Volterra integral equation

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) \neq 0. Taking the derivative of the first kind Volterra equation gives us:{df\over{dt}} = \int_{a}^{t}{\partial K\over{\partial t}}x(s)ds + K(t,t)x(t)Dividing through by K(t,t) yields:x(t) = {1\over{K(t,t)}}{df\over{dt}} - \int_{a}^{t}{1\over{K(t,t)}}{\partial K\over{\partial t}}x(s)dsDefining \widetilde{f}(t) = {1\over{K(t,t)}}{df\over{dt}} and \widetilde{K}(t,s) = -{1\over{K(t,t)}}{\partial K\over{\partial 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 \Delta x is given by:\int_{a}^{b}f(x)dx \approx {\Delta x\over{2}}\left[f(x_{0}) + 2\sum_{i=1}^{n-1} f(x_{i}) + f(x_{n}) \right ]Assuming equal spacing for the subintervals, the integral component of the Volterra equation may be approximated by:\int_{a}^{t}K(t,s)x(s)ds \approx {\Delta s\over{2}}\left[K(t,s_{0})x(s_{0}) + 2K(t,s_{1})x(s_{1}) + \cdots + 2K(t,s_{n-1})x(s_{n-1}) + K(t,s_{n})x(s_{n}) \right ]Defining x_{i} = x(s_{i}), f_{i} = f(t_{i}), and K_{ij} = K(t_{i},s_{j}), we have the system of linear equations:\begin{aligned} x_{0} &= f_{0} \ x_{1} &= f_{1} + {\Delta s\over{2}}\left(K_{10}x_{0} + K_{11}x_{1} \right ) \ x_{2} &= f_{2} + {\Delta s\over{2}}\left(K_{20}x_{0} + 2K_{21}x_{1} + K_{22}x_{2} \right ) \ &\vdots \ x_{n} &= f_{n} + {\Delta s\over{2}}\left(K_{n0}x_{0} + 2K_{n1}x_{1} + \cdots + 2K_{n,n-1}x_{n-1} + K_{nn}x_{n} \right ) \end{aligned}This is equivalent to the matrix equation:x = f + Mx \implies x = (I-M)^{-1}fFor 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 \psi(u) = \mathbb{P}\tau(u) , where u is the initial surplus and \tau(u) is the time of ruin. In the [classical model of ruin theory, the net cash position X_{t} is a function of the initial surplus, premium income earned at rate c, and outgoing claims \xi:X_{t} = u + ct - \sum_{i=1}^{N_{t}}\xi_{i}, \quad t \geq 0 where N_{t} is a Poisson process for the number of claims with intensity \lambda. Under these circumstances, the ruin probability may be represented by a Volterra integral equation of the form:\psi(u) = {\lambda\over{c}}\int_{u}^{\infty}S(x)dx + {\lambda\over{c}}\int_{0}^{u}\psi(u-x)S(x)dx where S(\cdot) is the survival function of the claims distribution.

Stochastic Volterra Equation

We say that the process X_t is solution to a stochastic Volterra equation if X_t = X_0 + \int_0^t K(t-s)b(x_s)ds + \int_0^t K(t-s) \sigma(X_s)dW_s

where we assume X_t 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.

References

References

1. Polyanin, Andrei D.. (2008). "Handbook of Integral Equations". Chapman and Hall/CRC.
2. Inaba, Hisashi. (2017). "Age-Structured Population Dynamics in Demography and Epidemiology". Springer.
3. Brunner, Hermann. (2017). "Volterra Integral Equations: An Introduction to Theory and Applications". Cambridge University Press.
4. (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.
5. (5 February 2020). "Dynamics of a microswimmer–microplatelet composite". Physics of Fluids.
6. (February 20, 2010). "Lecture Notes on Risk Theory". University of Kent.
7. Abi Jaber et altri. (23 October 2019). "Affine Volterra processes".