The Parametrix Method for Skew Diffusions
Tóm tắt
In this article, we apply the parametrix method in order to obtain the existence and the regularity properties of the density of a skew diffusion and provide a Gaussian upper bound. This expansion leads to a probabilistic representation.
Tài liệu tham khảo
Andersson, P., Kohatsu-Higa, A.: Unbiased simulation of stochastic differential equations using parametrix expansions (2015)
Bally, V., Kohatsu-Higa, A.: A probabilistic interpretation of the parametrix method. Ann. Appl. Probab. 25, 3095–3138 (2015)
Corielli, F., Foschi, P., Pascucci, A.: Parametrix approximation of diffusion transition densities. SIAM J. Financ. Math. 1, 833–867 (2010)
Étoré, P.: On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients. Electron. J. Probab. 11, 249–275 (2006)
Étoré, P., Martinez, M.: Exact simulation of one-dimensional stochastic differential equations involving the local time at zero the unknown process. Monte Carlo Methods Appl. 19, 41–71 (2013)
Friedman, A.: Partial Differential Equations of Parabolic Type. Dover, New York (1964)
Gairat, A., Shcherbakov, V.: Density of skew Brownian motion and its functionals with application in finance. Preprint arXiv:1407.1715v2 (2014)
Harrison, J.M., Shepp, L.A.: On skew Brownian motion. Ann. Probab. 9, 309–313 (1981)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1998)
Kestelman, H.: Modern Theories of Integration, 2nd edn. Dover, New York (1960)
Krylov, N.V.: On Itô stochastic integral equations. Theory Probab. Appl. 14 (2), 330–336 (1969)
Kulik, A.M.: On the solution of a one-dimensional stochastic differential equation with singular drift coefficient. Ukr. Math. J. 56(5), 774–789 (2004)
Le Gall, J.F.: One-dimensional stochastic differential equations involving the local times of the unknown process. Stoch. Anal. Appl. Lect. Notes Math. 1095, 51–82 (1984)
Lejay, A.: On the constructions of the skew Brownian motion. Probab. Surv. 3, 413–466 (2006)
Lejay, A., Martinez, M.: A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients. Ann. Appl. Probab. 16(1), 107–139 (2006)
Martinez, M., Talay, D.: One-dimensional parabolic diffraction equations: pointwise estimates and discretization of related stochastic differential equations with weighted times. Elect. J. Probab. 17(27), 1–32 (2012)
Menozzi, S.: Parametrix techniques and martingale problems for some degenerate Kolmogorov equations. Electron. Commun. Probab. 16, 234–250 (2011)
Nakao, S.: On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations. Osaka J. Math. 9, 513–518 (1972)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 2nd edn. Grundlehren der Mathematischen Wissenschaften (1999)