Abstract:
Vovk and Shafer, [41], introduced game-theoretic framework for probability in
mathematical finance. This is a new trend in financial mathematics in which no
probabilistic assumptions on the space of price paths are made. The only assumption
considered is the no-arbitrage opportunity widely accepted by the financial
mathematics community. This approach rests on game theory rather than measure
theory. We deal with various properties and constructions of quadratic variation
for model-free càdlàg price paths and integrals driven by such paths. Quadratic
variation plays an important role in the analysis of price paths of financial securities
which are modelled by Brownian motion and it is sometimes used as the measure of
volatility (i.e. risk). This work considers mainly càdlàg price paths rather than just
continuous paths. It turns out that this is a natural settings for processes with jumps.
We prove the existence of partition independent quadratic variation. In addition,
following assumptions as in Revuz and Yor’s book, the existence and uniqueness of
the solutions of SDEs with Lipschitz coefficients, driven by model-free price paths
is proven.
