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pachaquis3s

pachaquis3s

Answered question

2022-06-27

Setting We work on a filtered probability space ( Ω , F , ( F ) t [ 0 , T ] , P ). Let ( X n ) n and X be finite variation processes such that X n converges pointwise to X, i.e.
lim n X t n ( ω ) = X t ( ω )
for all ( ω , t ) Ω × [ 0 , T ], and such that ( X n ) n converges pointwise to X (where t denotes the total variation on [0,t]).
Question Does this imply that sup n X n < pointwise?

Answer & Explanation

Alisa Gilmore

Alisa Gilmore

Beginner2022-06-28Added 22 answers

Yes, this follows from the fact that lim n ( X n ) n = X < pointwise. This implies there exists N = N ( ω ) such that for all n N we have | X n ( ω ) X ( ω ) | < 1. Therefore X n ( ω ) < 1 + X ( ω ) for all n N, and hence
sup n X n ( ω ) sup n N X n ( ω ) + sup n N X n ( ω ) sup n N X n ( ω ) + 1 + X ( ω ) < .

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