Partial Integro-Differential Equations (PIDEs)

• Partial integro-differential equations (PIDEs) are equations that blend local differential operators with nonlocal integral terms, capturing jump processes in stochastic models.
• They are applied in robust probability and uncertainty quantification by using sublinear expectations and frameworks like Peng’s G‐expectation to address non-separable uncertainty sets.
• Recent recursive approximation schemes provide explicit error bounds and convergence rates, enabling quantitative analysis in robust limit theorems and numerical simulations.

Partial integro-differential equations (PIDEs) are a broad class of equations that combine the structure of partial differential equations (PDEs) with nonlocal integral terms, often originating from jump processes in stochastic modeling. In their most general form, PIDEs govern the evolution of a function $u(t,x)$ over time and space, incorporating both local differential operators and nonlocal operators associated with jump measures. Fully nonlinear, second-order PIDEs are central to modern analysis in sublinear expectations, stochastic control, and robust limit theorems, as typified by robust central limit theory, laws of large numbers under model uncertainty, and robust stable limit theorems. Classical analytic and numerical results for PIDEs have limited applicability when the uncertainty set is non-separable or degenerate. Recent advances develop approximation schemes and error bounds suited to these challenging regimes, particularly within Peng’s $G$-expectation and nonlinear Feynman–Kac frameworks.

1. Formulation of Fully Nonlinear Second-Order PIDEs

Consider the canonical fully nonlinear second-order PIDE on $[0,1]\times\mathbb{R}^{3d}$:

$\sup_{\nu\in\Theta}\Big\{\partial_{t}u(t,x,y,z) - \LL^{\nu}u(t,x,y,z)\Big\} = 0,$

with terminal condition $u(0,x,y,z)=\phi(x,y,z)$, where $u:[0,1]\times\R^{3d}\to\R$ and the “uncertainty index”

$\nu=(F_\mu, q, Q)\in\Theta\subset\LL\times\R^d\times\S^d_+,$

with:

• $\LL$: a compact convex set of $\alpha$-stable Lévy measures, $F_\mu$ having the spectral form $F_\mu(d\lambda)=\int_{S}\mu(d\theta)\int_{0}^{\infty}_{\{\lambda=r\theta\}}\frac{dr}{r^{1+\alpha}}$, $\alpha\in(1,2)$.
• $q\in\R^d$: drift.
• $Q\in\S^d_+$: diffusion, possibly degenerate.
• $\Theta$: non-separable and not a simple product set.

The nonlocal–differential operator is: $\LL^{\nu}u = \int_{\mathbb{R}^d}\left[u(t,x,y,z+\lambda)-u(t,x,y,z)-\langle D_z u(t,x,y,z),\lambda \rangle_{|\lambda|\le 1}\right]F_\mu(d\lambda) + \langle D_y u,q\rangle + \frac12\mathrm{tr}\left[D_x^2 u\, Q\right].$ Because $\Theta$ is not a Cartesian product, existing analytical and numerical splitting techniques do not apply.

2. Probabilistic Representation via Sublinear Expectation

To treat fully nonlinear PIDEs where the nonlinearity is posed as a supremum over a non-separable uncertainty set, one employs the framework of sublinear expectations:

• On $(\Omega,\mathcal{H},\widehat{\mathbb{E}})$, increments $\{(X_i,Y_i,Z_i)\}$ are i.i.d. with law uncertain in $\Theta$.
• On $$(\widetilde{\Omega},\widetilde{\mathcal{H}},\widetilde{\mathbb{E}})$$, a canonical nonlinear Lévy process $(\xi_t,\eta_t,\zeta_t)_{t\in[0,1]}$ is constructed with increments matching the scaling dictated by Brownian, deterministic, and $\alpha$-stable jump processes.

The solution $u$ admits the nonlinear Feynman–Kac formula: $$u(t,x,y,z) = \widetilde{\mathbb{E}}\left[\phi(x+\xi_t, y+\eta_t, z+\zeta_t)\right],$$ where under $\widetilde{\mathbb{E}}$, increments are generated according to the maximal-variance triplet(s) in $\Theta$. Equivalently,

$$u(t,x,y,z) = \sup_{P\in \mathcal{P}} E^P\left[\phi(X^{t,x,y,z}_1)\right],$$

where under each probability $P$, $X^{t,x,y,z}$ solves an SDE with jumps whose generator matches $\LL^\nu$.

3. Recursion-Based Approximation Scheme

A practical approximation uses a recursive, piecewise-constant time-stepping scheme:

• Fix mesh size $h>0$.
• For $t \leq h$, set $u_h(t,x,y,z) = \phi(x,y,z)$.
• For $h \le t \le 1$,

$$u_h(t,x,y,z) = \widehat{\mathbb{E}}\bigl[u_h(t-h, x+h^{1/2}X, y+hY, z+h^{1/\alpha}Z)\bigr].$$

Here, $(X,Y,Z)$ are sampled according to the maximal law in $\Theta$ encoded via the sublinear expectation $\widehat{\mathbb{E}}$.

The recursion can be rewritten as a monotone scheme: $$S\bigl(h,x,y,z, u_h(t,x,y,z), u_h(t-h, \cdot)\bigr) = 0,$$

$$S(h,x,y,z,p,v) = \frac1h \Bigl(p - \widehat{\mathbb{E}}\big[v(x+h^{1/2}X, y+hY, z+h^{1/\alpha}Z)\big]\Bigr).$$

The strong nonlinearity and the nonlocal ($\alpha$-stable) jump integral are handled implicitly via the sublinear expectation, which, in computation, requires evaluation over all extremal measures specified by $\Theta$.

4. Error Analysis and Convergence Rates

Let $u$ be the unique viscosity solution and $u_h$ the output of the recursive scheme. Under moment and consistency assumptions ($$\mathbb{E}[|X|^3]<\infty, \mathbb{E}[|Y|^2]<\infty, \mathbb{E}[|Z|^\delta]<\infty$$ for some $\delta \in (\frac{3}{4}\alpha \vee 1, \alpha)$, plus a uniform Lévy–Khintchine-type consistency), explicit error bounds are established:

$$u(t,x,y,z) - u_h(t,x,y,z) \le 2C_0 h^{1/2} + \min_{\epsilon>0}\{4C_0 \epsilon + E_1(\epsilon,h)\},$$

$$u_h(t,x,y,z) - u(t,x,y,z) \le C_0 h^{1/2} + \min_{\epsilon>0}\{4C_0 \epsilon + E_2(\epsilon,h)\}.$$

Selecting $\epsilon\sim h^\gamma$ to balance terms yields uniform convergence: $\|u-u_h\|_\infty \le C h^\beta,$ where

$$\beta = \min\Bigl\{\frac{4\delta-3\alpha}{2\alpha(2\delta+3)}, \frac{2-\alpha}{2\alpha}, \frac{q_0}{2}\Bigr\},$$

with $q_0$ a tail-regularity index depending on the increment laws. The scheme thus features an identifiable order of convergence related to the underlying jump activity and moment assumptions.

5. Application: Universal Robust Limit Theorems

The critical application of this framework is to robust limit theorems under sublinear expectations:

• For i.i.d. sums $S_n^1 = \sum_{i=1}^n X_i$, $S_n^2 = \sum_{i=1}^n Y_i$, $S_n^3 = \sum_{i=1}^n Z_i$,
• The recursive scheme yields

$$u_{1/n}(1,0,0,0) = \widehat{\mathbb{E}}\left[\phi\left(\frac{S_n^1}{\sqrt n}, \frac{S_n^2}{n}, \frac{S_n^3}{n^{1/\alpha}}\right)\right],$$

with the error

$$\left|\widehat{\mathbb{E}}\left[\phi(\cdots)\right] - \widetilde{\mathbb{E}}\left[\phi(\xi_1,\eta_1,\zeta_1)\right]\right| \le C n^{-\beta}.$$

This result quantifies, with explicit Berry–Esseen-type rates, the convergence in law under sublinear expectation, simultaneously encompassing:

• Peng’s robust central limit theorem ($\alpha=2$),
• The sublinear law of large numbers (for $Y$),
• The $\alpha$-stable law (for $Z$), all under a non-separable, potentially degenerate uncertainty set $\Theta$.

6. Mathematical and Practical Implications

This approach provides several key methodological advances for the paper and application of PIDEs in robust probability and uncertainty quantification:

• Treatment of PIDEs where the uncertainty in the Lévy measure, drift, and diffusion is non-separable and non-product, beyond the applicability of splitting or standard monotone schemes.
• All nonlocal and nonlinearities are encoded within the sublinear expectation; numerical implementation requires maximizing expectations over the admissible laws in $\Theta$—a significant but manageable computational overhead when $\Theta$ is convex and compact.
• The explicit convergence rate $\beta$ is crucial for assessing the speed of convergence in robust limit theorems and for calibrating mesh size $h$ in practical computations.

7. Extensions and Context Within Nonlinear PDE Theory

This probabilistic approximation and error analysis framework sits at the intersection of several advanced topics:

• Nonlinear Feynman–Kac formulas for viscosity solutions of PIDEs;
• Peng’s $G$-expectation theory for sublinear expectations and model uncertainty;
• Asymptotic statistics (Berry–Esseen bounds) in nonclassical environments;
• Recursive numerical schemes for nonlocal and fully nonlinear equations.

These methods open the door to quantitative analysis and simulation in robust probability, risk, and finance, especially when uncertainty is imposed not just on volatility but on jump structure, diffusion, and drift in a genuinely non-separable manner.