Weak Convergence of Stochastic Processes Whose Trajectories Have No Discontinuities of the Second Kind and the “Heuristic” Approach to the Kolmogorov-Smirnov Tests

Abstract

In 1934 A. N. Kolmogorov proved the following Theorem. I/(t) is a separable (see [1]) stochastic process, (1) Ml(tl)-(t)i < clq-t,.I+, where p > O, r > 0 and C is a constant independent o/ t, then the trajectories o/ the process are continuous with probability 1% A generalization of this theorem is the following proposition which was suggested to the author by A. N. Kolmogorov" Theorem 1. I/ (t) is a separable stochastic process, 0 < _ 1, andand C is a constant independent o/t, then the trajectories o/ the process have no discontinuities o/ the second kind with probability 1. PROOF. It is easy to show that if (t) satisfies (2), any denumerable set which is dense in the rel line may be tken as the set relative to which the process is separable, for example, the set 0 of rational binary fractions. By A+ we will mean the event [(k/2) --((k+ 1)/2n)[ < Ln, where exp [--(r log 2)/2(p + q)], L+q C. Further, we introduce the events A_tO A+, the events Dn= B, and the event D k=l It follows easily from inequality (2) hat P(} < -n(+q)2-n(l+r); P{Dn} < n/(1--), where 2 -< 1, and finally P(D} 1. I remains o show hat he set D has no trajectory with discontinuities of the second kind which is separable relative Lemma 1. I/ x(t) is a /unction separable relative to F. and x(t) A+t D n+l, thenmax ()--< +1 << () max (t)--XJ < 1" <<Poo. Since (t) is a unction which is separable relative to , it is sufficient to prove the lemma for rational binary values of . Let us assume under the induction hypothesis that ()

Differentiability of multiplicative processes related to branching random walks

“…= [a, b] a non-trivial compact subinterval of I and choose an integer 0 such that 2-(~+D/~ By the proof of Theorem 5.1 of [9] and the hypothesis (ii)(a), for all t ~ K, So, hypothesis (ii)(b) together with Lemma 2.2 applied with A 1 = AZ = A3 = At and m = mK yield a constant C1 such that for all (t, h) E K x R+ such that t + h E K Then, the Kolmogorov-Tchentov Theorem [16] yields a continuous modification t )2014~ Yt of t Yt.…”

Super-replication with nonlinear transaction costs and volatility uncertainty

“…Now observe that, when taking the Q N -expectation, the contribution from the summand for j = 1 can be dropped since it has vanishing F N n -conditional expectation due to the martingale property of M N under Q N . From (3.12) Let us next focus on Law(M N | Q N ) N =1,2,... for which we will verify Kolmogorov's tightness criterion (see [5]) on C[0, 1]. To this end, recall that M N n+j /M N n+j−1 − 1 = O(1/ √ N ) and so the quadratic variation of M N satisfies…”

Functional Central Limit Theorems for Rough Volatility

“…Not only is the solution to (1.4) continuous, but ( 1 2 − ε)-Hölder continuous for any ε ∈ (0, 1 2 ) as a consequence of the Kolmogorov-Centsov theorem [16]. Existence and precise meaning of G α Y is delicate, and is treated below.…”