Local linearization method: Difference between revisions

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''High Order Local Linear discretization'' of the SDE ''(7.1)'' at each point <math>t_{n+1}\in \left( t\right) _{h} </math> is then defined by the recursive expression <ref name=":4">de la Cruz H.; Biscay R.J.; Jimenez J.C.; Carbonell F.; Ozaki T. (2010). "High Order Local Linearization methods: an approach for constructing A-stable high order explicit schemes for stochastic differential equations with additive noise". BIT Numer. Math. 50 (3): 509–539. [http://doi.org/10.1007%2Fs10543-010-0272-6 [doi:10.1007/s10543-010-0272-6]]. S2CID 119834289.</ref>

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* Pope D.A. (1963) introduces the LL discretization for ODEs and the LL scheme based on Taylor expansion. <ref name=″:18″> Pope, D. A. (1963). An exponential method of numerical integration of ordinary differential equations. Communications of the ACM, 6(8), 491-493. [https://doi.org/10.1145%2F366707.367592 doi:10.1145/366707.367592]</ref>

* Ozaki T. (1985) introduces the LL method for the integration and estimation of SDEs. The term "Local Linearization" is used for first time. <ref name=″:19″> Ozaki, T. (1985). 2 Non-linear time series models and dynamical systems. Handbook of statistics, 5, 25-83.[https://doi.org/10.1016/S0169-7161(85)05004-0 <small>doi:10.1016/S0169-7161(85)05004-0</small>]</ref>

* Biscay R. et al. (1996) reformulate the strong LL method for SDEs.<ref name="″:20″"> Biscay, R., Jimenez, J. C., Riera, J. J., & Valdes, P. A. (1996). Local linearization method for the numerical solution of stochastic differential equations. Annals ofInst. theStatis. Institute of Statistical Mathematics,Math. 48(4), 631-644.[[doi:10.1007/BF00052324|<small>doi:10.1007/BF00052324</small>]] </ref>

* Shoji I. and Ozaki T. (1997) reformulate the weak LL method for SDEs.<ref name="″:21″"> Shoji, I., & Ozaki, T. (1997). Comparative study of estimation methods for continuous time stochastic processes. JournalJ. ofTime timeSeries series analysis,Anal. 18(5), 485-506.[[doi:10.1111/1467-9892.00064|<small>doi: 10.1111/1467-9892.00064</small>]]</ref>

* Hochbruck M. et al. (1998) introduce the LL scheme for ODEs based on Krylov subspace approximation. <ref name="″:22″">Hochbruck, M., Lubich, C., & Selhofer, H. (1998). Exponential integrators for large systems of differential equations. SIAM J. Scient. Comput. 19(5), 1552-1574.[[doi:10.1137/S1064827595295337|<small>doi:10.1137/S1064827595295337</small>]]</ref>

* Jimenez J.C. (2002) introduces the LL scheme for ODEs and SDEs based on rational Padé approximation. <ref name="″:23″">Jimenez, J. C. (2002). A simple algebraic expression to evaluate the local linearization schemes for stochastic differential equations. AppliedAppl. MathematicsMath. Letters, 15(6), 775-780.[[doi:10.1016/S0893-9659(02)00041-1|<small>doi:10.1016/S0893-9659(02)00041-1</small>]]</ref>

* Carbonell F.M. et al. (2005) introduce the LL method for RDEs. <ref name="″:24″">Carbonell, F., Jimenez, J. C., Biscay, R. J., & De La Cruz, H. (2005). The local linearization method for numerical integration of random differential equations. BIT Num. Math. 45(1), 1-14. [[doi:10.1007/S10543-005-2645-9|doi:10.1007/s10543-005-2645-9]]</ref>

* Jimenez J.C. et al. (2006) introduce the LL method for DDEs. <ref name=":13" />