## Doctoral Theses

### Pathwise Stochastic Calculus of Continuous Semimartingales.

2-28-1981

2-28-1982

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Rao, B. V. (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

Stochastic integration with respect to Brownian motion was introduced by Ito. Stochastic integration with respect to martingales (and seminartingales) was developed by Kunita-Watanable [24 Fisk [9), Courrege D] and Meyer [33]. In this thesis, we study the path wise stochastic calculus restricting ourselves to continuous semimartingales. Here is a brief summary of our results.In Chapter I, we obtain a pathwise formula for the quadratic variation process < M > of a continuous local martingale M. Recall theat < M > is the natural increasing process in the Doob-Meyer decomposition of M. By a part wice formula for M> we mean a formula describing cxplicitly a w-path of M> in terms of the corresponding w- path of M.. Observe that existence of such a formula already implies that M> depends neither on the underl, ing probability nor on the underlying filtration. The proof of our formula for M> is simple and does not aseume the existence of M>, thus providing a simple proof of the existence as well of M>. Proceeding as in Kunita-Watanable [24, we also deduce that is the only continuous increasing process M> A such thet M2 is a local martingale. We also give an elementary proof of the lesser known fact that almost every path of a continuous local martingale has the property : On any interval, it is either a constant or of unbounded variation. This result also provides a proof of uniqueness of M.If M is a continuous local martingale such that aM> (t) Ä‘t, then intogral vith respect to can be defined as in the Brownian motion case for all progressively measurable integrands. For a general continuous local martingale M, we show that by applying a random time change we can reduce it to the previous case. Once stochastic integration with respect to a continuous local martingale is done, extensions to cover continuous semi martingale integrates and to allow vector or matrix valued integrals and s are immediate. Then using random time change; and Doobs maximal inequality, we obtain an estimate on the throw of stochastic integrals. Then, we obtain a path wise formula; for the stochastic integral of a r.c.1.1. process by showing that the Rienann sums calculated for appropriate randon partitions do indeed converge to the stochastic integral. Then we shall prove a path wise version of the well known It's formula. The proof highlights the path wise nature of the stochastic integrals.

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