On backward Kolmogorov equation related to CIR process

We consider the existence of a classical smooth solution to the backward Kolmogorov equation \begin{align*} \begin{cases} \partial_t u(t,x)=Au(t,x),& x\ge0,\ t\in[0,T],\\ u(0,x)=f(x),& x\ge0, \end{cases} \end{align*} where $A$ is the generator of the CIR process, the solution to the stochastic differential equation \begin{equation*} X^x_t=x+\int_0^t\theta \bigl(\kappa-X^x_s\bigr)\,ds+\sigma\int _0^t\sqrt {X^x_s} \,dB_s, \quad x\ge0,\ t\in[0,T], \end{equation*} that is, $Af(x)=\theta(\kappa-x)f'(x)+\frac{1}{2}\sigma^2xf''(x)$, $x\ge0$ ($\theta,\kappa,\sigma>0$). Alfonsi \cite{Alfonsi} showed that the equation has a smooth solution with partial derivatives of polynomial growth, provided that the initial function $f$ is smooth with derivatives of polynomial growth. His proof was mainly based on the analytical formula for the transition density of the CIR process in the form of a~rather complicated function series. In this paper, for a CIR process satisfying the condition $\sigma^2\le4\theta\kappa$, we present a direct proof based on the representation of a CIR process in terms of a~squared Bessel process and its additivity property.Comment: Published at https://doi.org/10.15559/18-VMSTA98 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/

Teigiamų difuzijų teigiamos aproksimacijos

For positive diffusions, we construct split-step second-order weak approximations preserving the positivity property. For illustration, we apply the construction to some popular stochastic differential equations such as Verhulst, CIR, and CKLS equations.Teigiamoms difuzijoms sukonstruotos teigiamos antrosios eilės silpnosios aproksimacijos. Naujoji konstrukcija iliustruojama taikymu stochastinėms diferencialinėms Verhulsto, CIR ir CKLS lygtims

Silpnųjų aproksimacijų taikymas opcionų kainų skaičiavimui Hestono modelyje

We apply weak split-step approximations of the Heston model for evaluation of put and call option prices in this model.Naudodami [2] straipsnyje pasiūlytą silpnąją atskyrimo (split-step) aproksimaciją diskrečiais atsitiktiniais dydžiais Hestono modeliui, vertiname pirkimo (call) bei pardavimo (put) opcionų kainas. Aproksimuojant gautas kainas palyginame su kainomis, gautomis naudojant [1] straipsnyje aprašytą formulę

Weak approximations of the Wright–Fisher process

In this paper, we construct first- and second-order weak split-step approximations for the solutions of the Wright–Fisher equation. The discretization schemes use the generation of, respectively, two- and three-valued random variables at each discretization step. The accuracy of constructed approximations is illustrated by several simulation examples

Wright-Fisher lygties silpnosios aproksimacijos

We construct weak approximations of the Wright-Fisher model and illustrate their accuracy by simulation examples.Sukonstruota silpnoji pirmos eilės aproksimacija stochastinei Wright-Fisher lygčiai. Pavyzdžiais iliustruojamas jos tikslumas

Second-Order Weak Approximations of CKLS and CEV Processes by Discrete Random Variables

In this paper, we construct second-order weak split-step approximations of the CKLS and CEV processes that use generation of a three−valued random variable at each discretization step without switching to another scheme near zero, unlike other known schemes (Alfonsi, 2010; Mackevičius, 2011). To the best of our knowledge, no second-order weak approximations for the CKLS processes were constructed before. The accuracy of constructed approximations is illustrated by several simulation examples with comparison with schemes of Alfonsi in the particular case of the CIR process and our first-order approximations of the CKLS processes (Lileika– Mackevičius, 2020)