Numerical Euler-type methods for stohastic differental equations with delay

[A. M. Petrović]

[7], 163 lista

Biografija: listovi 163-[164]. Bibliografija: listovi 157-162.

Stochastic analysis

In general case, stochastic differential equations (SDEs) are not explicitly solvable, such tht studying numerical methods of approximation to their solutions is of great importance. The subject of the doctoral dissertation are numerical Euler-type methods for neutral SDEs with time-dependent delay (and Markovian switching). Most of the results are obtained assuming that the drift and diffusion coefficients of the considered equations grow superlinearly. The explicit numerical methods, such as the classical and truncated Euler-Maruyama (EM) methods, as well as semi-implicit Euler method, were sudied. Classes of equations for wich the explicit methods converge in the L p –sense for p ˃ 0 and order of L q-convergence of the truncated EM method for q ˃ 2 are determined. Beside the classes of equations for wich the classical and truncated EM methods converge, those for wich the mentioned methods diverge in a strong L p-sense on a finite time interval are also determined. Also, the L p-divergence of the semi-implicit Euler method for a class of neutral SDEs with time-dependent delay is proved. The techniques applied in proofs are influenced by the type of considered equation and the assumptions on its coefficients, as well as on the neutral term. Considerations are completed by examples and numerical simulations.

neutralne stohastičke diferencijalne jednačine, vremenski zavisno kašnjenje, proces Markova, L q-konvergencija, klasična i sečena metoda Ojler-Marujame, semiimplicitna Ojlerova metoda, uslov Hasminskog, L p-divergencija, superlinearni rast

neutral stochastic differential equations, time-dependent delay, Markov process, L q-convergence, classical and truncated Euler-Maruyama methods, semi-implicit Euler method, Khasminskii condition, L p-divergence, super-linear growth

517.91:519.216(043.3)

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