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Multivalued backward stochastic differential equations with time delayed generators

100%
Diomande B., Maticiuc L.
Open Mathematics
|
2014
|
tom 12
|
nr 11
1624-1637
EN
Our aim is to study the following new type of multivalued backward stochastic differential equation: $$\left\{ \begin{gathered} - dY\left( t \right) + \partial \phi \left( {Y\left( t \right)} \right)dt \ni F\left( {t,Y\left( t \right),Z\left( t \right),Y_t ,Z_t } \right)dt + Z\left( t \right)dW\left( t \right), 0 \leqslant t \leqslant T, \hfill \\ Y\left( T \right) = \xi , \hfill \\ \end{gathered} \right.$$ where ∂φ is the subdifferential of a convex function and (Y t, Z t):= (Y(t + θ), Z(t + θ))θ∈[−T,0] represent the past values of the solution over the interval [0, t]. Our results are based on the existence theorem from Delong & Imkeller, Ann. Appl. Probab., 2010, concerning backward stochastic differential equations with time delayed generators.
2
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Dynamical behavior for a stochastic two-species competitive model

100%
Xu C., Liao M.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1258-1266
EN
This paper deals with a stochastic two-species competitive model. Some very verifiable criteria on the global stability of the positive equilibrium of the deterministic system are established. An example with its computer simulations is given to illustrate our main theoretical findings.
3
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Calibration and simulation of Heston model

84%
Mrázek M., Pospíšil J.
Open Mathematics
|
2017
|
tom 15
|
nr 1
679-704
EN
We calibrate Heston stochastic volatility model to real market data using several optimization techniques. We compare both global and local optimizers for different weights showing remarkable differences even for data (DAX options) from two consecutive days. We provide a novel calibration procedure that incorporates the usage of approximation formula and outperforms significantly other existing calibration methods. We test and compare several simulation schemes using the parameters obtained by calibration to real market data. Next to the known schemes (log-Euler, Milstein, QE, Exact scheme, IJK) we introduce also a new method combining the Exact approach and Milstein (E+M) scheme. Test is carried out by pricing European call options by Monte Carlo method. Presented comparisons give an empirical evidence and recommendations what methods should and should not be used and why. We further improve the QE scheme by adapting the antithetic variates technique for variance reduction.
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