Scaling limits for the discretization error of a Brownian stochastic integral
Pub. online: 4 November 2026
Type: Research Article
Open Access
Open Access
Received
15 June 2026
15 June 2026
Accepted
19 October 2026
19 October 2026
Published
4 November 2026
4 November 2026
Abstract
In this note, we derive a large-deviation-type scaling limit for the discretization error of a Brownian stochastic integral. Somewhat surprisingly, and to the best of our knowledge, this result has not been previously obtained in the literature.
References
Bertsimas, D., Kogan, L., Lo, A.W.: When is time continuous? J. Financ. Econ. 55(2), 173–204 (2000). MR1886691. https://doi.org/10.1142/9789812810663_0003
Billingsley, P.: Convergence of Probability Measures, 2nd ed. Wiley, New York (1999). MR1700749. https://doi.org/10.1002/9780470316962
Cai, J., Fukasawa, M., Rosenbaum, M., Tankov, P.: Optimal discretization of hedging strategies with directional views. SIAM J. Financ. Math. 7(1), 34–69 (2016). MR3461312. https://doi.org/10.1137/151004306
Cohen, A., Dolinsky, Y.: A scaling limit for utility indifference prices in the discretised Bachelier model. Finance Stoch. 26(2), 335–358 (2022). MR4397933. https://doi.org/10.1007/s00780-022-00473-y
Dolinsky, Y., Zhang, X.: Scaling limits for exponential hedging in the Brownian framework. SIAM J. Control Optim. 64(2), 748–762 (2026). MR5054594. https://doi.org/10.1137/25M1737043
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley (1986). MR0838085. https://doi.org/10.1002/9780470316658
Fukasawa, M.: Discretization error of stochastic integrals. Ann. Appl. Probab. 21(4), 1436–1465 (2011). MR2857453. https://doi.org/10.1214/10-AAP730
Fukasawa, M.: Efficient discretization of stochastic integrals. Finance Stoch. 18(1), 175–208 (2014). MR3146491. https://doi.org/10.1007/s00780-013-0215-6
Geiss, S.: Weighted BMO and discrete time hedging within the Black–Scholes model. Probab. Theory Relat. Fields 132(1), 13–38 (2005). MR2136865. https://doi.org/10.1007/s00440-004-0389-0
Geiss, C., Geiss, S.: On an approximation problem for stochastic integrals where random time nets do not help. Stoch. Process. Appl. 116(3), 407–422 (2006). MR2199556. https://doi.org/10.1016/j.spa.2005.10.002
Geiss, S., Nguyen, T.: On Riemann–Liouville type operators, bounded mean oscillation, gradient estimates and approximation on the Wiener space. Stoch. Process. Appl. 187, 104651 (2025). MR4894808. https://doi.org/10.1016/j.spa.2025.104651
Geiss, S., Toivola, A.: Weak convergence of error processes in discretizations of stochastic integrals and Besov spaces. Bernoulli 15(4), 925–954 (2009). MR2597578. https://doi.org/10.3150/09-BEJ197
Geiss, S., Toivola, A.: On fractional smoothness and ${L_{p}}$-approximation on the Gaussian space. Ann. Probab. 43(2), 605–638 (2015). MR3306001. https://doi.org/10.1214/13-AOP884
Gobet, E., Makhlouf, A.: The tracking error rate of the delta-gamma hedging strategy. Math. Finance 22(2), 277–309 (2012). MR2897386. https://doi.org/10.1111/j.1467-9965.2010.00466.x
Gobet, E., Temam, E.: Discrete time hedging errors for options with irregular payoffs. Finance Stoch. 5(3), 357–367 (2001). MR1849426. https://doi.org/10.1007/PL00013539
Hayashi, T., Mykland, P.A.: Evaluating hedging errors: an asymptotic approach. Math. Finance 15(2), 309–343 (2005). MR2132193. https://doi.org/10.1111/j.0960-1627.2005.00221.x
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd ed. Springer (1991). MR1121940. https://doi.org/10.1007/978-1-4612-0949-2
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd ed. Springer (1999). MR1725357. https://doi.org/10.1007/978-3-662-06400-9
