Show that (2.3) and (2.4) are equivalent. Let be a martingale. Show that for all n > m. Let are independent random variables with zero mean and variance a) Show that is a martingale (w.r.t....

Show that (2.3) and (2.4) are equivalent.

Let

be a martingale. Show that

for all n > m.

Let

are independent random variables with zero mean and variance

a) Show that

is a martingale (w.r.t. the history generated by the process itself).

b) Use (2.7) to find the predictable variation process

c) Use (2.8) to find the optional variation process