Explicit positivity preserving numerical method for linear stochastic volatility models driven by $\alpha$-stable process

Hongjiong Tian; Wei Liu; Xiaotong Li; Xuerong Mao; Yue Wu

REVIEW 3 major objections 6 minor 29 references

Explicit positivity preserving numerical method for linear stochastic volatility models driven by $\alpha$-stable process

T0 review · 3 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper proves that an explicit, positivity-preserving Euler-Maruyama scheme for linear α-stable stochastic volatility models converges in L^q with strong order 1/α.

desk verdict A useful explicit positivity-preserving scheme for α-stable-driven SDEs, with a plausible 1/α rate, but the key projection lemma is genuinely under-proved and the numerics don't match the assumptions. read the letter →

arxiv 2502.00788 v1 pith:3EAMUQDJ submitted 2025-02-02 math.PR

Xiaotong Li , Wei Liu , Xuerong Mao , Hongjiong Tian , Yue Wu This is my paper · ORCID

classification math.PR MSC 60G5160H3565C30

keywords stochasticvolatilitymodelsα-stableprocesspositivitypreservingEuler-MaruyamaschemestrongconvergenceLévypositivesolutionexplicit

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces a linear stochastic volatility model driven by an α-stable Lévy process, $dx(t) = (\mu - \lambda x(t))dt + \kappa x(t-)dL_\alpha(t)$, and proves that under a lower bound on negative jumps the model has a unique positive global solution. It then constructs an explicit Euler-Maruyama scheme that preserves positivity by replacing any value below the step size $\Delta$ with $\Delta$ itself. The main result is that this scheme converges in $L^q$, for every $q \in [1, \alpha)$, with strong order $1/\alpha$: $\sup_{0 \le t \le T} E|x(t) - \tilde{X}_\Delta(\eta(t))|^q \le C\Delta^{q/\alpha}$. Because the scheme is explicit and positivity is built in, it offers a practical way to simulate fat-tailed volatility paths without solving implicit equations at each step.

What carries the argument

The load-bearing object is the projected explicit Euler-Maruyama scheme (4.1): at each step the raw update $X_{k+1}$ is floored at the step size, $\tilde{X}_{k+1} = X_{k+1} \vee \Delta$, and the floored value is the one used in the drift and diffusion of the next step. This floor keeps the numerical process strictly positive. The convergence argument works through a continuous-time interpolation $\tilde{x}_\Delta(t)$ that agrees with $X_\Delta$ at grid points, and measures three discrepancies: the extrapolation error between $\tilde{x}_\Delta$ and the true solution, the within-step error between $\tilde{x}_\Delta$ and $X_\Delta$, and the projection error between $X_\Delta$ and its floor. The rate $1/\alpha$ emerges from the maximal inequality for $\alpha$-stable integrals and the $q$-th moment growth of stable increments, which together convert a stochastic integral with respect to $L_\alpha$ into a term of size $\Delta^{q/\alpha}$.

What would settle it

Run the same scheme on a spectrally negative $\alpha$-stable process whose Lévy measure includes jumps below $-1/\kappa$. If the true SDE still has a positive solution and the scheme still converges at order $1/\alpha$, then Assumption 2.2 is not needed and the theorem's stated hypothesis is too strong; if positivity fails or the error slope on a log-log plot departs from $1/\alpha$, the assumption is doing real work.

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Extended reading notes

Core claim

The paper's central claim is that positivity and strong convergence of order $1/\alpha$ can be achieved simultaneously by a fully explicit numerical method for SDEs driven by symmetric $\alpha$-stable noise with $\alpha \in (1,2)$. The proposed method updates $X_{k+1} = X_k + (\mu - \lambda \tilde{X}_k)\Delta + \kappa \tilde{X}_k \Delta L_k$ and then sets $\tilde{X}_{k+1} = X_{k+1} \vee \Delta$, so the numerical solution is never below the step size. Theorem 4.5 states that for step sizes $\Delta \in (0, (\mu-1)/\lambda \wedge 1/\lambda)$ and $q \in [1, \alpha)$, the $L^q$ error between the true solution $x(t)$ and the projected numerical solution $\tilde{X}_\Delta(\eta(t))$ is bounded by $C\Delta^{q/\alpha}$. The proof combines Ito's formula with a truncated Lévy measure, a maximal inequality for $\alpha$-stable stochastic integrals, and a nonlinear Gronwall inequality; the error decomposes into the continuous-time extrapolation error, the jump-induced error, and the projection bias, each of which is shown to be $O(\Delta^{q/\alpha})$.

Load-bearing premise

Everything rests on Assumption 2.2, which says the negative jumps of the driving stable process are never smaller than $-1/\kappa$; if arbitrarily large negative jumps are allowed, the true solution may be pushed to or through zero, and the positivity theorem plus the convergence target would no longer hold.

Editorial extensions

If this is right

By construction, $\tilde{X}_\Delta(\eta(t)) \ge \Delta > 0$, so the simulated volatility path never crosses zero.
The $L^q$ error bound $\sup_{0\le t\le T} E|x(t)-\tilde{X}_\Delta(\eta(t))|^q \le C\Delta^{q/\alpha}$ holds uniformly over the whole interval $[0,T]$, not just at grid points.
The explicit update requires only one stable increment per step, so the method is as cheap as the classical Euler-Maruyama scheme while preserving positivity.
The same rate $1/\alpha$ is obtained for every $q \in [1,\alpha)$, with a constant independent of the step size.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the $1/\alpha$ rate is the minimax rate for any method driven by $\alpha$-stable increments, then this explicit scheme is rate-optimal; the paper proves only the upper bound.
The mismatch between the theory, which truncates negative jumps, and the simulations, which use the full symmetric stable measure, suggests the truncation may be a proof device rather than a computational requirement; testing on a process with genuinely unbounded negative jumps would settle this.
The same floor construction should extend to super-linear mean-reverting drifts, a future direction the paper names, but the Ito estimates would need modification because the drift is no longer linear.
For option pricing, positive heavy-tailed volatility paths can now be simulated explicitly, opening Monte Carlo pricing for models such as $\alpha$-CIR without implicit solves.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take. The contribution is real: a fully explicit EM scheme with a positivity floor for a linear volatility SDE driven by an α-stable process, and they claim strong L^q order 1/α. That's a natural extension of the implicit schemes by Li-Taguchi and Li-Liu, and I don't know of an explicit version in the literature. The existence and moment-bound sections are standard but mostly correct, and the numerical experiments show nice slopes.

The soft spots are not cosmetic. First, in Theorem 3.1's J22 estimate, they drop the positive term −0.5 log(1+κz) for negative large jumps and then bound the remaining integral. Dropping a positive term gives a lower bound, not an upper bound, so the displayed inequality is reversed. It's probably fixable, but as written it's wrong.

The bigger issue is Lemma 4.3 and its use in Theorem 4.4. The proof uses the stable tail asymptotic P(ΔL_k < x) ∼ |x|^{-α} at a threshold that approaches −1/κ, exactly where the truncated Lévy measure has no mass and the power-law asymptotic stops being valid. The argument also needs a uniform-in-k bound on the probability of falling below Δ; the proof gives only a pointwise limit, no rate, and no union bound over the N steps. Without something like sup_k P(XΔ(t_k)<Δ) ≤ CΔ, the ε-δ step in Theorem 4.4 can't be completed. This is the load-bearing gap for the 1/α claim.

There's also a mismatch between theory and simulation. Assumption 2.2 truncates negative jumps, but the numerical section appears to use standard symmetric α-stable increments. The scheme itself is positive by construction, but the experiment no longer tests the theorem's assumptions. They need to simulate the truncated process.

All that said, I think the underlying claim is likely true — the rate is what you'd expect for EM with α-stable noise and a small perturbation from the floor. The paper deserves a serious referee, because the idea is useful and the problems are repairable. Send it out; expect major revision but don't desk reject.

Who this is for: people working on numerical methods for jump SDEs in mathematical finance. It's a subfield-level contribution, not a breakthrough.

Referee Report

3 major / 6 minor

Summary. This paper studies a scalar linear stochastic volatility model dx(t)=(µ-λx(t))dt+κx(t-)dL_α(t) driven by an α-stable process with α∈(1,2), under an assumption that negative jump sizes are bounded below by -1/κ. The authors prove existence and uniqueness of a global positive solution (Theorem 3.1), introduce an explicit Euler-Maruyama scheme with a floor projection X_{k+1}↦X_{k+1}∨Δ, establish uniform moment bounds, and claim strong convergence of order 1/α in L^q for q∈[1,α) (Theorem 4.5). Numerical experiments with several parameter sets are reported as supporting the predicted rate.

Significance. If the main theorem were fully proved, the paper would make a useful contribution: an explicit, positivity-preserving EM scheme for a linear α-stable-driven SDE with the optimal strong order 1/α, in line with known rates for EM for stable noise. The argument is constructive, contains no fitted parameters, and the claimed order is empirically falsifiable; the simulations are consistent with the slope 1/α. However, the proof of the key projection lemma is currently invalid, and the model is not consistently defined (truncated Lévy measure vs full stable measure used in lemmas and simulations). These issues are local in nature but must be repaired before the result can be accepted.

major comments (3)

[§2, Eq. (2.2) / Assumption 2.2] The paper never defines the truncated Lévy measure that Assumption 2.2 requires. Eq. (2.2) states ν(dz)=C_α|z|^{-α-1}dz on R\{0}, the full symmetric stable measure, and Lemma 2.4 and Lemma 2.5 are stated for the full α-stable process. If negative jumps are restricted to z>-1/κ, then the driver in (2.1) is no longer the α-stable process used in the lemmas and in the Section 5 simulations; the proofs and the numerics describe different processes. Please state the truncated measure explicitly (e.g., ν(dz)=C_α|z|^{-α-1}1_{z>-1/κ}dz), give the corresponding Itô formula, and adapt all tail estimates and simulation increments accordingly.
[Lemma 4.3] The proof applies the full-stable tail asymptotic P(ΔL_k<x)~|x|^{-α} to the threshold ((1-μ)Δ-(1-λΔ)X_k)/(κX_k). As Δ→0 this threshold tends to -1/κ, which is the boundary of the support under Assumption 2.2, and when X_k is of order Δ the threshold can differ from -1/κ by O(1) (e.g., X_k=Δ gives threshold ≈ -μ/κ). The asymptotic is therefore not applicable, and the conclusion '→P(ΔL_k<-1/κ)→0' is not justified. The lemma also gives no quantitative rate and no bound uniform in k, although Theorem 4.4 requires, for each fixed δ, a control of P(|X_Δ(η(s))-tilde X_Δ(η(s))|≥δ) uniformly in s∈[0,T] to close the ε-δ argument in (4.6). A correct proof should provide a uniform bound such as sup_k P(X_Δ(t_{k+1})<Δ)≤CΔ.
[Theorem 3.1, estimate of J22] The line 'J22 < ∫ ...' for z∈(-1/κ,-1) drops the term -0.5 log(1+κz), which is positive on this interval; dropping it makes the expression smaller, so the displayed inequality is reversed. The same issue affects the subsequent bound. This is repairable by adding the omitted integral as a constant, but as written the proof of the key Lyapunov estimate (3.5) contains a false inequality.

minor comments (6)

[§4, Eq. (4.1)] Clarify that X_Δ(t_{k+1}) itself is not projected; only \tilde X_Δ is, so X_Δ can be negative in intermediate steps. The notation \tilde X_Δ(t_{k+1})=X_Δ(t_{k+1})∨Δ is clear, but the text 'positivity preserving' should be stated for the projected variable.
[Lemma 4.3 statement] The assertion 'P(X_Δ(t_{k+1})<Δ)=0, as Δ→0' is a limit statement; write '→0'.
[§5] The simulations do not specify the skewness β of the stable increments, nor how the truncation z>-1/κ is implemented. Since the theoretical results rely on the truncation, please state the simulation procedure for the truncated process.
[§5, Tables 1-3] Some entries are formatted inconsistently (e.g., Table 2, α=1.1 row uses 0.0551×10^{-3}); please unify the notation.
[Lemma 4.1 / Theorem 4.4] The displayed nonlinear Gronwall bound contains e(t) and e1(t) with different definitions; please verify the application against [27], in particular the denominator 1-[1-e1(t)]^{q/α}.
[References] Reference [3] is an unpublished preprint; if a peer-reviewed version exists, cite it.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the 1/α convergence result is derived from standard stable-process inequalities and explicit assumptions, not from the paper's own conclusions.

full rationale

A full walk of the derivation chain shows no step in which an output is equivalent to an input by construction or by self-citation. Theorem 3.1 obtains positivity from the explicit truncation Assumption 2.2 rather than assuming positivity itself; the assumption is a stated hypothesis, not a renamed conclusion. The numerical scheme (4.1)-(4.2) is defined independently of the target error bound. Lemma 4.2 and Theorem 4.4 use only standard external ingredients: Lemma 2.4 (stable moment bound), Lemma 2.5 (maximal inequality for stable integrals), Hölder's inequality, and the nonlinear Gronwall inequality; generic constants absorb dependencies on parameters, and no quantity is fitted to the error reported in Theorem 4.5. Lemma 4.3, the most delicate supporting estimate, has a proof gap: after Assumption 2.2 truncates the Lévy measure below -1/κ, the proof invokes the unrestricted stable tail P(ΔL_k < x) ∼ |x|^{-α}, and no uniform-in-k rate is given. This is a correctness defect in a supporting estimate, not a circularity: the conclusion of Theorem 4.5 does not coincide with any assumed input or fitted value. References [15] and [16] are prior schemes cited for design inspiration, and the convergence theorems are proved in this paper rather than imported from a self-citation; no uniqueness theorem or ansatz is smuggled in via citation. The Section 5 simulations independently generate paths and compare against a finest-step reference; the observed slopes confirm but do not define the theoretical rate. Thus no circularity pattern is present, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper leverages standard stochastic calculus facts for Lévy processes and two explicit modeling assumptions. No free parameters are fitted to data. The main modeling assumption is the truncation of negative jumps, which is essential for positivity but is not reflected in the numerical implementation.

assumptions (6)

standard math Itô formula for semimartingales with jumps (Lemma 2.3)
Used in the proofs of Theorem 3.1 and Lemma 3.2 to compute the evolution of the Lyapunov function and moments.
standard math Moment bound for α-stable increments: E|Lα(t)|^q ≤ C t^{q/α} for q ∈ [1,α) (Lemma 2.4)
Quoted from Samorodnitsky-Taqqu; used in Lemma 4.2 to control the stochastic integral term.
standard math Maximal inequality for stochastic integrals w.r.t. α-stable processes (Lemma 2.5)
Quoted from Zhang [28]; used in Lemmas 4.1 and Theorem 4.4 to bound the L^q norm of stable integrals.
standard math Nonlinear Gronwall inequality from Willett-Wong [27]
Used in Lemmas 4.1 and Theorem 4.4 to close the moment estimates.
domain assumption Assumption 2.1: µ > 1, λ > 0, 0 < κ < 1, and λ > 2 κ^{0.5} Cα / (2α - 1)
Parameter constraints needed for the Lyapunov estimate in Theorem 3.1, ensuring the drift dominates the large-jump contribution.
domain assumption Assumption 2.2: negative jumps of the driving process satisfy z > -1/κ
Ensures the multiplicative factor 1 + κz stays positive so the solution remains positive. This is load-bearing for the model but is not part of the standard α-stable process definition, and the paper does not consistently enforce it in the simulations.

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Cite this review

Pith. "Pith review of Explicit positivity preserving numerical method for linear stochastic volatility models driven by $\alpha$-stable process." pith.science (2026). https://pith.science/paper/3EAMUQDJ

@misc{pith2026250200788,
  author       = {Pith},
  title        = {Pith review of: Explicit positivity preserving numerical method for linear stochastic volatility models driven by $\alpha$-stable process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3EAMUQDJ}},
  note         = {Machine review of arXiv:2502.00788}
}

abstract

In this paper, we introduce a linear stochastic volatility model driven by $\alpha$-stable processes, which admits a unique positive solution. To preserve positivity, we modify the classical forward Euler-Maruyama scheme and analyze its numerical properties. The scheme achieves a strong convergence order of $1/\alpha$. Numerical simulations are presented at the end to verify theoretical results.

Figures

Figures reproduced from arXiv: 2502.00788 by the authors.

**Figure 2.** Errors versus stepsize ∆ on log-log scale with [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** Errors versus stepsize ∆ on log-log scale with [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Errors versus stepsize ∆ on log-log scale with [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]

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Reference graph

Works this paper leans on

29 extracted references · 28 canonical work pages

[1]

Applebaum, L´ evy Processes and Stochastic Calculus, Cambr idge Univer- sity Press, 2nd Ed., 2009

D. Applebaum, L´ evy Processes and Stochastic Calculus, Cambr idge Univer- sity Press, 2nd Ed., 2009

work page 2009
[2]

Bergomi, Stochastic Volatility Modeling, CRC press, 2015

L. Bergomi, Stochastic Volatility Modeling, CRC press, 2015

work page 2015
[3]

Butkovsky, K

O. Butkovsky, K. Dareiotis, M. Gerencs´ er, Strong rate of co nvergence of the Euler scheme for SDEs with irregular drift driven by L´ evy noise, arXiv preprint arXiv:2204.12926

arXiv
[4]

H. Cao, H. Ouyang, Y. Li, X. Li, Y. Chen, The power law character istics of stock price jump intervals: an empirical and computational expe rimental study, Entropy, 20(4) (2018), 304

work page 2018
[5]

Coﬃe, Numerical approximation of a hybrid Poisson-jump Ait-S ahalia- type interest rate model with delay, Stoch

E. Coﬃe, Numerical approximation of a hybrid Poisson-jump Ait-S ahalia- type interest rate model with delay, Stoch. Models, 40 (3), (2024 ), 583–616

work page 2024
[6]

Z. Fu, Z. Li, Stochastic equations of non-negative processes w ith jumps, Stochastic Process. Appl., 120(3) (2010), 306–330

work page 2010
[7]

Huang, Z.-W

X. Huang, Z.-W. Liao, The Euler-Maruyama method for S(F)DEs w ith H¨ older drift andα-stable noise, Stoch. Anal. Appl., 36(1) (2018), 28–39

work page 2018
[8]

J. Hull, A. White, The pricing of options on assets with stochastic v olatilities, The Journal of Finance, 42(2) (1987), 281–300

work page 1987

Show all 29 references

[9]

Y. Jiao, C. Ma, S. Scotti, Alpha-CIR model with branching proces ses in sovereign interest rate modelling, Finance Stoch., 21(3) (2017), 7 89–813

2017
[10]

Janicki, A

A. Janicki, A. Weron. Simulation and Chaotic Behavior of Alpha-St able Stochastic Processes, CRC Press, 2021

2021
[11]

Kabaˇ sinskas, S

A. Kabaˇ sinskas, S. Rachev, L. Sakalauskas, W. Sun, I. Belovas, Alpha-stable paradigm in ﬁnancial markets, J. Comput. Anal. Appl. 11 (4) (2009) , 641– 668

2009
[12]

Z. Lei, S. Gan, J. Liu, First order strong approximation of Ait–S ahalia-type interest rate model with Poisson jumps, Numer. Algorithms, 94(1) (2023), 93–130

2023
[13]

A. L. Lewis, Option Valuation under Stochastic Volatility, Finance Press, 2000

2000
[14]

Z. Li, C. Ma, Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model, Stochastic Process. Appl., 125(8) (20 15), 3196– 3233. 23
[15]

L. Li, D. Taguchi, On a positivity preserving numerical scheme fo r jump- extended CIR process: the alpha stable case, BIT Numerical Math ematics, 59(3) (2019), 747–774

2019
[16]

L. Li, G. Liu, A positivity preserving numerical scheme for the alp ha-CEV process, arXiv preprint arXiv:2103.13002 (2023)

2023 arXiv
[17]

Mikuleviˇ cius, F

R. Mikuleviˇ cius, F. Xu, On the rate of convergence of strong Euler ap- proximation for SDEs driven by Levy processes, Stochastics, 90( 4) (2018), 569–604

2018
[18]

C. L. Nikias, M. Shao, Signal Processing with Alpha-Stable Distrib utions and Applications, Wiley-Interscience, 1995

1995
[19]

O. M. Pamen, D. Taguchi, Strong rate of convergence for the Euler- Maruyama approximation of SDEs with H¨ older continuous drift coeﬃ cient, Stochastic Process. Appl., 127(8) (2017), 2542–2559

2017
[20]

R. D. Pierce, Application of the positive alpha-stable distribution , In Pro- ceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics (1997), 420–424

1997
[21]

Øksendal, A

B. Øksendal, A. Sulem, Applied Stochastic Control of Jump Diﬀus ion, Springer, Berlin, 2000

2000
[22]

Samorodnitsky and M

G. Samorodnitsky and M. S. Taqqu. Stable Non-Gaussian Rando m Pro- cesses: Stochastic models with inﬁnite variance. Chapman and Hall, N ew York, 1994

1994
[23]

Sato, L´ evy Processes and Inﬁnitely Divisible Distributions, C ambridge University Press, Cambridge, 2013

K. Sato, L´ evy Processes and Inﬁnitely Divisible Distributions, C ambridge University Press, Cambridge, 2013

2013
[24]

L. O. Scott, Option Pricing when the variance changes randomly : theory, estimation, and an application, J. of Fin. and Quantitative Analysis, 1 2 (4) (1987), 419–438

1987
[25]

N. N. Taleb, Statistical Consequences of Fat Tails: Real World P reasymp- totics, Epistemology, and Applications, arXiv preprint arXiv:2001.10 488, 2020

2001
[26]

J. B. Wiggins, Option values under stochastic volatility, Journal of Financial Economics, 19 (1987), 351–372

1987
[27]

Willett, J

D. Willett, J. Wong, On the discrete analogues of some generaliza tions of Gronwall’s inequality, Monatsh. Math., 69 (1965), 362–367. 24

1965
[28]

Zhang, Derivative formulas and gradient estimates for SDEs driven by α-stable processes, Stochastic Process

X. Zhang, Derivative formulas and gradient estimates for SDEs driven by α-stable processes, Stochastic Process. Appl., 123 (2013), 1213 –1228

2013
[29]

Y. Zhao, X. Wang, M. Wang, On the backward Euler method for a gener- alized Ait-Sahalia-type rate model with Poisson jumps, Numer. Algor ithms, 87 (2021), 1321–1341. 25

2021

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Applebaum, L´ evy Processes and Stochastic Calculus, Cambr idge Univer- sity Press, 2nd Ed., 2009

D. Applebaum, L´ evy Processes and Stochastic Calculus, Cambr idge Univer- sity Press, 2nd Ed., 2009

work page 2009

[2] [2]

Bergomi, Stochastic Volatility Modeling, CRC press, 2015

L. Bergomi, Stochastic Volatility Modeling, CRC press, 2015

work page 2015

[3] [3]

Butkovsky, K

O. Butkovsky, K. Dareiotis, M. Gerencs´ er, Strong rate of co nvergence of the Euler scheme for SDEs with irregular drift driven by L´ evy noise, arXiv preprint arXiv:2204.12926

arXiv

[4] [4]

H. Cao, H. Ouyang, Y. Li, X. Li, Y. Chen, The power law character istics of stock price jump intervals: an empirical and computational expe rimental study, Entropy, 20(4) (2018), 304

work page 2018

[5] [5]

Coﬃe, Numerical approximation of a hybrid Poisson-jump Ait-S ahalia- type interest rate model with delay, Stoch

E. Coﬃe, Numerical approximation of a hybrid Poisson-jump Ait-S ahalia- type interest rate model with delay, Stoch. Models, 40 (3), (2024 ), 583–616

work page 2024

[6] [6]

Z. Fu, Z. Li, Stochastic equations of non-negative processes w ith jumps, Stochastic Process. Appl., 120(3) (2010), 306–330

work page 2010

[7] [7]

Huang, Z.-W

X. Huang, Z.-W. Liao, The Euler-Maruyama method for S(F)DEs w ith H¨ older drift andα-stable noise, Stoch. Anal. Appl., 36(1) (2018), 28–39

work page 2018

[8] [8]

J. Hull, A. White, The pricing of options on assets with stochastic v olatilities, The Journal of Finance, 42(2) (1987), 281–300

work page 1987