A change of variable formula with Itô correction term

Abstract

We consider the solution u(x, t) to a stochastic heat equation. For fixed x, the process F (t) = u(x, t) has a nontrivial quartic variation. It follows that F is not a semimartingale, so a stochastic integral with respect to F cannot be defined in the classical Itô sense. We show that for sufficiently differentiable functions g(x, t), a stochastic integral g(F (t), t) dF (t) exists as a limit of discrete, midpoint-style Riemann sums, where the limit is taken in distribution in the Skorokhod space of cadlag functions. Moreover, we show that this integral satisfies a change of variable formula with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent of F . . This reprint differs from the original in pagination and typographic detail. 1 2 K. BURDZY AND J. SWANSON 0, whereẆ is a space-time white noise on R × [0, ∞). That is, u(x, t) = R×[0,t]

A change of variable formula for the 2D fractional Brownian motion of Hurst index bigger or equal to 1/4

“…In [4], Coutin and Qian have shown that the rough paths theory of Lyons [13] can be applied to the 2D fractional Brownian motion B = (B (1) , B (2) ) under the condition that its Hurst parameter H (supposed to be the same for the two components) is strictly bigger than 1/4. Since this seminal work, several authors have recovered this fact by using different routes (see e.g.…”

“…We refer to [11] for an exhaustive study of this notion. Now, let us introduce the following object: (1) k/n , B (2) k/n ) + ∂f ∂x (B (1) (k+1)/n , B (2) k/n ) 2 B (1) (k+1)/n − B (1) k/n + nt −1 k=0 ∂f ∂y (B (1) k/n , B (2) k/n ) + ∂f ∂y (B (1) k/n , B (2) (k+1)/n ) 2 B (2) (k+1)/n − B (2) k/n . (1.1) If I n (t) defined by (1.1) does not converge in probability but converges stably, we denote the limit by…”

“…Our main result is as follows: Let also B = (B (1) , B (2) ) denote a 2D fractional Brownian motion of Hurst index H ∈ (0, 1), and t > 0 be a fixed time. (1) t B (2) t using the integral defined in Definition 1.1.…”

On Simpson’s Rule and Fractional Brownian Motion with $$H = 1/10$$ H = 1 / 10

“…The 'Midpoint' sum, ⌊ can be shown to converge in probability for fBm with H > 1/4 (see [11]). The end point case H = 1/4 was considered in papers by Burdzy and Swanson [1], and Nourdin and Réveillac [7]. These papers proved the change-of-variable formula…”

“…For some T > 0, let B = {B H t , 0 ≤ t ≤ T } be a fractional Brownian motion with Hurst parameter H. That is, B is a centered Gaussian process with covariance R(s, t) given in (1). Let E denote the set of R-valued step functions on [0, T ].…”