By Charles R. MacCluer

ISBN-10: 0486439011

ISBN-13: 9780486439013

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**Example text**

XSk-, ((xSk-)d)Sk-) ^ j,. 1). Uniqueness is obvious. We have completed the proof. Let {a^} be a sequence of positive real numbers, {Sk) a sequence of nondecreasing stopping times. Let M e X and 1 < p < «>. , 39 II ( G(XM, uM) - G(YM. v M) ) 1]] 0. Sk )| II (fP < ak(||X -Y ||^+ (3) G(0, 0) % 0, Sk )1 ^ ||U-V| all k = 1 ,2 ,.... s. j. , II ( 'F(XM, (XM)**) - (YM)d) )Sk- II ^ ^ P k ( ||X - Y ||^ + ||X d - Y d ||^ ) . 1 there. 8 Let Mj e M q>Gfi W^Xia^}, {8^}), 'F g V^({pk}, {8^}), where l < i < m , l

We are done. So far the solutions of SDES are all cadlag processes. But sometimes we need to have semimartingale solutions. 1). We denote by 9 ^ the family of n-dimensional semimartingales. Let 1 < p < 00 and M e M. Let be a sequence of positive real numbers and {S^} a nondecreasing sequence of stopping times. Denote by M ^({ak)> iS^^}) the set of all mappings F P such that (1) for any X e and stopping time T, F(X)1 jo. , l l ( F ( X W ) _ F ( Y T k ( X ) - ) ) i „ o s ^ j 11^ (3) F(0) 1)] 0, Sfc II ^ < akIIX-Yll^; for all k = 1, 2 ,....

Thus llxj^i'-11^ > jP(Qk) for all j = 1,2,... 22). s. and complete the proof of existence. We shall now prove uniqueness. M i. i=l in X. Set (2k = Tk(X) A Tk(Y) for k = 1,2,... s. мf^■. , which means X = Y. The proof of uniqueness is also complete. From the proof we have the following estimate of the solution immediately. s. (1 - a )/(m Cp Kj ), such that for all 1 < i ^ m and k > 1, and the unique solution X in X of the g equation m X . M, i=l satisfies the estimate II x “ “- II, 1 1 - a -mCpbKj 8 ^ -1 m (Kj + mCpbKi + 2 C p K i £ | | M f ^ k - l l ^ j for all k = 1 , 2 .....

### Boundary value problems and Fourier expansions by Charles R. MacCluer

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