Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter January 23, 2022

Solving equations with semimartingale noise

  • Jonathan Gutierrez-Pavón EMAIL logo and Carlos G. Pacheco


In this work we focus on a method for solving equations with a coefficient formally given in terms of the derivative of a continuous semimartingale. This generalizes the case of coefficients being the white noise. The idea for solving the equation is to find explicitly the inverse of the ill-posed differential operator, which boils down to finding the associated Green kernel. To find the kernel we give explicitly two homogeneous solutions in terms of the so-called Dolean–Dade exponential. The general idea to define rigorously differential operators lies on dealing with them through bilinear forms. We give several examples with explicit calculations.

MSC 2010: 60K37; 60H25

Communicated by Anatoly F. Turbin


[1] T. Brox, A one-dimensional diffusion process in a Wiener medium, Ann. Probab. 14 (1986), no. 4, 1206–1218. 10.1214/aop/1176992363Search in Google Scholar

[2] J. Gutierrez-Pavón and C. G. Pacheco, The killed Brox diffusion, preprint (2019), 10.1080/15326349.2022.2074460Search in Google Scholar

[3] J. Gutierrez-Pavón and C. G. Pacheco, Inverting weak random operators, Random Oper. Stoch. Equ. 27 (2019), no. 1, 53–63. 10.1515/rose-2019-2003Search in Google Scholar

[4] K. Kawazu and H. Tanaka, A diffusion process in a Brownian environment with drift, J. Math. Soc. Japan 49 (1997), no. 2, 189–211. 10.1142/9789812778550_0028Search in Google Scholar

[5] C. G. Pacheco, Green kernel for a random Schrödinger operator, Commun. Contemp. Math. 18 (2016), no. 5, Article ID 1550082. 10.1142/S0219199715500820Search in Google Scholar

[6] P. E. Protter, Stochastic Integration and Differential Equations, Stoch. Model. Appl. Probab. 21, Springer, Berlin, 2005. 10.1007/978-3-662-10061-5Search in Google Scholar

[7] J. A. Ramírez and B. Rider, Diffusion at the random matrix hard edge, Comm. Math. Phys. 288 (2009), no. 3, 887–906. 10.1007/s00220-008-0712-1Search in Google Scholar

[8] F. Russo and G. Trutnau, Some parabolic PDEs whose drift is an irregular random noise in space, Ann. Probab. 35 (2007), no. 6, 2213–2262. 10.1214/009117906000001178Search in Google Scholar

[9] Z. Shi, A local time curiosity in random environment, Stochastic Process. Appl. 76 (1998), no. 2, 231–250. 10.1016/S0304-4149(98)00036-2Search in Google Scholar

[10] A. V. Skorohod, Random Linear Operators, Math. Appl. (Soviet Series), D. Reidel, Dordrecht, 1984. 10.1007/978-94-009-6063-3Search in Google Scholar

[11] M. Talet, Annealed tail estimates for a Brownian motion in a drifted Brownian potential, Ann. Probab. 35 (2007), no. 1, 32–67. 10.1214/009117906000000539Search in Google Scholar

Received: 2020-04-10
Accepted: 2021-09-04
Published Online: 2022-01-23
Published in Print: 2022-03-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 5.12.2023 from
Scroll to top button