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Additional resources for Seminar on Stochastic Processes, 1987
L. (a,~J. Right continuous quasimartingales. Proposition. on each bounded interval. Assume X is a quasimartingale X is right continuous in the mean if and only if it is riqht continuous in probability. If X has a right continuous modification, then it is right continuous in the mean. The above proposition is due to Orey [15J for the scalar case. The proof for the vector case is similar.
X(n) can be decomposed uniquely (up to an evanescent set) in the form x(n)= M(n)+ A(n), where M(n) is an E-valued martingale of class (D) and A(n) ~ Evalued, predictable, cadlag process with integrable 41 variation and with A(n)=O. VI below. V. Lemma. Let Y be an E-valued, adapted process I such that Yt€LE(P) for each t. 4(0,00] and satisfies Proof. I). 2) limUs~y«s,t]XF)=O We shall prove that for s Processes with finite variation. The following theorem is new even in the scalar case. We remark that a modification of a process with finite variation does not necessarily have finite variation. 5. Theorem. Assume X has finite variation Ixi and that IXlt is integrable for every t. Then X+ (which exists everywhere) has integrable variation on bounded intervals. Moreover, X+ is a modification of X if and only if condition (R) holds. Proof. The existence of X+ and X_ follows from the inequality UXt - Xs" " Ixl t - Ixls for s-< t.
Processes with finite variation. The following theorem is new even in the scalar case. We remark that a modification of a process with finite variation does not necessarily have finite variation. 5. Theorem. Assume X has finite variation Ixi and that IXlt is integrable for every t. Then X+ (which exists everywhere) has integrable variation on bounded intervals. Moreover, X+ is a modification of X if and only if condition (R) holds. Proof. The existence of X+ and X_ follows from the inequality UXt - Xs" " Ixl t - Ixls for s-< t.