arXiv subjects

Plamen Turkedjiev

Publications and source records attributed to Plamen Turkedjiev.

3 recordsLinked to original sources

arXiv2016

Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression

We design a numerical scheme for solving a Dynamic Programming equation with Malliavin weights arising from the time-discretization of backward stochastic differential equations with the integration by parts-representation of the $Z$-component by (Ann. Appl. Probab. 12 (2002) 1390-1418). When the sequence of conditional expectations is computed using empirical least-squares regressions, we establish, under general conditions, tight error bounds as the time-average of local regression errors only (up to logarithmic factors). We compute the algorithm complexity by a suitable optimization of the parameters, depending on the dimension and the smoothness of value functions, in the limit as the number of grid times goes to infinity. The estimates take into account the regularity of the terminal function.

math.ST↗

arXiv2014

Multilevel approximation of backward stochastic differential equations

We develop a multilevel approach to compute approximate solutions to backward differential equations (BSDEs). The fully implementable algorithm of our multilevel scheme constructs sequential martingale control variates along a sequence of refining time-grids to reduce statistical approximation errors in an adaptive and generic way. We provide an error analysis with explicit and non-asymptotic error estimates for the multilevel scheme under general conditions on the forward process and the BSDE data. It is shown that the multilevel approach can reduce the computational complexity to achieve precision $ε$, ensured by error estimates, essentially by one order (in $ε^{-1}$) in comparison to established methods, which is substantial. Computational examples support the validity of the theoretical analysis, demonstrating efficiency improvements in practice.

math.PR↗

arXiv2014

Two algorithms for the discrete time approximation of Markovian backward stochastic differential equations under local conditions

Two discretizations of a class of locally Lipschitz Markovian backward stochastic differential equations (BSDEs) are studied. The first is the classical Euler scheme which approximates a projection of the processes Z, and the second a novel scheme based on Malliavin weights which approximates the mariginals of the process Z directly. Extending the representation theorem of Ma and Zhang leads to advanced a priori estimates and stability results for this class of BSDEs. These estimates are then used to obtain competitive convergence rates for both schemes with respect to the number of points in the time-grid. The class of BSDEs considered includes Lipschitz BSDEs with fractionally smooth terminal condition as well as quadratic BSDEs with bounded, Hölder continuous terminal condition (for bounded, differentiable volatility), and BSDEs related to proxy methods in numerical analysis.

math.PR↗