Numerical method for nonlinear Kolmogorov PDEs via sensitivity analysis

TL;DR

This work analyzes the sensitivity of nonlinear Kolmogorov PDEs to small nonlinearities arising from drift and diffusion uncertainty in an epsilon-neighborhood of baseline parameters. It derives a first-order expansion v^epsilon = v^0 + epsilon partial_epsilon v^0, with partial_epsilon v^0 expressed as an expectation involving linear PDE solutions w and their Jacobians, all grounded in Feynman–Kac representations. A Monte Carlo based numerical scheme is then developed to approximate the linear and correction terms, accompanied by rigorous error and complexity analysis and demonstrated effectiveness in high dimensions up to 100. The framework combines weak/strong formulations of uncertainty via semimartingale measures, BSDE techniques, and viscosity solution theory to provide a scalable, robust method for solving nonlinear Kolmogorov PDEs in settings with parameter misspecification.

Abstract

We examine nonlinear Kolmogorov partial differential equations (PDEs). Here the nonlinear part of the PDE comes from its Hamiltonian where one maximizes over all possible drift and diffusion coefficients which fall within a $\varepsilon$-neighborhood of pre-specified baseline coefficients. Our goal is to quantify and compute how sensitive those PDEs are to such a small nonlinearity, and then use the results to develop an efficient numerical method for their approximation. We show that as $\varepsilon\downarrow 0$, the nonlinear Kolmogorov PDE equals the linear Kolmogorov PDE defined with respect to the corresponding baseline coefficients plus $\varepsilon$ times a correction term which can be also characterized by the solution of another linear Kolmogorov PDE involving the baseline coefficients. As these linear Kolmogorov PDEs can be efficiently solved in high-dimensions by exploiting their Feynman-Kac representation, our derived sensitivity analysis then provides a Monte Carlo based numerical method which can efficiently solve these nonlinear Kolmogorov equations. We establish an error and complexity analysis for our numerical method. Moreover, we provide numerical examples in up to 100 dimensions to empirically demonstrate the applicability of our numerical method.

Figures (3)

• Figure 1: Comparative analysis between the approximated solution $v^0+\varepsilon\cdot \partial_\varepsilon v^0$ and the actual counterpart $v^\varepsilon$ over varying $\varepsilon$.
• Figure 2: Stability analysis with respect to the model parameters $N,M_0,M_1,M_2$ in Algorithm \ref{['alg:ours']}. We fix $\varepsilon=0.1$ and $\gamma=\eta=1$, and consider the basis parameters $N=100$, $M_0=2.4\times 10^6$ and $M_1=M_2= 2.4\times 10^4$, as specified in Section \ref{['sec:ex1']}. In each plot, all but one parameters are fixed (e.g. in the left hand image, $M_0,M_1,M_2$ are fixed and $N$ is varied). The plots features blue lines to represent the average of estimates $v^0+\varepsilon \cdot \partial_\varepsilon v^0$ across 400 independent runs of the Python code. Moreover, the grey dashed lines depict the $L^2$ errors across the 400 runs, where the benchmark for these errors is the average value (i.e., $v^0+\varepsilon \cdot \partial_\varepsilon v^0=14.0601$) under the basis parameters.
• Figure 3: The running maximum for the error between the approximation of $v^0$ and $\partial_{\varepsilon} v^0$ across $k$ runs with $k\in \{1,\dots,1000\}$, where the benchmark for these errors is the average value for the scaled solution for $d=1$ according to \ref{['eq:univ0']}-\ref{['eq:univ2']} across the 1,000 runs. $(b^o,\sigma^o)$ are generated randomly for every run and every $d$.

Theorems & Definitions (44)

• Remark 2.3
• Remark 2.5
• Proposition 2.6
• Theorem 2.7
• Remark 2.8
• Theorem 2.9
• Remark 2.10
• Lemma 4.1
• proof
• Remark 4.2