From Increment Moments to Regular Paths
A process is specified by its random variables at fixed times, but many questions concern its paths: can we choose a version whose sample paths are continuous, or even Hölder continuous? Moment estimates for increments provide a way to answer these questions. The key is to control many short increments at once, and then use a summable probability bound to obtain a pathwise estimate.
Work on a probability space \((\Omega,\mathcal{F},\mathbb{P})\), and let \((X_t)_{0\le t\le T}\) be a real-valued process, where \(T>0\). A process \((Y_t)_{0\le t\le T}\) is a modification of \(X\) if \(\mathbb{P}(Y_t=X_t)=1\) for each fixed \(t\). The exceptional null set is allowed to depend on \(t\). For \(0<\gamma\le1\), a function \(f:[0,T]\to\mathbb{R}\) is \(\gamma\)-Hölder continuous if there is a finite constant \(K\) such that \(|f(t)-f(s)|\le K|t-s|^\gamma\) for all \(s,t\in[0,T]\).
The strict inequality \(\gamma<\beta/p\) matters: the argument gives summable bounds only below this threshold. Here is a proof, including the construction of the modification.
Proof. Fix \(0<\gamma<\beta/p\). At level \(n\), use the grid points \(kT2^{-n}\), \(k=0,\ldots,2^n\), and define \(A_n\) to be the largest absolute increment between adjacent grid points.
For each of the \(2^n\) adjacent pairs, Markov's inequality and the assumed moment bound give $$ \mathbb{P}\left(|X_{(k+1)T2^{-n}}-X_{kT2^{-n}}|>2^{-n\gamma}\right) \le C(T2^{-n})^{1+\beta}2^{np\gamma}. $$ Taking a union bound over the pairs gives $$ \mathbb{P}(A_n>2^{-n\gamma}) \le CT^{1+\beta}2^{-n(\beta-p\gamma)}. $$ Since \(\beta-p\gamma>0\), the sum over \(n\ge0\) of the right-hand side is a convergent geometric series. The Borel–Cantelli lemma therefore implies that, almost surely, \(A_n\le2^{-n\gamma}\) for all sufficiently large \(n\). In particular, $$ K(\omega)=\sup_{n\ge0}2^{n\gamma}A_n(\omega)<\infty $$ almost surely: the terms are eventually at most \(1\), and there are only finitely many earlier terms.
Let \(D=\bigcup_{n\ge0}\{kT2^{-n}:0\le k\le2^n\}\), the dense set of dyadic grid points. On the probability-one event where \(K<\infty\), take distinct \(u,v\in D\), and choose \(n\ge0\) so that $$ T2^{-(n+1)}<|u-v|\le T2^{-n}. $$ At level \(n\), the grid points immediately to the left of \(u\) and \(v\) are either the same or adjacent. Their process values differ by at most \(A_n\). To move from either point to \(u\) or \(v\), refine the grid one level at a time. At each refinement the grid point either stays fixed or moves by one adjacent increment, bounded by \(A_m\) at level \(m\). Because \(u,v\) are dyadic, these refinement sums have only finitely many nonzero terms. Thus $$ |X_u-X_v|\le A_n+2\sum_{m=n+1}^{\infty}A_m \le K2^{-n\gamma}\left(1+2\sum_{r=1}^{\infty}2^{-r\gamma}\right). $$ The series equals \(2^{-\gamma}/(1-2^{-\gamma})\), so its displayed coefficient is finite. Also \(2^{-n}<2|u-v|/T\). Consequently there is a finite random constant \(L\) such that $$ |X_u-X_v|\le L|u-v|^\gamma\qquad(u,v\in D). $$ This gives a uniformly continuous function on \(D\), which has a unique continuous extension to \([0,T]\). The same inequality passes to the extension by taking limits of dyadic points, so the extension is \(\gamma\)-Hölder continuous.
Call the extension \(Y\) on the probability-one event just constructed, and define \(Y_t=0\) for every \(t\) on its null complement. For fixed \(t\), choose dyadic \(d_n\to t\). The moment assumption gives, for every \(\varepsilon>0\), $$ \mathbb{P}(|X_{d_n}-X_t|>\varepsilon) \le C\varepsilon^{-p}|d_n-t|^{1+\beta}\longrightarrow0. $$ Thus \(X_{d_n}\to X_t\) in probability. By construction \(Y_{d_n}=X_{d_n}\) almost surely, while continuity gives \(Y_{d_n}\to Y_t\) almost surely and hence in probability. For any \(\varepsilon>0\), $$ \mathbb{P}(|Y_t-X_t|>\varepsilon) \le \mathbb{P}(|Y_{d_n}-Y_t|>\varepsilon/2) +\mathbb{P}(|X_{d_n}-X_t|>\varepsilon/2). $$ Both terms tend to zero. Therefore \(\mathbb{P}(Y_t=X_t)=1\). This holds for each fixed \(t\), which is precisely the modification property. \(\square\)
Applying the Criterion to Brownian Increments
For a Brownian process \((B_t)_{t\ge0}\), an increment \(B_t-B_s\), for \(s<t\), has a centered normal distribution with variance \(t-s\). For every positive integer \(m\), the \(2m\)-th moment of a centered normal variable with variance \(v\) is \((2m-1)!!v^m\), where \((2m-1)!!=1\cdot3\cdot5\cdots(2m-1)\). Hence $$ \mathbb{E}|B_t-B_s|^{2m}=(2m-1)!!\,|t-s|^m. $$ On a compact interval, this has the criterion’s form with \(p=2m\), \(1+\beta=m\), and \(\beta=m-1\), provided \(m>1\). It yields every Hölder order \(\gamma<(m-1)/(2m)=1/2-1/(2m)\). Increasing \(m\) brings this threshold arbitrarily close to \(1/2\).
Worked Example: The Fourth-Moment Bound
Take \(m=2\). A centered normal variable with variance \(v\) has fourth moment \(3v^2\), so for \(0\le s,t\le T\), $$ \mathbb{E}|B_t-B_s|^4=3|t-s|^2. $$ Here \(p=4\), \(1+\beta=2\), and therefore \(\beta=1\). The criterion gives a continuous modification whose paths are almost surely \(\gamma\)-Hölder for every \(\gamma<1/4\). In particular, choosing \(\gamma=1/8\) satisfies the strict condition \(1/8<1/4\).
Check: Substituting \(p=4\) and \(\beta=1\) into \(\beta/p\) gives \(1/4\), the claimed threshold. This single moment bound does not establish Hölder exponents between \(1/4\) and \(1/2\); higher moments are needed for those.
Worked Example: Obtaining Any Order Below One Half
Let \(0<\alpha<1/2\) be fixed. Choose an integer \(m>1\) large enough that \(\alpha<1/2-1/(2m)\); such an integer exists because \(1/(2m)\) tends to zero. For Brownian increments the \(2m\)-th moment bound has \(C=(2m-1)!!\), \(p=2m\), and \(\beta=m-1\). Thus the criterion applies with $$ \frac{\beta}{p}=\frac{m-1}{2m}=\frac12-\frac{1}{2m}>\alpha. $$ It follows that the continuous modification is almost surely \(\alpha\)-Hölder on \([0,T]\).
This statement can be made simultaneously for every \(\alpha<1/2\), not just one fixed \(\alpha\). First apply the result to a countable sequence of positive rational exponents increasing to \(1/2\), and intersect the corresponding probability-one events. The continuous modifications obtained for different exponents agree almost surely: at each rational time they both equal \(B_t\) almost surely; intersecting over the countably many rational times and using continuity makes them agree everywhere on \([0,T]\). The common version is Hölder for each exponent in the sequence. If \(\alpha\) is smaller than one of those exponents \(\gamma\), then \(|t-s|^\gamma\le T^{\gamma-\alpha}|t-s|^\alpha\), so it is also \(\alpha\)-Hölder.
Check: Each chosen \(m\) satisfies the strict inequality required by the criterion. The conclusion does not assert a Hölder bound of order \(1/2\).
Applying this construction on each interval \([0,M]\), for positive integers \(M\), gives a continuous modification on all nonnegative times. To see why the versions can be joined, any two continuous versions agree almost surely at each rational time in their shared interval, since both are modifications of the same process. A countable intersection gives agreement at all rational times at once, and continuity then gives agreement throughout the interval. The versions therefore patch together. The same countable-event argument for rational Hölder exponents shows that this global continuous version is almost surely Hölder of every order below \(1/2\) on each compact time interval.
Why the Strict Moment Condition Matters
The criterion requires an increment moment exponent \(1+\beta\) strictly greater than \(1\). The next example shows that a bound with exponent exactly \(1\) does not guarantee a continuous modification.
Worked Example: A Jump Process at the Endpoint
Let \(\Omega=[0,1]\) with its Borel sets and uniform probability, let \(U(\omega)=\omega\), and define \(X_t=\mathbf{1}_{\{U\le t\}}\) for \(0\le t\le1\). For \(0\le s<t\le1\), the increment has absolute value \(1\) exactly when \(U\in(s,t]\), an event of probability \(t-s\); otherwise it is zero. Consequently, for every \(p>0\), $$ \mathbb{E}|X_t-X_s|^p=t-s=|t-s|. $$ This has exponent \(1\), not \(1+\beta\) for any \(\beta>0\).
There cannot be a continuous modification. If \(Y\) were one, then at each rational \(q\in[0,1]\) we would have \(Y_q=X_q\) almost surely. Since there are countably many such \(q\), with probability one these equalities hold simultaneously for all rational \(q\). On that event, for any outcome with \(0<U<1\), rational times below \(U\) have value \(0\) and rational times above \(U\) have value \(1\). Such values arbitrarily close to \(U\) from opposite sides contradict continuity at \(U\). The event \(0<U<1\) has probability \(1\), so no continuous modification exists.
Check: The process has the endpoint moment bound for every positive \(p\), but the criterion cannot be applied because there is no positive \(\beta\). The example shows why that hypothesis cannot simply be removed.
Common pitfall. For Brownian motion the second moment is \(\mathbb{E}|B_t-B_s|^2=|t-s|\). It is incorrect to treat this as a bound with \(1+\beta=2\), or to infer a positive Hölder exponent from it: the actual exponent is \(1\), so \(\beta=0\), outside the theorem’s assumptions. Instead use the fourth moment, \(\mathbb{E}|B_t-B_s|^4=3|t-s|^2\). This gives \(\beta=1\) and the valid conclusion \(\gamma<1/4\). Higher even moments then extend the conclusion to every order below \(1/2\).
The criterion is a sufficient condition, not a necessary one: a process may have regular paths even when a particular moment estimate is too weak to prove it. Its practical value is that a uniform increment bound converts into a pathwise statement without assuming path continuity in advance. The modification step is essential: the theorem constructs a process agreeing almost surely at each fixed time, with improved paths, rather than asserting that every initially chosen version already has those paths.
Check Your Understanding
Use the criterion’s hypotheses carefully when answering.
- If \(\mathbb{E}|X_t-X_s|^6\le C|t-s|^4\), what are \(p\), \(\beta\), and the resulting range of Hölder exponents?
- Why does the dyadic probability bound have a summable series when \(\gamma<\beta/p\)?
- For Brownian motion, what Hölder exponents follow from the eighth-moment bound?
- Why does proving \(Y_t=X_t\) almost surely for each fixed \(t\) not itself prove that the original process \(X\) has continuous paths?
- For the jump process \(X_t=\mathbf{1}_{\{U\le t\}}\), compute \(\mathbb{E}|X_t-X_s|^p\) when \(s<t\), and explain why the criterion does not apply.