Itô Calculus without Probability in Idealized Financial Markets*

Abstract

We consider idealized financial markets in which price paths of the traded securities are càdlàg functions, imposing mild restrictions on the allowed size of jumps. We prove the existence of quadratic variation for typical price paths, where the qualification "typical" means that there is a trading strategy that risks only one monetary unit and brings infinite capital if quadratic variation does not exist. This result allows one to apply numerous known results in pathwise Itô calculus to typical price paths; we give a brief overview of such results.

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

“…Integration by parts formula for model-free price paths. The existence of quadratic variation for typical ω ∈ Ω in the sense outlined in the previous section is equivalent to the existence of quadratic variation in the sense of Norvaiša [12,Proposition 3]. This also allows (see [12,Sect.…”

“…The existence of quadratic variation for typical ω ∈ Ω in the sense outlined in the previous section is equivalent to the existence of quadratic variation in the sense of Norvaiša [12,Proposition 3]. This also allows (see [12,Sect. 7]) to apply Föllmer's pathwise Itô formula [4] for typical price paths in Ω and in particular to define the pathwise integral´…”

“…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, . .…”

“…∧t,π n k ∧t (ω) 2 t∈[0,T ]converges to the quadratic variation of ω i as n → +∞ (the convergence still holds though partitions (π n ) n∈N may be finer than the Lebesgue partitions of ω, see the proofs of [6, Corollary 3.11],[12, Theorem 2]). Finally notice that ∧t,π n k ∧t (ω) = 0<s≤t,|∆ω(s)|>ε ∆ω i (s) ∆ω j (s) .…”

Purely pathwise probability-free Ito integral

“…Other papers (such as Perkowski and Prömel [17] and Davis et al [6]) extend Föllmer's results by relaxing the regularity assumptions about f , which requires inclusion of local time. All these papers assume that ω possesses quadratic variation (defined in a pathwise manner), and this assumption is satisfied when ω is a typical price path (see, e.g., [20]; the existence of quadratic variation for such ω was established in, e.g., [24] and [23]; precise definitions will be given below). The existence of local times for typical continuous price paths follows from the main result of [24] (as explained in [17], p. 13) and was explicitly demonstrated, together with its several nice properties, in [17] (Theorem 3.5).…”