Numerical computation of probabilities for nonlinear SDEs in high dimension using Kolmogorov equation

Abstract

Stochastic Differential Equations (SDEs) in high dimension, having the structure of finite dimensional approximation of Stochastic Partial Differential Equations (SPDEs), are considered. The aim is to compute numerically expected values and probabilities associated to their solutions, by solving the associated Kolmogorov equations, with a partial use of Monte Carlo strategy - precisely, using Monte Carlo only for the linear part of the SDE. The basic idea was presented in Flandoli et al., JMAA (2020), but here we strongly improve the numerical results by means of a shift of the auxiliary Gaussian process. For relatively simple nonlinearities, we have good results in dimension of the order of 100.

Figures (8)

• Figure 1: Trajectories of $u(t,x_{0})$ for $t \in [0,1]$, and plot of the error in $\log_{10}$ scale as a function of the number of iterations. Left block: cubic bounded case \ref{['eq:polynomialbounded']} in dimension $d = 10$ without the use of the shift. We see that the solution exhibits oscillations in time making the result quite inaccurate. Right block: cubic bounded case \ref{['eq:polynomialbounded']} in dimension $d = 100$ with the addition of the shift. Here we see that, even if the dimension is much larger than the figure on Left, the result is much more stable, and the final error decreases down to the value $0.02$. For both cases the initial condition has been taken as $\phi(x) = \mathds{1}_{\left\| x \right\|_{2}\geq 1}$ and $x_{0} = \mathbf{e}$.
• Figure 2: Trajectories of $u(t,x_{0})$ for $t \in [0,1]$, and plot of the error in $\log_{10}$ scale as a function of the number of iterations. Strictly quadratic case \ref{['eq:quadraticsimple']} in dimension $d = 10$. We see that the error first rises up and reaches values of order $10^{4}$ while at the final iteration is of order $10^{-2}$. The initial condition has been taken as $\phi(x) = \mathds{1}_{\left\| x \right\|_{2}\geq 1}$ and $x_{0} = \mathbf{e}$.
• Figure 3: Trajectories of $u(t,x_{0})$ for $t \in [0,1]$, and plot of the error in $\log_{10}$ scale as a function of the number of iterations. Dyadic case \ref{['eq:dyadicmodel']} in dimension $d = 10$. The green line labeled by movmean means that a time average as been applied to smooth the solution. We see that the first iteration improve the result, even if further iterates degenerate. The initial condition has been taken as $\phi(x) = \frac{1}{d} \sum_{i=1}^{d}x_{i}$, the value of $\lambda$ is set to $1.1$, $F_{1} = 2$ and $x_{0} = \mathbf{e}_{1}$.
• Figure 4: First two Fourier components in the cubic bounded case \ref{['eq:polynomialbounded']}. Each square grid has a height corresponding to the probability for the variable $X^{x_0}_{t}$ to be inside that square. Each values has been computed by means of the approximation scheme described in Section \ref{['sec:iterationscheme']} by choosing the function $\phi$ as the indicator function of the corresponding square grid. In the figure on the right a density-like plot instead of a histogram has been used to give a better feeling of the resulting probability distribution.
• Figure 5: First two Fourier components in the cubic bounded case \ref{['eq:polynomialbounded']}. Each point correspond to an independent realization of the nonlinear process $X^{x_0}_{t}$ (blue) or of the linear one $Z^{x_0}_{t}$ (red). On the left we see the result by computing samples without the shifting the gaussian process while on the right we see the effects of the shift. As we can see on the right the samples coming from the gaussian process with shift are much closer to the samples of non linear process, providing a better initial approximation for the iteration scheme.

Theorems & Definitions (10)

• Lemma \oldthetheorem
• Lemma \oldthetheorem
• Corollary \oldthetheorem
• proof
• Theorem \oldthetheorem
• Remark \oldthetheorem
• Lemma \oldthetheorem
• proof
• proof : Proof of Lemma \ref{['thm-1']}
• proof : Proof of Lemma \ref{['lem-derivative']}