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Let $\sigma(t,\omega)$ be a progressively measurable function and $\mathbb{E}[\int_0^T \sigma_t^2\mathrm dt] < \infty$. Can we say that the Ito process $\int_0^t \sigma_s \mathrm dW_s$ is Hölder continuous? Can you please provide proof of it.

I was trying on the lines of Kolmogorov's continuity theorem, but I couldn't get correct bounds. For example,

$$\mathbb{E}\left[\left(\int_0^t\sigma_u\mathrm dW_u - \int_0^s\sigma_u\mathrm dW_u\right)^2\right] = \mathbb{E}\left[\int_s^t\sigma_u^2 \mathrm du\right]$$

So we need to bound the integral on the right hand side by $|t-s|^{1+\alpha}$. However, I think this may not be possible for general $\sigma$. Can some one please provide simple sufficient conditions?

On the larger context, I am actually interested to know if the Ito diffusions (solutions to SDE) are Holder continuous with exponents less than $1/2$.

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up vote 7 down vote accepted

Your answer depends, as you guessed on the process $u \mapsto \sigma_u$.

You can amplify your approach using the Ito-isometry with the BDG-inequality: \begin{align} \mathbb{E}[|\int_0^t\sigma_udW_u - & \int_0^s\sigma_u \,dW_u|^{2p}] \stackrel{\text{BDG}}{\leq} c(p) \mathbb{E}[(\int_s^t |\sigma_u|^2 \, du)^p] \\ & \leq c(p) \mathbb{E}[(t-s)^{p-1} \int_s^t |\sigma_u|^{2p} \, du] \quad (\text{Hölder-ineq. on} \int_s^t)\\ & = c(p) (t-s)^{p-1} \int_s^t \mathbb{E} [|\sigma_u|^{2p} ]\, du \, . \end{align} Here $(t-s)^{p-1}$ appears from the use of the Hölder inequality with the integrand $1$. Now you can think of specifying some integrability properties of $u\to \sigma_u$: bounded, uniformly in $L^{2p}$, etc. and continue with the Kolmogorov-Centsov theorem.

I think, however, as you asked for SDE, it will suffice to consider that situation: $$ dX_t = \sigma(X_t) \, dW_t $$ where $\sigma:\mathbb{R} \to \mathbb{R}$ and $W$ is a standard Brownian motion. If we assume a further condition, called the linear growth condition on $\sigma$: $$ |\sigma(x)| \leq c_1 (1+ |x|)\, , \quad x \in \mathbb{R} \, ,$$ your claim on almost Hölder $\frac{1}{2}$-regularity is true. In order to ensure existence of a global solution (i.e. for all times $t\geq 0$) this condition is very natural.

Explanation:\ You need control over $\sup_{s\leq u \leq t} \mathbb{E}[|\sigma(X_u)|^{2p}]$ in that case. Assuming this growth condition, one has the following estimate $$ \mathbb{E}[|\sigma(X_u)|^{2p}] \leq (2c_1)^{2p} (1+ \mathbb{E}[|X_u|^{2p}] ) . $$ So we need the $p$-th moment of $X_u$. We can obtain a bound on that again using BDG-inequality: \begin{align} \mathbb{E} [|X_u|^{2p}] & = \mathbb{E} [ |\int_0^u \sigma(X_v) \, dW_v |^{2p}] \\ & \leq c(p) \mathbb{E}[ (\int_0^u |\sigma(X_v)|^2 \, dv)^{p} ] \\ & \leq c(p) \mathbb{E} [u^{p-1} \int_0^u |\sigma(X_v)|^{2p} \, dv ] \quad (\text{Hölder-ineq. on} \int_0^u) \\ & \leq c(p) c_1^{2p}u^{p-1} \mathbb{E} [ \int_0^u (1+|X_v|)^{2p} \, dv ]\\ & \leq c(p) (2c_1)^{2p} u^{p-1} (u + \int_0^u \mathbb{E} [|X_v|^{2p}] \,dv ). \end{align} Now you need to apply Gronwall's lemma to obtain a bound C(c,c_1,c_0 ,p, u) on $\mathbb{E} [|X_u|^{2p}]$ with the property that $C(c,c_1,c_0,p, u) \leq C(c,c_1,c_0,p, t)$ for $u \leq t$. Then you can continue: \begin{align} \mathbb{E}[|X_t - X_s|^{2p}] & \leq c(p) (t-s)^{p-1} \int_s^t C(c,c_1,c_0,p,u) \, du \\ & \leq c(p) (t-s)^{p-1} \int_s^t C(c,c_1,c_0,p,t) \, du \\ & \leq c(p) C(c,c_1,c_0,p,t) (t-s)^p. \end{align} This allows you to apply Kolmogorov-Centsov.

This also works fine for SDEs including a drift term $+b(X_t)\, dt$.

As you ask for references: the main source of the description is an article by Dalang, however on SPDE-regularity: Theorem 13 in and also the appendix in Mytnik, Perkins and Sturm,2006.

It is not too unlikely that this kind of calculation can also be found in a textbook on stochastic analysis.

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I think a much quicker way is to apply a time change to convert the integral into a standard Brownian motion. Then, all the results on Brownian motion paths (Holder continuity, law of the iterated logarithm, etc) can be applied. – George Lowther Nov 23 '13 at 12:18

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