IS18: SDEs: Analysis, Approximation, Inference

date: 7/15/2025, time: 16:00-17:30, room: IM WS

Organizer: Oleksii Kulyk (Wrocław University of Science and Technology)

Chair: Oleksii Kulyk (Wrocław University of Science and Technology)

A Tail-Respecting Explicit Numerical Scheme for Lévy-Driven SDEs With Superlinear Drifts

Ilya Pavlyukevich (Friedrich Schiller University Jena)

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.

The Feynman-Kac formula for the gradient of the Dirichlet problem and its applications

Arturo Kohatsu-Higa (Ritsumeikan University)

The Feynman-Kac formula for the gradient of the Dirichlet problem and its applications

Let \(u(t,x )\) be the classical Dirichlet problem on a multi-dimensional smooth bounded domain \(D\) as follows: \[\begin{aligned} \partial_tu(t,x)=&\frac 12 a_{ij}(x)\partial^2_{ij}u(t,x)+b_i(x)\partial_iu(t,x)\\ u(t,x)=&0,\quad x\in \partial D\\ u(T,x)=&f(x).\end{aligned}\] Here \(f:\bar{D}\to \mathbb{R}\) is a smooth function with bounded derivatives and in the above equation we assume summation over repeated indices.

In our presentation we will discuss the Feynman-Kac formula for \(\nabla u(t,x)\) and some of its applications.

This is based on a series of ongoing research works with Dan Crisan, Fabio Antonelli, Jorge Gonzales Cazares, Ngoc Khue Tran and Hoang-Long Ngo.

Bibliography

\([1]\) Francesco Cosentino, Harald Oberhauser, Alessandro Abate (2023). Grid-Free Computation of Probabilistic Safety with Malliavin Calculus. IEEE Transactions on Automatic Control : ( Volume: 68, Issue: 10, October 2023)

\([2]\)E. Bandini, T. De Angelis, G. Ferrari. F. Gozzi (2022). Optimal dividend payout under stochastic discounting. Math. Financ, 32, 627-677.

\([3]\) Costantini, C., Gobet, E., El Karoui, N. (2006). Boundary Sensitivities for Diffusion Processes in Time Dependent Domains. Appl Math Optim 54, 159–187 (2006).

\([4]\) Crisan, D. and Kohatsu-Higa, A. A probabilistic representation of the derivative of a one dimensional killed diffusion semigroup and associated Bismut-Elworthy-Li formula. Preprint.

\([5]\) Crisan, D. and Kohatsu-Higa, A. Probabilistic representation of the gradient of a killed diffusion semigroup: The half-space case. Preprint.

\([6]\) E. Fournié, J.-M. Lasry, J. Lebuchoux, P.-L. Lions, and N. Touzi. (1999) Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch., 3(4):391–412, 1999.

\([7]\)T. Nakatsu. Integration by parts formulas concerning maxima of some SDEs with applications to study on density functions. Stoch. Anal. Appl., 34(2):293–317, 2016.

\([8]\) Taira, Kazuaki. (2020) Boundary value problems and Markov processes. Functional analysis methods for Markov processes. Third edition. Lecture Notes in Mathematics, 1499. Springer.

Statistical inference for locally stable regression

Hiroki Masuda (University of Tokyo)

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.