the optimal stopping time as the first exit time from this set [1, 2, 3]. This approach requires the convergence of exit times from certain sets; this property is used to study the convergence of optimal stopping times. In the current paper, we establish the convergence of exit times from a strip for stochastic processes that are solutions of stochastic differential equations with diffusion coefficients that satisfy the Yamada condition under the assumption that the coefficients converge. As an auxiliary result, we prove the uniform convergence in probability of pre-limit processes to the limit process.
2. Uniform convergence of solutions of stochastic differential equations whose coefficients satisfy the Yamada condition Consider a sequence of stochastic differential equations t t (1) Xn(t)=Xn(0) + bn(s, Xn(s)) ds + σn(s, Xn(s)) dW (s),n≥ 0,t≥ 0, 0 0 + with nonrandom initial conditions Xn(0). The coefficients bn,σn : R × R → R are measurable and {W (t),t≥ 0} is a Wiener process with respect to a filtration {Ft,t≥ 0} in the probability space (Ω,F,P). Assume that the coefficients of equation (1) satisfy the following Yamada conditions:
(Y1n) the coefficients bn and σn are continuous with respect to all their arguments; (Y2n) the coefficients grow at most linearly,
|bn(t, x)| + |σn(t, x)|≤L(1 + |x|),t≥ 0,x∈ R;
2010 Mathematics Subject Classification. Primary 60G40, 60H15, 60J60. Key words and phrases. Stopping times, stochastic differential equations, Yamada condition.
c 2014 American Mathematical Society 139
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(Y3n) the shift coefficient bn satisfies the Lipshitz condition,
|bn(t, x) − bn(t, y)|≤L |x − y| ,t≥ 0,x,y∈ R;
(Y4n) the Yamada condition holds; namely there exists an increasing function + + ρn : R → R such that −2 ∞ ρn (u) du = 0+ and
|σn(t, x) − σn(t, y)|≤ρn(|x − y|),t≥ 0,x,y∈ R. Definition 1. Let T>0. By LT (F, P)wedenotethespace 2 T × ⊗ × L2 Ω [0,T],F B([0,T]), P λ whose elements are stochastic processes such that T X2(t, ω) λ(dt) P(dω) < ∞. Ω 0
Conditions (Y1n)–(Y4n) yield the existence and uniqueness of a strong solution of T equation (1) belonging to the space L2 (F, P)foreachT>0 [4, 5].
Definition 2. Let τk be the first exit time from the interval (−k, k) for a stochastic process Y .Byτ = limk→∞ τk we denote the limit (finite or infinite) of the increasing sequence of exit times τk as k →∞. A process Y is called regular if P(τ = ∞)=1.
Remark 1. If conditions (Y1n)–(Y4n) hold, then the stochastic process Xn is regular and its paths are continuous [4, 5]. The following result contains conditions for the finiteness of moments of a diffusion process Xn. Lemma 1. Let the coefficients of stochastic differential equation (1) satisfy conditions (Y1n) − (Y4n). Then there is a constant C that does not depend on n and such that γ γ E |Xn(t)| < [|Xn(0)| + C · t] · exp [C · t] for all t ≥ 0 and γ>0. Proof. Note that the general case of the lemma follows from the particular one corre- sponding to even numbers γ =2p, p ∈ N. Consider the stopping time νk =inf{t: |Xn(t)| 2p ≥ k} and the function f(x)=x . Applying the Itˆoformulatof(Xn(t ∧ νk)) we obtain
f(Xn(t ∧ νk)) = f(X(0)) t∧ν k 1 + f (X (s)) · b (s, X (s)) + f (X (s)) · σ2 (s, X (s)) ds n n n 2 n n n 0 t∧ν k + f (Xn(s)) · σn(s, Xn(s)) dW (s). 0
By condition (Y 2n), 1 t∧ν 2 2 k 1 2p−1 E f (Xn(s)) · σn(s, Xn(s)) dW (s) ≤ 2pt 2 k L(1 + k) < ∞, 0
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whence t∧ν k E f (Xn(s)) · σn(s, Xn(s)) dW (s)=0 0 and 2p 2p E(Xn(t ∧ νk)) =(Xn(0)) t 2p−1 (2) + E 2p · bn(s ∧ νk,Xn(s ∧ νk)) · (Xn(s ∧ νk)) 0 − 2 ∧ ∧ ∧ 2p−2 + E p(2p 1)σn(s νk,Xn(s νk))(Xn(s νk)) ds. Now we estimate the expectations under the integral sign on the right hand side of equality (2). In the following chain of inequalities, we use condition (Y2n) and Lyapunov’s condition for expectations: 2p−1 E bn(s ∧ νk,Xn(s ∧ νk)) ·|Xn(s ∧ νk)| 2p−1 2p ≤ L · E |Xn(s ∧ νk)| + |Xn(s ∧ νk)| (3) 2p−1 2p 2p 2p ≤ L · E |Xn(s ∧ νk)| + E |Xn(s ∧ νk)| 2p ≤ L · 1+2· E |Xn(s ∧ νk)| .
Similarly to (3) we obtain a bound for the second expectation on the right hand side of equality (2): 2 ∧ ∧ | ∧ |2p−2 ≤ 2 · · | ∧ |2p E σn(s νk,Xn(s νk)) Xn(s νk) L 1+2 E Xn(s νk) . Thus, (2) can be rewritten as t 2p 2p 2 2p E(Xn(t ∧ νk)) ≤ (Xn(0)) + L(p +2p ) · t +2 E(Xn(s ∧ νk)) ds , 0 where L is a certain constant that depends only on L. Using the Gronwall inequality and passing to the limit as k →∞we get 2p 2p 2 2 E(Xn(t)) ≤ (Xn(0)) + L(p +2p ) · t · exp 2 · L p +2p · t , which completes the proof of the lemma.
Assume that
(4) bn(t, x) → b0(t, x),σn(t, x) → σ0(t, x),n→∞,
for all t ≥ 0andx ∈ R and that Xn(0) → X0(0) as n →∞. The following result concerning the convergence of solutions of stochastic differential equations (1) is proved in [6].
Theorem 1. Let conditions (Y1n)–(Y4n) hold for a sequence of stochastic differential equations (1).Ifrelation(4) holds, then
E[|Xn(t) − X0(t)|] → 0,n→∞, uniformly in finite intervals. Now we are in a position to establish the uniform convergence in probability of solu- tions of stochastic differential equations (1) to a limit process.
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Theorem 2. Let the coefficients of stochastic differential equation (1) satisfy condi- tions (Y1 )–(Y4 ). We also assume that convergence (4) holds. Then n n
P sup |Xn(t) − X0(t)| >ε −→ 0,n→∞, t∈[0,T ] for T>0 and ε>0. Proof. The assumptions of the theorem imply that equation (1) possesses a unique strong solution. The convergence of the initial conditions implies that there exists B>0such that |Xn(0)|
Moreover, H is determined completely by the constant L involved in condition (Y2n). It is clear that t2 t2 (Xn(t2) − Xn(t1)) = bn(u, Xn(u)) ds + σn(u, Xn(u)) dW (u). t1 t1 Then we use the inequality (a + b)4 ≤ 8a4 +8b4, Cauchy’s inequality, conditions imposed on the coefficients of equation (1), and inequality 4 b b E f(t) dW (t) ≤ 36(b − a) E |f(t)|4 dt. a a The latter inequality is valid for all measurable functions f(ω): [a, b] × Ω → R (see [7]). In what follows the symbol C denotes any constant whose precise value does not matter for the reasoning. Then 4 E(Xn(t2) − Xn(t1)) t2 4 t2 4 ≤ 8 E bn(u, Xn(u)) du +8E σn(u, Xn(u)) dW (u) t1 t1 t2 t2 3 4 4 ≤ C(t2 − t1) E[bn(u, Xn(u))] du + C(t2 − t1) E[σn(u, Xn(u))] du t1 t1 t2 t2 3 4 4 ≤ C(t2 − t1) E[1 + |Xn(u)|] du + C(t2 − t1) E[1 + |Xn(u)|] du. t1 t1
Lemma 1 implies that all moments of the process Xn are bounded, and this completes the proof of the inequality that had to be proved. In particular, the probability measures corresponding to stochastic processes Xn weakly converge. This implies that the sequence of probability measures correspond- ingtotheprocesses{Yn = Xn − X0} is weakly relatively compact.
Hence an arbitrary subsequence of the processes Ynk contains a further subsequence {Y } that weakly converges to some limit process: Y ⇒ Y∞ as l →∞.Theorem1 nkl nkl P implies that Y (t) −→ 0asl →∞for all t ≥ 0. Therefore Y∞(t)=0,t ≥ 0, and the nkl sequence of processes Yn weakly converges to 0. Since the weak limit of the processes is a constant, the processes Yn converge to 0 in probability in the uniform metric of the space C[0,T]. This implies that
P sup |Xn(t) − X0(t)| >ε −→ 0,n→∞, t∈[0,T ] which completes the proof of the theorem.
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3. Convergence of exit times of a diffusion process from a strip on bounded intervals We are going to establish a result on the convergence of exit times from a strip for pre-limit diffusion processes to the exit time from the same strip for the limit process. Let Xn be a solution of the following homogeneous stochastic differential equation: t t (5) Xn(t)=Xn(0) + bn(Xn(s)) ds + σn(Xn(s)) dW (s),n≥ 0, 0 0
with nonrandom initial conditions Xn(0) such that |Xn(0)|
(Y5n) the coefficients bn and σn are bounded; that is, there exists a constant K>0 such that
|bn(x)| + |σn(x)|≤K, x ∈ R,
(Y6n) the Yamada condition holds for the diffusion coefficient σn: |σn(x) − σn(y)|≤A |x − y|,x,y∈ R. Assume further that
(6) Xn(0) → X0(0),bn(x) → b0(x),σn(x) → σ0(x),n→∞, for t ≥ 0andx ∈ R. These conditions imply the existence and uniqueness of a strong solution belonging to T the space L2 (F, P) for all T>0 [4]. Theorem 2 also implies the uniform convergence in probability of the processes Xn to a limit process X0 in every compact set. Choose some l, r ∈ R such that l Theorem 3. Let the stochastic processes Xn be defined by equation (5). Assume that properties (Y 1n), (Y 3n), (Y 5n),and(Y 6n) hold. If convergence (6) is valid and there exists a constant a>0 such that |σn(x)| >afor x ∈ R and n ≥ 0, then the exit times converge in probability: ∀ [−∞,r] − [−∞,r] −→ →∞ ε>0: P τn,T τ0,T >ε 0,n . Proof. It is obvious that [−∞,r] − [−∞,r] [−∞,r] − [−∞,r] P τn,T τ0,T >ε = P τn,T τ0,T >ε (7) [−∞,r] − [−∞,r] + P τ0,T τn,T >ε . Consider the first term on the right hand side of (7). The random event [−∞,r] − [−∞,r] τn,T τ0,T >ε means that the difference between the time when the process Xn attains the level r and the time when the process X0 attains the same level is larger than ε.Inotherwords, ⎛ ⎞ [−∞,r] − [−∞,r] ⎝ [−∞,r] ⎠ P τn,T τ0,T >ε = P sup Xn(t) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 144 YU. S. MISHURA AND V. V. TOMASHYK Now ⎛ ⎞ ⎝ [−∞,r] ⎠ P sup Xn(t) = P sup |Xn(t) − X0(t)| >δ, ≤ ≤ [−∞,r] 0 t τ0,T +ε [−∞,r] sup Xn(t) + P sup |Xn(t) − X0(t)|≤δ, ≤ ≤ [−∞,r] 0 t τ0,T +ε [−∞,r] sup Xn(t) for all δ>0. Consider the term ⎛ ⎞ ⎝ [−∞,r] ⎠ P sup X0(t) [−∞,r] Note that X0 is a strong Markov process. Since τ0,T is a stopping time with respect to the natural filtration generated by this process, the strong Markov property is written as follows: ⎛ ⎞ ⎝ [−∞,r] ⎠ P sup X0(t) = P sup X0(t) If X0(0) = r,then t t sup (X0(t) − r)= sup b0(X0(s)) ds + σ0(X0(s)) dW (s) 0≤t≤ε 0≤t≤ε 0 0 t ≥ sup σ0(X0(s)) dW (s) − K · t . 0≤t≤ε 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use CONVERGENCE OF EXIT TIMES 145 Further, if δ1 > 0, then P sup (X0(t) − r) <δ,X0(0) = r 0≤t≤ε t ≤ P sup σ0(X0(s)) dW (s) − K · t <δ 0≤t≤ε 0 t = P sup σ0(r) · W (t)+ [σ0(X0(s)) − σ0(r)] dW (s) − K · t <δ 0≤t≤ε 0 t = P sup σ0(r) · W (t)+ [σ0(X0(s)) − σ0(r)] dW (s) − K · t <δ, 0≤t≤ε 0 t sup [σ0(X0(s)) − σ0(r)] dW (s) <δ1 (8) 0≤t≤ε 0 t + P sup σ0(r) · W (t)+ [σ0(X0(s)) − σ0(r)] dW (s) − K · t <δ, 0≤t≤ε 0 t sup [σ0(X0(s)) − σ0(r)] dW (s) ≥ δ1 0≤t≤ε 0 ≤ P sup [σ0(r) · W (t)] <δ+ δ1 + K · ε 0≤t≤ε t + P sup [σ0(X0(s)) − σ0(r)] dW (s) ≥ δ1 . 0≤t≤ε 0 We apply the Burkholder–Gundy inequality to the last term on the right hand side of (8): t P sup [σ0(X0(s)) − σ0(r)] dW (s) ≥ δ1 0≤t≤ε 0 2 t ≤ 1 · | − | 2 E sup σ0(X0(s)) σ0(r) dW (s) δ1 0≤t≤ε 0 ε ≤ C1 · − 2 2 E (σ0(X0(s)) σ0(r)) ds . δ1 0 As in the proof of Theorem 2 we denote by C all constants whose precise values do not matter. The Yamada condition implies that ε ε 2 E σ0(X0(s)) − σ0(X0(0)) ds ≤ E C |X0(s) − X0(0)| ds 0 0 ε s ε s ≤ E A |b0(X0(t))| dt ds + A σ0(X0(t)) dW (t) ds 0 0 0 0 1 ε s 2 2 2 ≤ C · ε + A E |σ0(X0(t))| dW (t) ds 0 0 1 ε s 2 2 2 2 3/2 ≤ C · ε + A E [σ0(X0(t))] dt ds ≤ C · ε + ε . 0 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 146 YU. S. MISHURA AND V. V. TOMASHYK Thus t − ≥ ≤ C · 2 3/2 P sup [σ0(X0(s)) σ0(r)] dW (s) δ1 2 ε + ε . 0≤t≤ε 0 δ1 Using the formula for the distribution of the supremum of a Wiener process (see [7]) we get 2 ∞ x2 P sup [σ (r) · W (t)] <δ+ δ + K · ε ≤ 1 − √ exp − dx, 0 1 · 0≤t≤ε 2π δ+δ1√+K ε 2 a· ε since σ0(r) >a. 5 Now we specify the constants δ and δ1.Putδ1 = ε 8 and δ = δ(ε)=ε.Then C 3 1 · 2 3/2 · 4 4 2 ε + ε = C ε + ε , δ1 √ √ δ + δ1 + K · ε 1 1 √ = · ε + ε 8 + K · ε . a · ε a Hence there exist numbers h(ε) > 0 such that h(ε) → 0asε → 0and P sup X0(t) Thus [−∞,r] − [−∞,r] P τn,T τ0,T >ε ⎛ ⎞ ≤ ⎝ | − | [−∞,r] ⎠ P sup Xn(t) X0(t) >δ,X0 τ0,T = r + h(ε). ≤ ≤ [−∞,r] 0 t τ0,T +ε By Theorem 2, ⎛ ⎞ ⎝ | − | [−∞,r] ⎠ P sup Xn(t) X0(t) >δ,X0 τ0,T = r −∞ 0≤t≤τ [ ,r]+ε 0,T n→∞ ≤ P sup |Xn(t) − X0(t)| >δ −−−−→ 0, 0≤t≤T +ε [−∞,r] ≤ since τ0,T T almost surely. Therefore −∞ −∞ lim P τ [ ,r] − τ [ ,r] >ε ≤ h(ε). n→∞ n,T 0,T Given γ>0chooseε > 0 such that 0 −∞ −∞ lim P τ [ ,r] − τ [ ,r] >ε ≤ γ n→∞ n,T 0,T for 0 <ε<ε0. It is obvious that −∞ −∞ lim P τ [ ,r] − τ [ ,r] >ε ≤ γ n→∞ n,T 0,T for ε>ε .Sinceγ>0 is arbitrary, we prove that 0 −∞ −∞ lim P τ [ ,r] − τ [ ,r] >ε =0. n→∞ n,T 0,T Similarly to the above result, one can prove that [−∞,r] − [−∞,r] −→ →∞ P τ0,T τn,T >ε 0,n . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use CONVERGENCE OF EXIT TIMES 147 Hence [−∞,r] − [−∞,r] −→ →∞ P τn,T τ0,T >ε 0,n . The proof of Theorem 3 is complete. Remark 2. It is not complicated to obtain a result similar to Theorem 3 for exit times [l,∞] ≥ τn,T , n 0. [−∞,r] [−∞,r] Remark 3. Since all the exit times τn,T and τ0,T with n>0 are bounded from above by the nonrandom time T , the Lebesgue dominated convergence theorem and [−∞,r] [−∞,r] Theorem 3 imply the convergence of exit times τn,T to τ0,T in the space Lp, p>1. The following result is a corollary of Theorem 3 and Remark 2. Theorem 4. Let stochastic processes Xn be defined by equation (5). Assume that con- ditions (Y 1n), (Y 3n), (Y 5n),and(Y 6n) are satisfied. If convergence (6) holds and there exists a constant a>0 such that |σn(x)| >afor x ∈ R and n ≥ 0, then the exit times from the strip [l, r] converge in the sense that ∀ [l,r] − [l,r] −→ →∞ ε>0: P τn,T τ0,T >ε 0,n . 4. Convergence of exit times of a diffusion process from a strip on the whole real line All processes and corresponding exit times have been considered in finite intervals [0,T], 0 Lemma 2. Let conditions (Y 1n)–(Y 4n) hold for the processes Xn(t). (a) If 0 y − 2bn(z) ∞ exp 2 dz dy = , −∞ 0 σn(z) then (9) P sup Xn(t)=+∞ =1,n≥ 0. t>0 (b) If ∞ y − 2bn(z) ∞ exp 2 dz dy = , 0 0 σn(z) then (10) P inf Xn(t)=−∞ =1,n≥ 0. t>0 [l,r] In other words, if at least one of the assumptions of Lemma 2 holds, then τn < ∞ almost surely for all n ≥ 0[9]. The following result is a generalization of Theorem 4 to the case of an infinite interval. In Theorem 5, we establish the convergence in probability of exit times from a strip and the convergence in probability of exit times from an infinite strip in the infinite interval. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 148 YU. S. MISHURA AND V. V. TOMASHYK Theorem 5. Let conditions (Y 1n), (Y 3n), (Y 5n),and(Y 6n) hold for the processes Xn, n ≥ 0.Then (a) if at least one of the assumptions of Lemma 2 holds for the process X0, then the [l,r] exit times τn , n ≥ 0, converge in the following sense: ∀ [l,r] − [l,r] −→ →∞ (11) ε>0: P τn τ0 >ε 0,n , (b) if the assumption of statement (a) of Lemma 2 holds for the process X0, then [−∞,r] the exit times τn , n ≥ 0, converge in the following sense: ∀ [−∞,r] − [−∞,r] −→ →∞ (12) ε>0: P τn τ0 >ε 0,n , [l,∞] (c) if the assumption of statement (b) of Lemma 2 holds, then the exit times τn , n ≥ 0, converge in the following sense: ∀ [l,∞] − [l,∞] −→ →∞ (13) ε>0: P τn τ0 >ε 0,n . Proof. We show the proof of statement (a) only, since the proofs of statements (b) and (c) are similar. ∈ [l,r] ∞ Choose some constants ε (0, 1) and δ>0. Lemma 2 implies that τ0 < almost surely, whence we conclude that δ ∃T>0: P τ [l,r] >T− 1 < . 0 2 By Theorem 4, there exists a number n0 such that δ P τ [l,r] − τ [l,r] <ε > 1 − n,T 0,T 2 for all n ≥ n0. Therefore [l,r] − [l,r] [l,r] − − (14) P τn,T τ0,T <ε,τ0 Now we estimate from above the probability on the left hand side of inequality (14). In − [l,r] [l,r] the interval [0,T 1), the exit time τ0,T coincides almost surely with the exit time τ0 . Hence [l,r] − [l,r] [l,r] − [l,r] − [l,r] [l,r] − P τn,T τ0,T <ε,τ0 [l,r] [l,r] Similarly, the exit time τn,T coincides almost surely with the exit time τn in the interval [0,T − 1+ε). Thus [l,r] − [l,r] [l,r] − [l,r] − [l,r] [l,r] − P τn,T τ0 <ε,τ0 Now we derive from (14) the following bound: − [l,r] − [l,r] [l,r] − ≤ [l,r] − [l,r] 1 δ for all n ≥ n0, which completes the proof of Theorem 5. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use CONVERGENCE OF EXIT TIMES 149 5. Concluding remarks We studied the problem of convergence of exit times from a strip for solutions of sto- chastic differential equations with non-Lipshitzian diffusion and nonrandom coefficients. The uniform convergence in probability of pre-limit processes was also obtained. For time homogeneous stochastic processes, the exit times from a strip related to pre-limit pro- cesses converged in probability to the exit time from the same strip for the limit process. The results were also obtained for the case of an infinite time and for the convergence of exit times from an infinite strip. Bibliography 1. I. I. Gihman and A. V. Skorohod, Controlled Stochastic Processes, “Naukova dumka”, Kiev, 1977; English transl., Springer-Verlag, Berlin, 1979. MR544839 (80h:60081) 2. E. B. Dynkin and A. A. Yushkevich, Markov Processes: Theorems and Problems, “Nauka”, Moscow, 1967; English transl., Plenum Press, New York, 1969. MR0242252 (39:3585a) 3. D. A. Darling, T. Ligget, and H. M. Taylor, Optimal stopping for partial sums, Ann. Math. Statist. 43 (1972), 1363–1368. MR0312564 (47:1121) 4. S. Watanabe and N. Ikeda, Stochastic Differential Equations and Diffusion Processes,North Holland Publ. Co., Amsterdam–Oxford–New York, 1981. MR637061 (84b:60080) 5. T. Yamada, Sur l’approximation des solutions d’equations differentielles stochastiques,Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (1976), 153–164. MR0413269 (54:1386) 6. Yu. S. Mishura, S. V. Posashkova, and G. M. Shevchenko, Properties of solutions of stochas- tic differential equations with nonhomogeneous coefficients and non-Lipschitz diffusion,Teor. Imovir. Mat. Stat. 79 (2008), 105–113; English transl. in Theory Probab. Math. Statist. 79 (2009), 117–126. MR2494541 (2010d:60136) 7. I. I. Gihman and A. V. Skorohod, Introduction to the Theory of Random Process, “Nauka”, Moscow, 1965; English transl., W. B. Saunders Company, Philadelphia, 1969. MR0247660 (40:923) 8. I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations, “Naukova dumka”, Kiev, 1968; English transl., Springer-Verlag, New York, 1972. MR0346904 (49:11625) 9. K. ItˆoandH.P.McKean,Diffusion Processes and Their Sample Paths, Springer, Berlin, 1965. MR0199891 Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine E-mail address: [email protected] Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine E-mail address: [email protected] Received 01/NOV/2012 Translated by N. 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