Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0510495

On Solutions of First Order Stochastic Partial Differential Equations

K. Hamza·F. C. Klebaner
View original source ↗Read source document ↗Export metadata ↓

Abstract

This note is concerned with an important for modelling question of existence of solutions of stochastic partial differential equations as proper stochastic processes, rather than processes in the generalized sense. We consider a first order stochastic partial differential equations of the form $\pd Ut = DW$, and $\pd Ut-\pd Ux= DW$, where $D$ is a differential operator and $W(t,x)$ is a continuous but non-differentiable function (field). We give a necessary and sufficient condition for stochastic equations to have solutions as functions. The result is then applied to the equation for a yield curve. Proofs are based on probability arguments.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

K. Hamza, F. C. Klebaner. 2005-10-24. On Solutions of First Order Stochastic Partial Differential Equations. https://arxiv.org/abs/math/0510495

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rough stochastic differential equations

We establish a simultaneous generalization of Itô's theory of stochastic and Lyons' theory of rough differential equations. The interest in such a unification comes from a variety of applications, including pathwise stochastic filtering, - control and the conditional analysis of stochastic systems with common noise.

math.PR↗
arXiv2026

Conditions for recurrence and transience for time-inhomogeneous random walks

The present paper extends the earlier results obtained by Abramov [`Conditions for recurrence and transience for time-inhomogeneous birth-and-death processes' \emph{Bull. Aust. Math. Soc.} \textbf{109} (2024), 393--402] for the case of time-inhomogeneous random walks, the increments of which take values in $\mathbb{R}$. By this, we give a full solution of the open problem formulated by Menshikov and Volkov [`Urn-related random walk with drift $ρx^α/t^β$' \emph{Electron. J. Probab.}, \textbf{13} (2008), paper No. 31, 944--960] that was partially solved in the aforementioned paper by Abramov.

math.PR↗
arXiv2026

Moments of traces of random symplectic matrices and hyperelliptic $L$-functions

We study matrix integrals of the form $$\int_{\mathrm{USp(2n)}}\prod_{j=1}^k\mathrm{tr}(U^j)^{a_j}\mathrm d U,$$ where $a_1,\ldots,a_r$ are natural numbers and integration is with respect to the Haar probability measure. We obtain a compact formula (the number of terms depends only on $\sum a_j$ and not on $n,k$) for the above integral in the non-Gaussian range $\sum_{j=1}^kja_j\le 4n+1$. This extends results of Diaconis-Shahshahani and Hughes-Rudnick who obtained a formula for the integral valid in the (Gaussian) range $\sum_{j=1}^kja_j\le n$ and $\sum_{j=1}^kja_j\le 2n+1$ respectively. We derive our formula using the connection between random symplectic matrices and hyperelliptic $L$-functions over finite fields, given by an equidistribution result of Katz and Sarnak, and an evaluation of a certain multiple character sum over the function field $\mathbb F_q(x)$. We apply our formula to study the linear statistics of eigenvalues of random unitary symplectic matrices in a narrow bandwidth sampling regime.

math.PR↗