On a probability space ( Ω , F , P ) consider a Brownian motion B t t ≥ 0 and a Lévy process ( Z t ) t ≥ 0 , independent of ( B t ) t ≥ 0 . We assume that both processes are one-dimensional and with the same state space. We denote by ( F t ) t ≥ 0 the augmented filtration, generated by the Brownian motion ( B t ) t ≥ 0 and ( Z t ) t ≥ 0 . Denote by N ( d t , d z ) the Poisson random measure on [ 0 , ∞ ) × R, associated with the process Z: N ( ( 0 , t ] , B ) : = # { s ∈ ( 0 , t ] : Δ Z s ∈ B } , B ∈ B ( R ∖ { 0 } ) , and by N ˜ ( d t , d z ) : = N ( d t , d z ) − d t ν ( d z ) the compensated Poisson random measure. Here ν ( · ) is the Lévy measure, associated with Z via the Lévy-Khinchin formula.

Throughout the paper we make the following standing assumptions:

Consider a Lévy-driven SDE of the type (2) X t = X 0 + ∫ 0 t a ( s , X s ) d s + ∫ 0 t b ( s , X s ) d B s + ∫ 0 t ∫ R c ( s , X s − , z ) N ˜ ( d s , d z ) , t > 0 , where a ( s , x ), b ( s , x ), c ( s , x , z ) are some deterministic measurable functions.

Protter and Talay [26] initiated the study of Euler schemes for such SDEs by analyzing errors for expectations of smooth test functions. Later, Jacod [10] extended this framework by proving limit theorems for Euler approximations and studying the asymptotic behavior of the normalized error process u n ( X t − X t n ) for some appropriate scaling sequence u n → ∞. Of special interest are the SDEs driven by α-stable processes. For α ∈ [ 1 , 2 ), strong convergence rates for the Euler–Maruyama approximation have been derived in [21, 22] and [17]; see also [20]. In parallel, Butkovsky et al. [5] investigated the strong convergence of the Euler–Maruyama scheme for multidimensional SDEs with β-Hölder drift ( β > 0) driven by an α-stable Lévy process with α ∈ ( 0 , 2 ]. In particular, they established strong L p -error bounds that are independent of p for α ∈ ( 2 / 3 , 2 ] and demonstrated the optimality of these rates.

From a practical standpoint, simulation of such SDEs is challenging because the exact distribution of Lévy increments is generally unknown except for some special cases. Moreover, Lévy processes can exhibit countably many small jumps in any given finite interval, making direct simulation infeasible. Such a behavior is known as infinite activity: (3) ∫ R ν ( d z ) = + ∞ .

For processes with finite activity, simulation reduces to that of a compound Poisson process, but infinite activity requires alternative approaches. Asmussen and Rosiński [1] addressed this issue by replacing the cumulative effect of small jumps (those below a fixed threshold δ AR ) with a Gaussian term whose variance matches that of the truncated small jumps. In the one-dimensional multiplicative case d X t = σ ( X t − ) d Z t , Fournier [7] analyzed the scheme with Gaussian replacement of jumps | z | ≤ δ AR and showed that choosing δ AR = n − 1 yields mean-square error O ( n − 1 ) at mean cost O ( n + ∫ | z | > 1 / n ν ( d z ) ); in particular, if ν ( · ) admits a density g ( z ) ≍ | z | − 1 − α near 0, the cost scales as O ( n + n α ), whereas omitting the small jumps allows us to reach the same accuracy, requiring ε = n − 1 / ( 2 − α ) and cost O ( n + n α / ( 2 − α ) ). Later, Bossy and Maurer [3] generalized that idea to the time-inhomogeneous jump-driven SDEs of type (2), and derived strong convergence rates for p ≥ 2 together with an optimal choice of the fixed threshold.

While fixed-grid discretization schemes are straightforward to implement, they ignore the timing of jump events. To address this issue, Platen [24] and Bruti-Liberati and Platen [25] developed the class of jump-adapted strong Taylor schemes, where the simulation grid is explicitly synchronized with the jump times of the driving process. Although this construction eliminates the need to evaluate multiple stochastic integrals, it becomes computationally infeasible for infinite-activity Lévy processes due to the accumulation of small jumps. To resolve this bottleneck, Kohatsu-Higa and Tankov [15] proposed a hybrid thresholding approach: they retain a non-uniform grid based only on large jumps (exceeding a fixed threshold) and approximate the contribution of the small jumps between these points using a Brownian motion with matched variance.

The aim of this paper is to derive the strong convergence rates, relying on the dynamic cutting procedure, which we hope to be an accurate and effective tool for approximation of the process X. Dynamic cutting was introduced in [12, 11] for deriving the representation of transition probability density of a Lévy process and its small-time approximation, and developed later in [13, 14], where the transition probability of the solution to a Lévy-driven SDE or, more generally, some Lévy-type process, is constructed. The density in this case is written as a convolution of some “proxy” function and the kernel (which depends on this “proxy”). Such a representation is needed, for example, when one wants to get (upper or lower) estimates of functionals of type 𝔼 x f ( X t ), where f is some (continuous) function. The transition probability density of a (time-inhomogeneous) process with the triplet ( 0 , 0 , 𝟙 | u | ≤ ϕ ( t ) ν ( y , d u ) ) plays the role of the “proxy”, where 𝟙 | u | ≤ ϕ ( t ) ν ( y , d u ) is the Lévy measure, restricted to some time-dependent domain, which explains the name “dynamic cutting”; here y is some fixed point, and ϕ is the cutting level, depending on the measure ν. Although the analytical results from the above papers give a closed form of the transition probability density, it is technically hard to implement.

In this paper we want to elaborate a similar idea for simulations. Namely, we approximate the underlying Lévy process by a time-inhomogeneous compound Poisson process (which we abbreviate below as ICPP), obtained from Z by deleting small jumps together with their compensator in a time–dependent way. In contrast to [1], where the small jumps of fixed size were removed, we delete the jumps whose size depends on time (see Figure 1 and (5) for the definition of functions τ ± ). Such a construction is more flexible especially if implemented on a small-time interval.

Using the Euler scheme with the jump-adapted (or “adjusted”) grid, we derive (cf. Theorem 3.1) the strong convergence rates in L p for p > max { α , 1 }, where α is a parameter, determined by the behavior of the Lévy measure ν ( · ) near the origin.

We compare the dynamic cutting scheme with the classical Asmussen–Rosiński (AR) truncation under matched simulation cost (i.e., with the expected number of retained jumps made equal). On the test example the dynamic cutting (DC) approach demonstrates consistently smaller L 1 –errors, which are still of the same order, when the jump coefficient has a time singularity near 0 (e.g., | c ( t , x , z ) | ≲ t − σ g ( x , z ) with σ > 0). We have taken σ close to the critical values, because on them the difference between the methods is more distinct. If such a singularity is absent, DC (for given parameters) does not show a uniform advantage over AR. We also discuss the role of parameters ε and h used in the cutting function.

Dynamic cutting could be applied to models with singularities, for instance, to stochastic integrals of the form ∫ 0 t K ( t , s ) d Z s , where the kernel is singular. In such a situation dynamic cutting allows a wider range of kernel singularities.

Recall the definition of α, cf. (9). Let (26) θ ∗ : = p σ − ε ( p − α ) α , p < 2 , p σ − ε p ( 2 − α ) 2 α , p ≥ 2 . Further, let β : = min ( γ , 1 / 2 , 1 − χ ), κ : = min { 1 , 2 ( 1 − θ ∗ ) }, and (27) R p ( n , h ) : = n − β + n − κ p h 2 ( σ p − θ ∗ ) p ( 1 + 𝟙 { θ ∗ = 1 2 } ln n ) , p < 2 , n − β + n − 1 p h σ p − θ ∗ p + n − κ p h 2 ( σ p − θ ∗ ) p ( 1 + 𝟙 { θ ∗ = 1 2 } ln n ) , p ≥ 2 . If (19) is satisfied, we put χ = 0. Theorem 3.1.

Suppose that Assumptions 1 – 2 are satisfied. Let p and ζ be the parameters appearing in (13) – (15), and σ ≥ 0 be the parameter from (18). Suppose that θ ∗ < ζ − 1 ζ and either a) or b) hold true: a)

p ≥ 2;

We provide simple examples, where Assumptions 1 and 2 are satisfied. Example 4.1.

Let ν ( d u ) = ln | u | ρ | u | 1 + α 𝟙 0 < | u | < 1 d u , α ∈ ( 0 , 2 ) , ρ > − 1 . For 0 < r < 1, ψ L , + ( 1 / r ) = ∫ 0 r | u / r | 2 ν + ( d u ) = r − 2 ∫ 0 r u 1 − α | ln u | ρ d u . The function L ( u ) : = ln u ρ , u > 0, is slowly varying at 0. Then substituting u = 1 / t and applying Karamata’s theorem (see [2, Prop.1.5.8]), we obtain ∫ 0 r u 1 − α L ( u ) d u ∼ r 2 − α 2 − α L ( r ) , r → 0 , hence (31) ψ L , + ( 1 / r ) ∼ 1 2 − α r − α | ln r | ρ , r → 0 . On the other hand, by [2, Prop.1.5.10], (32) N + ( r ) = ∫ r 1 ln u ρ u 1 + α d u ∼ 1 α r − α | ln r | ρ , r → 0 .

Therefore ψ L , + ( 1 / r ) N + ( r ) ⟶ α 2 − α ∈ ( 0 , ∞ ) , so ψ L , + ( 1 / r ) ≍ N + ( r ), which verifies Assumption 1 for ψ L , + and N + . By symmetry of the Lévy measure, Assumption 1 is satisfied also for the pair ψ L , − and N − .

Note that the same result holds true for the measure ν ( d u ) = ln | u | ρ | u | 1 + α d u , u ∈ R , α ∈ ( 0 , 2 ) , ρ > − 1 . Indeed, the argument for ψ L , + remains the same, and the asymptotic behavior of N ( r ) at 0 in the case of non-cut measure is clearly the same as that of the cut one. The restriction ρ > − 1 is caused by the singularity of ln | u | ρ for ρ ≤ − 1 and can be removed by replacing ln | u | with ln ( 1 + | u | ). In this case the asymptotic behavior (31) and (32) remains valid for all ρ ∈ R.

Next, for simplicity assume that a ( t , x ) = b ( t , x ) = 0, and for σ ∈ [ 0 , 1 ) define c ( t , x , z ) : = t − σ tanh ( x ) min { | z | , 1 } , t ∈ ( 0 , T ] , x , z ∈ R . Then c ( t , 0 , z ) = 0 and ∂ x c ( t , x , z ) = t − σ min { | z | , 1 } sech 2 ( x ). Thus, | ∂ x c ( t , x , z ) | ≤ t − σ min { | z | , 1 } = : L c ( t , z ) , so that condition (12) is clearly satisfied. Assumption 2.4 is clearly satisfied, since | tanh ( x ) | ≤ 1. Let us check the integrability assumptions (13) and (14) for c ( t , x , z ) and the α-stable Lévy measure. If p > α, it is easy to see that ∫ R min { 1 , | z | p } ν ( d z ) < ∞, and ∫ B c ( t , h ) L c p ( t , z ) ν ( d z ) ≲ t − σ p , t ∈ ( 0 , T ], uniformly in h. Hence, (14) is satisfied whenever ζ σ p < 1. Finally, (20) is satisfied when α < 1 and σ < 1. Thus, for c ( t , x , z ) and ν as above all conditions of Assumption 2 are satisfied, if α ∈ ( 0 , 1 ). If we modify c ( t , x , z ) in such a way that it is anti-symmetric in z, then Assumption 2 is satisfied for all α ∈ ( 0 , 2 ).

Let η ( t ) : = sup { t n : t n ≤ t }, where t i ∈ Π, and recall that ρ ( t ) = ♯ { k : t k ≤ t , t k ∈ Π }. Note that η ( t ) = t i − 1 for each t ∈ [ t i − 1 , t i ), 1 ≤ i ≤ ρ ( T ), and by definition η ( t ) , ρ ( t ) are F t – measurable. Write the Peano scheme with frozen coefficients for (2): (34) X ˜ t = x + ∫ 0 t a ( η ( s ) , X ˜ η ( s ) ) d s + ∫ 0 t b ( η ( s ) , X ˜ η ( s ) ) d B s + ∫ 0 t ∫ − ∞ + ∞ c ( s , X ˜ η ( s − ) , z ) N ˜ ( d s , d z ) . Applying Remark 2.3 to the process X ˜, which is a solution to an SDE with piece-wise constant coefficients, one can see that X ˜ possesses p – moments for p as in the assumptions of Theorem 3.1.

We distinguish two cases p < 2 and p ≥ 2, having in mind that p belongs to the intervals, defined in Theorem 3.1.

One can also rewrite (25) using the step function η ( t ): (35) X η ( t ) τ = X 0 τ + ∫ 0 η ( t ) a ( η ( s ) , X η ( s ) τ ) d s + ∫ 0 η ( t ) b ( η ( s ) , X η ( s ) τ ) d B s + ∫ 0 η ( t ) ∫ B c ( s , h ) c ( s , X η ( s − ) τ , z ) N ˜ ( d s , d z ) . Split (36) sup s ≤ t X η ( s ) − X η ( s ) τ L p ( Ω ) ≤ sup s ≤ t X η ( s ) − X ˜ η ( s ) L p ( Ω ) + sup s ≤ t X ˜ η ( s ) − X η ( s ) τ L p ( Ω ) .