Linearization of one-dimensional nonautonomous jump-diffusion stochastic differential equations

Ünal, Gazanfer; Iyigunler, Ismail; Khalique, Chaudry Masood

doi:10.2991/jnmp.2007.14.3.9

Cited by 3 publications

(4 citation statements)

References 12 publications

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“…Thus the main strategy for finding exact solutions of a scalar SDE consists in looking for invertible change of both the dependent and dependent variables that convert the underlying SDE into a linear one. Such an approach was pioneer by Gard [12] and extended by several researchers [13,14,15,16,17,18].…”

Section: Hypercomplexification Of Stochastic Differential Equationsmentioning

confidence: 99%

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Hypercomplexification of SDEs Exact integration of some systems of stochastic differential equations through hypercomplexification

Soh,

Mahomed

2017

Preprint

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We leverage commutative hypercomplex analysis to find closed-form solutions of some systems of stochastic differential equations. Specifically, we obtain necessary and sufficient conditions under which a system of stochastic differential equations can be transformed into a scalar one involving processes valued in a commutative hypercomplex. In the event the targeted scalar stochastic differential equation is solved by quadratures, we recover the solution of the original system by projecting the solution of the scalar stochastic differential equation along the units of the underlying commutative hypercomplex. The conversion of a system of stochastic differential equations involving real-valued processes into a scalar one written in terms of hypercomplex-valued processes is termed hypercomplexification. Both hypercomplexification and its reverse are mediated by the analyticity of stochatic differential equations data. They may be iterated in order to generate higherdimensional integrable systems of stochastic differential equations and solve them. We showcase the utility of hypercomplexification by treating several examples including linear, and linearizable systems of stochastic differential equations and stochastic Lotka-Volterra systems. Although we consider only random systems driven by white noises, hypercomplexification is fundamentally algebraic, and it readily extends to stochastic systems involving other types of disturbances.

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Section: Hypercomplexification Of Stochastic Differential Equationsmentioning

confidence: 99%

“…Thus the strategy often adopted for finding exact solutions of a nonlinear scalar SDE is to ascertain whether it can be invertibly mapped to a linear SDE. Such an approach was pioneered by Gard [12] and extended by various researchers [13,14,15,16,17,18].…”

Section: Introductionmentioning

confidence: 99%

Hypercomplexification of SDEs Exact integration of some systems of stochastic differential equations through hypercomplexification

Soh,

Mahomed

2017

Preprint

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“…, (  and the quadratic variation Let us seek a solution to the system of PDEs (27) and (28) together with (29). Equation (29) Since t  has a continuous part, (30) implies that it is in the form…”

Section: Solution Of the Linear Equationmentioning

confidence: 99%

“…In this paper, we derive the conditions for linearizability of (2) and solve the arising linear stochastic differential equation using integrating factor method. Linearization problem has been considered in [29,30] for equations including a single Poisson jump term, which serve as preliminaries of the present work. We extend and generalize these results by considering Equation 2and demonstrate identification of a stochastic integrating factor for solving the linearized equation.…”

Section: Introductionmentioning

confidence: 99%

Exact Solvability of Stochastic Differential Equations Driven by Finite Activity Levy Processes

Iyigunler

Çaǧlar

Ünal

2012

MCA

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We consider linearizing transformations of the one-dimensional nonlinear stochastic differential equations driven by Wiener and compound Poisson processes, namely finite activity Levy processes. We present linearizability criteria and derive the required transformations. We use a stochastic integrating factor method to solve the linearized equations and provide closed-form solutions. We apply our method to a number ofstochastic differential equations including Cox-Ingersoll-Ross short-term interest rate model, log-mean reverting asset pricing model and geometric Ornstein-Uhlenbeck equation all with additional jump terms. We use their analytical solutions to illustrate the accuracy of the numerical approximations obtained from Euler and Maghsoodi discretization schemes. The means of the solutions are estimated through Monte Carlo method.

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Linearization criteria for systems of two second-order stochastic ordinary differential equations

Mkhize

Govinder

Moyo

et al. 2017

Applied Mathematics and Computation

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Linearization of one-dimensional nonautonomous jump-diffusion stochastic differential equations

Cited by 3 publications

References 12 publications

Hypercomplexification of SDEs Exact integration of some systems of stochastic differential equations through hypercomplexification

Hypercomplexification of SDEs Exact integration of some systems of stochastic differential equations through hypercomplexification

Exact Solvability of Stochastic Differential Equations Driven by Finite Activity Levy Processes

Linearization criteria for systems of two second-order stochastic ordinary differential equations

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