By Thomas Craig

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**Extra info for A Treatise on Linear Differential Equations, Volume I: Equations with Uniform Coefficients**

**Example text**

1), we still need some lemmas. 4 (Yan (1980)) Let M e ,Ai and H be a local ^-predictable process. Then H e £(M). 5 Let M e M, l ^ p < o o , a > 0 . I f G e 'lV’j^(a), then F(X,Xd) € £(M) forallXeX. Proof. L e t X e X. Define stopping times Tk = infit: IX(t)l>k or IXd(t)|^k, t^O}, Then IIX^*'" 11^ ^ k and ||(Xd)^k- | | ^ ^ k. k = 1,2,.... Using the assumptions, we have | | ( F ( X , X d ) - F ( 0 , 0 ))ll],o,T^ll II = II ( F(xTk-, (Xd)Tk-) _ F(0 , 0 ) ) 1 „ 0, < II ( F(XTk-, (Xd)Tk-) _ F(0 , 0 ) ) II ^ )] II ^ < a d l X ^ k - 11^ + ||(Xd)Tk- 11^ ) < 2 ak.

5. 1 Let 1 < i < m. s. Assume (1) for any X, y G R", g i ( . s. 1). 1. right continuous adapted process. 1). s. 1. Let {a^} be a sequence of positive real numbers and {S^} a nondecreasing sequence of stopping times. Assume (1) for any X, y e R", g i ( . s. , s, X(s-), Xd(s-)) dMi(s). s. s. on {(0 : Tk(X) S ^un Sk }. 1. s. s.. 45 (4) ( E sup |A(t-d(t))P ^ ten / tGD < 00 for some l

Such that Tj^ > Tj^_| whenever Tj^_| < c» and I _ mV i II ^ ^ b for all 1 ^ i ^ m and 1 ^ k ^ 1 . 21) Proof, Since Mj G W“ , there exists a decomposition M| = N| + Aj of Mj with Nj a local martingale and Aj a process of finite variation such that joo(Nj, Aj) < oo. , m }, k > 1 . It is clear that whenever Tj^_j < oo for all k > 1. s. Otherwise, we have P(Tk < 00 ) > 0 f or a l l k= 1 , 2 ...... j) S ( P + Z J I) (®) Z Z [ [ N i,N i] T . _j ) V ( J IdAi(s)l)^]) (co) > ( k b V 4 ) ^ / ^ = -T-yic. That produces m ~ m Ell > Z jJ N i,A i) = i=l i=l t Ni .