On the quadratic variation process of a continuous Martingale
Rajeeva Laxman Karandikar · Illinois Journal of Mathematics · 1983
In this article we give a simple proof of the existence of the quadratic variation process of a continuous local martingale by providing an explicit expression for it.Let (l, 3) be a fixed measurable space and let 3 ((t)t>o be an increasing family of sub o--fields of 3. Let M be a continuous 3 adapted process such that M(0) 0.Let Kn(t, w) j, if there exists ti such that and,X"(t, w) X'(t-, w), and X(t, w) X"(t / U(w), w).w)} THEOREM.X is a continuous (g adapted increasing process.Further, for all P such that (M(t), cgt, P) is a local martingale, (M2(t) X(t), (4t, P) is also a local martingale.Proof.Fix a P such that M is a P-local Martingale.Let {T' > 1}, n > 1, be defined by T 0, T'+ inf{t > TT'IM(t) M(TT)I > 2-"}.