arXiv subjects

Ruishu Liu

Publications and source records attributed to Ruishu Liu.

5 recordsLinked to original sources

arXiv2025

Unconditionally positivity-preserving explicit order-one strong approximations of financial SDEs with non-Lipschitz coefficients

In this paper, we are interested in positivity-preserving approximations of stochastic differential equations (SDEs) with non-Lipschitz coefficients, arising from computational finance and possessing positive solutions. By leveraging a Lamperti transformation, we develop a novel, explicit, and unconditionally positivity-preserving numerical scheme for the considered financial SDEs. More precisely, an implicit term $c_{-1}Y_{n+1}^{-1}$ is incorporated in the scheme to guarantee unconditional positivity preservation, and a corrective operator is introduced in the remaining explicit terms to address the challenges posed by non-Lipschitz (possibly singular) coefficients of the transformed SDEs. By finding a unique positive root of a quadratic equation, the proposed scheme can be explicitly solved and is shown to be strongly convergent with order $1$, when used to numerically solve several well-known financial models such as the CIR process, the Heston-3/2 volatility model, the CEV process and the Aït-Sahalia model. Numerical experiments validate the theoretical findings.

math.NA↗

arXiv2025

Explicit modified Euler approximations of the Aït-Sahalia type model with Poisson jumps

This paper focuses on mean-square approximations of a generalized Aït-Sahalia interest rate model with Poisson jumps. The main challenge in the construction and analysis of time-discrete numerical schemes is caused by a drift that blows up at the origin, highly nonlinear drift and diffusion coefficients and positivity-preserving requirement. Due to the presence of the Poisson jumps, additional difficulties arise in recovering the exact order $1/2$ of convergence for the time-stepping schemes. By incorporating implicitness in the term $α_{-1}x^{-1} $ and introducing the modifications functions $f_h$ and $g_h$ in the recursion, a novel explicit Euler-type scheme is proposed, which is easy to implement and preserves the positivity of the original model unconditionally, i.e., for any time step-size $h>0$. A mean-square convergence rate of order $1/2$ is established for the proposed scheme in both the non-critical and general critical cases. Finally, numerical experiments are provided to confirm the theoretical findings.

math.NA↗

arXiv2024

An unconditional boundary and dynamics preserving scheme for the stochastic epidemic model

In the present article, we construct a logarithm transformation based Milstein-type method for the stochastic susceptible-infected-susceptible (SIS) epidemic model evolving in the domain (0,N). The new scheme is explicit and unconditionally boundary and dynamics preserving, when used to solve the stochastic SIS epidemic model. Also, it is proved that the scheme has a strong convergence rate of order one. Different from existing time discretization schemes, the newly proposed scheme for any time step size h > 0, not only produces numerical approximations living in the entire domain (0,N), but also unconditionally reproduces the extinction and persistence behavior of the original model, with no additional requirements imposed on the model parameters. Numerical experiments are presented to verify our theoretical findings.

math.NA↗

arXiv2024

Unconditionally positivity-preserving approximations of the Ait-Sahalia type model: Explicit Milstein-type schemes

The present article aims to design and analyze efficient first-order strong schemes for a generalized Aït-Sahalia type model arising in mathematical finance and evolving in a positive domain $(0, \infty)$, which possesses a diffusion term with superlinear growth and a highly nonlinear drift that blows up at the origin. Such a complicated structure of the model unavoidably causes essential difficulties in the construction and convergence analysis of time discretizations. By incorporating implicitness in the term $α_{-1} x^{-1}$ and a corrective mapping $Φ_h$ in the recursion, we develop a novel class of explicit and unconditionally positivity-preserving (i.e., for any step-size $h>0$) Milstein-type schemes for the underlying model. In both non-critical and general critical cases, we introduce a novel approach to analyze mean-square error bounds of the novel schemes, without relying on a priori high-order moment bounds of the numerical approximations. The expected order-one mean-square convergence is attained for the proposed scheme. The above theoretical guarantee can be used to justify the optimal complexity of the Multilevel Monte Carlo method. Numerical experiments are finally provided to verify the theoretical findings.

math.NA↗

arXiv2024

Unconditionally positivity-preserving explicit Euler-type schemes for a generalized Ait-Sahalia model

The present work is devoted to strong approximations of a generalized Aït-Sahalia model arising from mathematical finance. The numerical study of the considered model faces essential difficulties caused by a drift that blows up at the origin, highly nonlinear drift and diffusion coefficients and positivity-preserving requirement. In this paper, a novel explicit Euler-type scheme is proposed, which is easily implementable and able to preserve positivity of the original model unconditionally, i.e., for any time step-size $h >0$. A mean-square convergence rate of order $0.5$ is also obtained for the proposed scheme in both non-critical and general critical cases. Our work is motivated by the need to justify the multi-level Monte Carlo (MLMC) simulations for the underlying model, where the rate of mean-square convergence is required and the preservation of positivity is desirable particularly for large discretization time steps. Numerical experiments are finally provided to confirm the theoretical findings.

math.NA↗