IS18: SDEs: Analysis, Approximation, Inference
Organizer: Oleksii Kulyk (Wrocław University of Science and Technology)
A Tail-Respecting Explicit Numerical Scheme for Lévy-Driven SDEs With Superlinear Drifts
Ilya Pavlyukevich
We present an explicit numerical approximation scheme, denoted by \(\{X^n\}\), for the effective simulation of solutions \(X\) to a multivariate stochastic differential equation \[X_t=x + \int_0^t A(X_s)\,\mathrm{d} s + \int_0^t a(X_{s})\,\mathrm{d} s + \int_0^t b(X_{s})\,\mathrm{d} B_s + \int_0^t c(X_{s-})\,\mathrm{d} Z_s\] driven by a \(d\)-dimensional standard Brownian motion \(B\), \(d\in\mathbb N\), and an independent \(d\)-dimensional Lévy process \(Z\).
We assume that
1) the function \(A\) is superlinearly dissipative, i.e., there are \(\kappa\in(1,\infty)\) and \(C\in(0,\infty)\) such that \[\langle A(x),x\rangle \leq -C\|x\|^{1+\kappa}+C,\quad x\in\mathbb R^d,\] and satisfies certain conditions on its derivatives;
2) the functions \(a\), \(b\) and \(c\) are bounded and globally Lipschitz continuous;
3) There is \(p\in(0,\infty)\) such that the Lévy measure \(\nu\) of \(Z\) satisfies \[\int_{\|z\|>1}\|z\|^p\nu(\mathrm{d} z)<\infty.\]
Under these assumptions, we show that for any \(q\in (0,p+\kappa-1)\), the strong \(L^q\)-convergence \[\sup_{t\in[0,T]}\mathbf E \|X^n_t-X_t\|^q=\mathcal O (h_n^{\gamma})\] holds true, in particular, our numerical scheme preserves the \(q\)-moments of the solution beyond the order \(p\). Additionally, for any \(q\in (0,p)\) we establish strong uniform convergence: \[\mathbf E\sup_{t\in[0,T]} \|X^n_t-X_t\|^q=\mathcal{O} ( h_n^{\delta_q^\mathrm{uc}} ).\] In both cases we determine the convergence rates.
In the special case of SDEs driven solely by a Brownian motion, our numerical scheme preserves super-exponential moments of the solution.
The scheme \(\{X^n\}\) is realized as a combination of a well-known Euler method with a Lie–Trotter type splitting technique.
Bibliography
\([1]\) O. Aryasova, O. Kulyk and I. Pavlyukevich. “A tail-respecting explicit numerical scheme for Lévy-driven SDEs with superlinear drifts.” arXiv:2504.07255 math.PR, 2025.
\([2]\) A. Kulik and I. Pavlyukevich. “Moment bounds for dissipative semimartingales with heavy jumps.” Stochastic Processes and their Applications, Vol. 141, 2021. pp. 274–308.
Statistical inference for locally stable regression
Hiroki Masuda
We propose a class of flexible continuous-time location-scale regression models driven by a locally stable system noise. Our primary interest is statistical inference based on a high-frequency sample over a fixed time domain. We construct a tailor-made quasi-likelihood function and show some asymptotic distributional results of the associated estimator: different from the diffusion-type models, the setting allows us to consistently estimate the trend, scale, and activity index. We will also present a method for relative model comparison. Some possible extensions and refinements will be mentioned.