Numerically solving the integral equation $b(t) = c \int_0^\infty K( t, u, b(t), b(t+u)) du$

Question

I am reading an article where the author numerically solves the following integral equation for $b(t)$ (sec 7, p. 25): $$ b(t) = c \int_0^\infty K( t, u, b(t), b(t+u)) du, \qquad t \in [0, \infty), \qquad\qquad (1) $$ where $$ K(t, s, b(t), b(t+s)) := \Phi(d_2) - \Phi(d_1), \\[5mm] d_{1,2} := \frac {\pm \Phi^{-1}(b(t+s)) \gamma(t) \sqrt{1 + s \gamma^2(t)} - m(t)(1 + \gamma^2(t))} {\gamma^2(t) \sqrt{s + s^2 \gamma^2(t)}}, \\[5mm] \gamma(t) := \frac{\gamma}{\sqrt{1 + t\gamma^2}}, \\[5mm] m(t) := \Phi^{-1}(b(t)) \gamma(t), $$ in the case where $c = \frac 1 2$ and $\gamma = 1$. In the above $\Phi(\cdot)$ is the distribution function for the standard normal distribution.

Can someone suggest how one can approach numerically solving the integral equation $(1)$, I am not sure how to start. I have been looking through the Handbook of Integral Equations, but I don't think $(1)$ is a well-known type.

For context: this equation arises when trying to solve Bayesian sequential testing problem for the sign of the drift of a Brownian motion.

Answer 1

You may use a Gauss Laguerre quadrature to calculate it approximativelly. The quadrature is designed for integration kernels of following form

$$\int_{0}^{\infty} e^{-x} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i),$$

where

$$w_i = \frac {x_i} {\left(n + 1\right)^2 \left[L_{n+1}\left(x_i\right)\right]^2}$$

and $L_n$ is the n-th Laguerre polynomial.

A similar quadrature is the Gauss Hermite quadrature, however over the whole domain. The quadrature is designed for integration kernels of following form

$$\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i),$$

where

$$w_i = \frac {2^{n-1} n! \sqrt{\pi}} {n^2[H_{n-1}(x_i)]^2}$$

and $H_n$ is the n-th Hermite polynomial.

NOTE:

• The given quadrature rules should rather be considered as an indication. From my knownledge, they are the only ones which can handle definite ranges including $\mp \infty$. In your case, you have to consider that the quadrature is applied only on the $s$ or $u$ dependency. The $t$ dependency is not influenced by the quadrature and can be applied point-wise at a desired time, e.g. $t=1$. However, i am not quite sure if it is possible to substitute your explicit expression in the required quadrature forms. In your case you might set $f(x)=1$.

• As suggested by user619894 you still have to iterate.

Answer 2

Partial answer:

If you are lucky, a naive iteration method might work: Make an initial guess for $b(t)$, say $b_0(t)=1$, and then iterate

$b_{n+1}(t) = c \int_0^\infty K( t, u, b_n(t), b_n(t+u)) du, \qquad t \in [0, \infty)$

The numerical evaluation of the integral and the points $t_i$ at which the integral should be evaluated are tricky, and I can't help. Also, it is possible that the iteration will not converge, and you may need more sophisticated methods.