I am trying To establish tightness for a sequence of approximations. The limiting process resides in the skorokhod space. I am using the euler maruyama scheme to approximate the sde ie.
S(ti+1)=S{ti}+μ△S{ti}+σ√(△)z{i+1}S(ti)
I need to establish the following condition:
Condition 1) Lim_{N→∞}sup_{n}P{sup_{t≤T}|Xⁿ(t)|≥N}=0
(The euler scheme is piecwise constant and therefore cadlag my mistake)
If anyone knows of any papers with formal proofs for weak convergence in the skorokhod space let me know, i have been unable to find any for weak convergence of simulated option functionals when no analytic solution exists.