50 citations · 202 across the 30 of their papers we have counts for
32 papers
Linear systems, determinants and solutions of the Kadomtsev-Petviashvili equation
Gordon Blower, Simon J. Malham
Let be a linear system in continuous time with input and output space and state space . The scattering (or impulse response) functions $ϕ_{(x)}(t)…
The noncommutative KP hierarchy and its solution via descent algebra
Gordon Blower, Simon J. A. Malham
We give the solution to the complete noncommutative Kadomtsev--Petviashvili (KP) hierarchy. We achieve this via direct linearisation which involves the Gelfand--Levitan--Marchenko…
Direct Inversion for Chi-Square Sampling, the Squared Bessel Process, and Applications
Simon J. A. Malham, Anke Wiese, Yifan Xu
In this paper we derive a new direct inversion method to simulate squared Bessel processes. Since the transition probability of these processes can be represented by a non-central…
Direct linearisation of the non-commutative Kadomtsev-Petviashvili equations
Gordon Blower, Simon J. A. Malham
We prove that the non-commutative Kadomtsev-Petviashvili (KP) equation and a `lifted' modified Kadomtsev-Petviashvili (mKP) equation are directly linearisable, and thus integrable…
Coagulation, non-associative algebras and binary trees
Simon J. A. Malham
We consider the classical Smoluchowski coagulation equation with a general frequency kernel. We show that there exists a natural deterministic solution expansion in the non-associa…
The algebraic structure of the non-commutative nonlinear Schrodinger and modified Korteweg-de Vries hierarchy
Gordon Blower, Simon J. A. Malham
We prove that each member of the non-commutative nonlinear Schrodinger and modified Korteweg--de Vries hierarchy is a Fredholm Grassmannian flow, and for the given linear dispersio…
Applications of Grassmannian flows to coagulation systems
Anastasia Doikou, Simon J. A. Malham, Ioannis Stylianidis +1
We demonstrate how many classes of Smoluchowski-type coagulation models can be realised as multiplicative Grassmannian flows and are therefore linearisable, and thus integrable in…
The non-commutative Korteweg--de Vries hierarchy and combinatorial Poppe algebra
Simon J. A. Malham
We give a constructive proof, to all orders, that each member of the non-commutative potential Korteweg-de Vries hierarchy is a Fredholm Grassmannian flow and is therefore linearis…
Integrability of local and nonlocal non-commutative fourth order quintic nonlinear Schrodinger equations
Simon J. A. Malham
We prove integrability of a generalised non-commutative fourth order quintic nonlinear Schrodinger equation. The proof is relatively succinct and rooted in the linearisation method…
Series expansions and direct inversion for the Heston model
Simon J. A. Malham, Jiaqi Shen, Anke Wiese
Efficient sampling for the conditional time integrated variance process in the Heston stochastic volatility model is key to the simulation of the stock price based on its exact dis…
Grassmannian flows and applications to non-commutative non-local and local integrable systems
Anastasia Doikou, Simon J. A. Malham, Ioannis Stylianidis
We present a method for linearising classes of matrix-valued nonlinear partial differential equations with local and nonlocal nonlinearities. Indeed we generalise a linearisation p…
Time evolution in quantum systems and stochastics
Anastasia Doikou, Simon J. A. Malham, Anke Wiese
The time evolution problem for non-self adjoint second order differential operators is studied by means of the path integral formulation. Explicit computation of the path integral…
Applications of Grassmannian flows to integrable systems
Anastasia Doikou, Simon J. A. Malham, Ioannis Stylianidis +1
We show how many classes of partial differential systems with local and nonlocal nonlinearities are linearisable in the sense that they are realisable as Fredholm Grassmannian flow…
Stochastic analysis & discrete quantum systems
Anastasia Doikou, Simon J. A. Malham, Anke Wiese
We explore the connections between the theories of stochastic analysis and discrete quantum mechanical systems. Naturally these connections include the Feynman-Kac formula, and the…
Partial differential systems with nonlocal nonlinearities: Generation and solutions
Margaret Beck, Anastasia Doikou, Simon J. A. Malham +1
We develop a method for generating solutions to large classes of evolutionary partial differential systems with nonlocal nonlinearities. For arbitrary initial data, the solutions a…
Grassmannian flows and applications to nonlinear partial differential equations
Margaret Beck, Anastasia Doikou, Simon J. A. Malham +1
We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, fro…
Algebraic Structures and Stochastic Differential Equations driven by Levy processes
Charles Curry, Kurusch Ebrahimi-Fard, Simon J. A. Malham +1
We construct an efficient integrator for stochastic differential systems driven by Levy processes. An efficient integrator is a strong approximation that is more accurate than the…
The exponential Lie series for continuous semimartingales
Kurusch Ebrahimi-Fard, Simon J. A. Malham, Frederic Patras +1
We consider stochastic differential systems driven by continuous semimartingales and governed by non-commuting vector fields. We prove that the logarithm of the flowmap is an expon…
Flows and stochastic Taylor series in Ito calculus
Kurusch Ebrahimi-Fard, Simon J. A. Malham, Frederic Patras +1
For stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the Ito flow map is given. A similar formula is also obtained for solutions of…
Levy Processes and Quasi-Shuffle Algebras
Charles Curry, Kurusch Ebrahimi-Fard, Simon J. A. Malham +1
We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfill the…
Computing the Maslov index for large systems
Margaret Beck, Simon J. A. Malham
We address the problem of computing the Maslov index for large linear symplectic systems on the real line. The Maslov index measures the signed intersections (with a given referenc…
Algebraic structure of stochastic expansions and efficient simulation
Kurusch Ebrahimi-Fard, Alexander Lundervold, Simon J. A. Malham +2
We investigate the algebraic structure underlying the stochastic Taylor solution expansion for stochastic differential systems.Our motivation is to construct efficient integrators.…
Evans function and Fredholm determinants
Issa Karambal, Simon J. A Malham
We explore the relationship between the Evans function, transmission coefficient and Fredholm determinant for systems of first order linear differential operators on the real line.…
Efficient almost-exact Levy area sampling
Simon J. A. Malham, Anke Wiese
We present a new method for sampling the Levy area for a two-dimensional Wiener process conditioned on its endpoints. An efficient sampler for the Levy area is required to implemen…