Volterra integral equation

Short description: Operator equation in the style of Fredholm theory

Not to be confused with Product integral.

In mathematics, the Volterra integral equations are a special type of integral equations, named after Vito Volterra.[1] They are divided into two groups referred to as the first and the second kind.

A linear Volterra equation of the first kind is

f(t)=∫atK(t,s)x(s)ds

where f(t) is a given function and x(t) is to be determined. A linear Volterra equation of the second kind is

x(t)=f(t)+∫atK(t,s)x(s)ds.

In operator theory, and in Fredholm theory, the corresponding operators are called Volterra operators. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian.

A linear Volterra integral equation is a convolution equation if

x(t)=f(t)+∫t0tK(t−s)x(s)ds.

The function K in the integral is called the kernel. Such equations can be analyzed and solved by means of Laplace transform techniques.

For a weakly singular kernel of the form K(t,s)=(t2−s2)−α with 0<α<1, Volterra integral equation of the first kind can conveniently be transformed into a classical Abel integral equation.

The Volterra integral equations were introduced by Vito Volterra and then studied by Traian Lalescu in his 1908 thesis, Sur les équations de Volterra, written under the direction of Émile Picard. In 1911, Lalescu wrote the first book ever on integral equations.

Volterra integral equations find application in demography as Lotka's integral equation,[2] the study of viscoelastic materials, in actuarial science through the renewal equation,[3] and in fluid mechanics to describe the flow behavior near finite-sized boundaries.[4][5]

## Contents

* 1 Conversion of Volterra equation of the first kind to the second kind
* 2 Numerical solution using trapezoidal rule
* 3 Application: Ruin theory
* 4 Stochastic Volterra Equation
* 5 See also
* 6 References
* 7 Further reading
* 8 External links

## Conversion of Volterra equation of the first kind to the second kind

A linear Volterra equation of the first kind can always be reduced to a linear Volterra equation of the second kind, assuming that K(t,t)≠0. Taking the derivative of the first kind Volterra equation gives us:dfdt=∫at∂K∂tx(s)ds+K(t,t)x(t)Dividing through by K(t,t) yields:x(t)=1K(t,t)dfdt−∫at1K(t,t)∂K∂tx(s)dsDefining f~(t)=1K(t,t)dfdt and K~(t,s)=−1K(t,t)∂K∂t completes the transformation of the first kind equation into a linear Volterra equation of the second kind.

## Numerical solution using trapezoidal rule

A standard method for computing the numerical solution of a linear Volterra equation of the second kind is the trapezoidal rule, which for equally-spaced subintervals Δx is given by:∫abf(x)dx≈Δx2[f(x0)+2∑i=1n−1f(xi)+f(xn)]Assuming equal spacing for the subintervals, the integral component of the Volterra equation may be approximated by:∫atK(t,s)x(s)ds≈Δs2[K(t,s0)x(s0)+2K(t,s1)x(s1)+⋯+2K(t,sn−1)x(sn−1)+K(t,sn)x(sn)]Defining xi=x(si), fi=f(ti), and Kij=K(ti,sj), we have the system of linear equations:x0=f0x1=f1+Δs2(K10x0+K11x1)x2=f2+Δs2(K20x0+2K21x1+K22x2)⋮xn=fn+Δs2(Kn0x0+2Kn1x1+⋯+2Kn,n−1xn−1+Knnxn)This is equivalent to the matrix equation:x=f+Mx⟹x=(I−M)−1fFor well-behaved kernels, the trapezoidal rule tends to work well.

## Application: Ruin theory

One area where Volterra integral equations appear is in ruin theory, the study of the risk of insolvency in actuarial science. The objective is to quantify the probability of ruin ψ(u)=ℙ[τ(u)<∞], where u is the initial surplus and τ(u) is the time of ruin. In the classical model of ruin theory, the net cash position Xt is a function of the initial surplus, premium income earned at rate c, and outgoing claims ξ:Xt=u+ct−∑i=1Ntξi,t≥0where Nt is a Poisson process for the number of claims with intensity λ. Under these circumstances, the ruin probability may be represented by a Volterra integral equation of the form[6]:ψ(u)=λc∫u∞S(x)dx+λc∫0uψ(u−x)S(x)dxwhere S(⋅) is the survival function of the claims distribution.

## Stochastic Volterra Equation

We say that the process Xt is solution to a stochastic Volterra [7] equation if Xt=X0+∫0tK(t−s)b(xs)ds+∫0tK(t−s)σ(Xs)dWs

where we assume Xt to be adapted to the filtration of a probability space with the usual conditions. One model using this framework is for example the Rough Bergomi Model.

## See also

* Fredholm integral equation
* Integral equation
* Integro-differential equation

## References

1. ↑ Polyanin, Andrei D.; Manzhirov, Alexander V. (2008). Handbook of Integral Equations (2nd ed.). Boca Raton, FL: Chapman and Hall/CRC. ISBN 978-1584885078.
2. ↑ Inaba, Hisashi (2017). "The Stable Population Model". Age-Structured Population Dynamics in Demography and Epidemiology. Singapore: Springer. pp. 1–74. doi:10.1007/978-981-10-0188-8_1. ISBN 978-981-10-0187-1.
3. ↑ Brunner, Hermann (2017). Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics. Cambridge, UK: Cambridge University Press. ISBN 978-1107098725.
4. ↑ Daddi-Moussa-Ider, A.; Vilfan, A.; Golestanian, R. (6 April 2022). "Diffusiophoretic propulsion of an isotropic active colloidal particle near a finite-sized disk embedded in a planar fluid–fluid interface". Journal of Fluid Mechanics 940: A12. doi:10.1017/jfm.2022.232.
5. ↑ Daddi-Moussa-Ider, A.; Lisicki, M.; Löwen, H.; Menzel, A. M. (5 February 2020). "Dynamics of a microswimmer–microplatelet composite". Physics of Fluids 32 (2): 021902. doi:10.1063/1.5142054.
6. ↑ "Lecture Notes on Risk Theory". University of Kent. February 20, 2010. pp. 17-22. https://www.kent.ac.uk/smsas/personal/lb209/files/risk-notes-10.pdf.
7. ↑ Abi Jaber et altri (23 October 2019). "Affine Volterra processes". https://arxiv.org/pdf/1708.08796.

## Further reading

* Traian Lalescu, Introduction à la théorie des équations intégrales. Avec une préface de É. Picard, Paris: A. Hermann et Fils, 1912. VII + 152 pp.
* Hazewinkel, Michiel, ed. (2001), "Volterra equation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/v096840
* Weisstein, Eric W.. "Volterra Integral Equation of the First Kind". http://mathworld.wolfram.com/VolterraIntegralEquationoftheFirstKind.html.
* Weisstein, Eric W.. "Volterra Integral Equation of the Second Kind". http://mathworld.wolfram.com/VolterraIntegralEquationoftheSecondKind.html.
* Integral Equations: Exact Solutions at EqWorld: The World of Mathematical Equations
* Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 19.2. Volterra Equations". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8. http://apps.nrbook.com/empanel/index.html#pg=992.

## External links

* IntEQ: a Python package for numerically solving Volterra integral equations

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