Solve the Following Steps Properly: In what follows we let T > 0 and we consider a

Solve the Following Steps Properly:

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In what follows we let T > 0 and we consider a filtration (Ft)te[0,7] on [0,T] with Fo = {0,2} and a probability measure P on (N, FT). An (Ft) te[0,7]-adapted process (Mt) te[0,7] is called a (true) martingale on [0,7] if Text i) E[M] <∞ for all t= [0, T], ii) E[Mt | Fs] = Ms, for all 0 < s <t. An (Ft) te[0,T]-adapted process (Mt)te [0,7] is called a supermartingale on [0,7] if i) E[M] <∞ for all t= [0, T], ii) E[Mt | Fs] ≤ Ms, for all 0 < s <t. An (Ft) te[0,7]-adapted process (Mt)te[0,1] is called a local martingale on [0, T] if there exists a nondecreasing sequence (Tn)n21 of [0, T]-valued stopping times such that i) limn→∞ Tn = T almost surely, ii) for all n ≥ 1 the stopped process (Mt) te [0, 7] is a (true) martingale under P. A local martingale on [0, 7] which is not a true martingale is called a strict local martingale. In what follows we let T > 0 and we consider a filtration (Ft)te[0,7] on [0,T] with Fo = {0,2} and a probability measure P on (N, FT). An (Ft) te[0,7]-adapted process (Mt) te[0,7] is called a (true) martingale on [0,7] if Text i) E[M] <∞ for all t= [0, T], ii) E[Mt | Fs] = Ms, for all 0 < s <t. An (Ft) te[0,T]-adapted process (Mt)te [0,7] is called a supermartingale on [0,7] if i) E[M] <∞ for all t= [0, T], ii) E[Mt | Fs] ≤ Ms, for all 0 < s <t. An (Ft) te[0,7]-adapted process (Mt)te[0,1] is called a local martingale on [0, T] if there exists a nondecreasing sequence (Tn)n21 of [0, T]-valued stopping times such that i) limn→∞ Tn = T almost surely, ii) for all n ≥ 1 the stopped process (Mt) te [0, 7] is a (true) martingale under P. A local martingale on [0, 7] which is not a true martingale is called a strict local martingale.