Purely pathwise probability-free Ito integral

Abstract

This paper gives several simple constructions of the pathwise Itô integral t 0 φ dω for an integrand φ and a price path ω as integrator, with φ and ω satisfying various topological and analytical conditions. The definitions are purely pathwise in that neither φ nor ω are assumed to be paths of stochastic processes, and the Itô integral exists almost surely in a non-probabilistic financial sense. For example, one of the results shows the existence of t 0 φ dω for a càdlàg integrand φ and a càdlàg integrator ω with jumps bounded in a predictable manner.

One-dimensional game-theoretic differential equations

“…In the spirit of stochastic Itô integration, it is possible to develop a game-theoretic integration theory for typical paths, see [PP16,Vov16,LPP18] and also [SV19,Chapter 14]. Like the classical Itô integration ([RY99, Chapter IV]), the game-theoretic integration requires as a fundamental ingredient the existence of quadratic variation for typical paths.…”

“…A more general integration theory for typical paths was developed in [PP16], [Vov16] and [ LPP18] providing, e.g., more sophisticated continuity estimates for the model-free Itô integral and integration for not necessarily continuous integrands, and not necessarily continuous typical paths as integrators.…”

“…for instance [KS88,RY99]. Game-theoretic ideas coming in particular from mathematical finance, indeed, allowed to set up a rather mature game-theoretic (or model-free) Itô integration for typical paths, see [PP16,Vov16,LPP18]. Since the stochastic Itô integration was originally motivated by stochastic differential equations, it rose the natural question whether the game-theoretic Itô integration is sufficiently powerful to treat differential equations driven by typical paths.…”

On the quadratic variation of the model-free price paths with jumps

“…Assume that lim n→+∞ d ∞ H n , S i + S j = 0 and lim n→+∞ d ∞ J n , S i − S j = 0 then for any ǫ ∈ (0, 1), (13) follows immediately from (12) and Theorem 5 applied to F t = 0, 0, . .…”

Stochastic integration and differential equations for typical paths

“…Using their outer measure they constructed a model-free stochastic integral which is continuous for typical price paths and connected their typical paths with rough paths by demonstrating that every typical price path possess an Itô rough path. Moreover, [36,37] provide several additional constructions of model-free stochastic integrals for typical paths. In [29,30] Itô calculus with respect to the so-called G-Brownian motion as integrator has been developed, which is motivated from financial mathematics when investigating pricing and portfolio optimization problems under volatility uncertainty.…”