Spectral solution of delayed random walks

We develop a spectral method for computing the probability density function for delayed random walks; for such problems, the method is exact to machine precision and faster than existing approaches. In conjunction with a step function approximation and the weak Euler-Maruyama discretization, the spectral method can be applied to nonlinear stochastic delay differential equations (SDDE). In essence, this means approximating the SDDE by a delayed random walk, which is then solved using the spectral method. We carry out tests for a particular nonlinear SDDE that show that this method captures the solution without the need for Monte Carlo sampling.

Figure 1

Snapshots at different values of time $n$ show machine precision agreement between the densities computed using the spectral method (each plotted with a different marker) and an enumerative exact method (each plotted using the same gray scale and color as the corresponding marker). Computed densities are for the random walk Eq. (4) with delay $\ell =5$ and two types of Bernoulli steps $K_n$: outcomes $\{2,-2\}$ with probabilities $\{0.7,0.3\}$ when $Y_n\ge Y_{n-5}$, and outcomes $\{1,-1\}$ with probabilities $\{0.9,0.1\}$ when $Y_n<Y_{n-5}$. Initial conditions were $Y_n=0$ for $n\le \ell$.

Figure 2

For different delays $\ell$, densities computed at time $n=60$ using the spectral method (plotted using solid markers) agree to machine precision with densities computed using an enumerative exact method (plotted using lines of the same gray scales and colors as markers). Computed densities are for five different versions of the random walk Eq. (4), each with a different delay $\ell \in \{1,2,3,4,5\}$. The Bernoulli steps $K_n$ are as follows: outcomes $\{1,-1\}$ with probabilities $\{0.3,0.7\}$ when $Y_n\ge Y_{n-\ell }$, and outcomes $\{2,-2\}$ with probabilities $\{0.9,0.1\}$ when $Y_n<Y_{n-\ell }$. Initial conditions were $Y_n=0$ for $n\le \ell$.

Figure 3

Infinity norm errors $\parallel f_{\mathrm{IFFT}}{(n,s)-f(n,s)\parallel }_{\infty }$ between the spectral and exact pdfs are at the level of machine precision, both as a function of time $n$ and delay $\ell$. Each solid square, one per value of time step $n$, corresponds to the ${\parallel ·\parallel }_{\infty }$ error between the solid markers (spectral) and lines (exact) in Fig. 1. Each solid circle, one per value of delay $\ell$, corresponds to the ${\parallel ·\parallel }_{\infty }$ error between the solid markers (spectral) and lines (exact) in Fig. 2. All parameters are as in Figs. 1 and 2, respectively.

Figure 4

For the nonlinear SDDE Eq. (11), the accuracy of the cdf computed via the spectral method increases as we increase the number of piecewise constant branches $R$ used to approximate the $\tanh$ function. In the top panel, we plot in solid gray a fine-scale reference cdf obtained through Monte Carlo (MC) simulation with $\Delta t=0.04$ and ${10}^8$ sample paths. In dot-dashed gray and solid black, we plot the cdfs obtained using the spectral method with $R=3$ and $R=5$ approximations, respectively. In the bottom panel, we plot the pointwise errors between the spectral method cdfs and the MC cdf. The maximum error decreases from $0.0727$ to $0.0337$ as we go from $R=3$ to $R=5$. All plots are at time $t=2$ for zero initial conditions.