IS18: SDEs: Analysis, Approximation, Inference
Organizer: Oleksii Kulyk (Wrocław University of Science and Technology)
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.
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.
The Feynman-Kac formula for the gradient of the Dirichlet problem and its applications
Arturo Kohatsu-Higa
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.