Strong approximation for stochastic Volterra equations by compound Poisson processes

This paper establishes a strong approximation scheme for stochastic differential and Volterra equations with merely measurable or singular time-dependent coefficients by replacing the Brownian motion with a compound Poisson process, proving explicit convergence rates and demonstrating superior stability compared to traditional Euler-Maruyama methods.

The Gist

Imagine you are trying to navigate a ship through a stormy sea. The sea represents time, and your ship is a mathematical model describing how something changes (like a stock price, a virus spread, or a chemical reaction).

Usually, to predict where your ship will be, you use a map with a grid. You check your position every hour, every minute, or every second. This is how most computer simulations work today (a method called Euler–Maruyama). You look at the map, see the wind and waves at that exact second, and take a step.

The Problem:
Sometimes, the "weather" in your model is crazy. It might have:

1. Sudden jumps: The wind instantly switches from calm to a hurricane.
2. Singularities: The wind gets infinitely strong for a split second before calming down.
3. Rough patches: The wind changes so erratically that checking it at a specific, pre-planned time (like exactly 12:00 PM) might catch you right in the middle of a chaotic spike, giving you a terrible reading.

If you use the standard "grid" method, you might keep landing your checks exactly on these bad spots, causing your simulation to crash or give wildly wrong answers. It's like trying to measure the temperature of a pot of boiling water by sticking a thermometer in at the exact moment a bubble bursts; you'll get a bad reading every time.

The Solution: The "Random Clock"
The authors of this paper, Zhang and Zhao, propose a completely different way to navigate. Instead of checking the weather on a fixed schedule (12:00, 12:01, 12:02...), they use a random clock.

Imagine your ship doesn't move in a straight line. Instead, it moves in a series of jumps.

• The Jumps: The ship waits for a random amount of time (like waiting for a bus that arrives unpredictably).
• The Randomness: When the "bus" (a Poisson process) arrives, the ship jumps.
• The Magic: Because the jumps happen at random times, it is statistically very unlikely that you will keep landing on the exact same "bad" spots in the weather. You might hit a rough patch once, but the next jump will likely land you in calm water.

Furthermore, instead of just jumping forward, the ship also gets a little "kick" from the wind (the Brownian motion) at every jump. The authors found a clever way to combine these random jumps with the wind kicks so that, on average, the ship follows the exact same path as the original, smooth model.

The Compound Poisson Approximation
The technical name for this method is a Compound Poisson Approximation. Here is the simple breakdown:

• Compound: The ship's movement is a mix of two things: the random waiting time (the clock) and the random wind kick (the jump size).
• Poisson: This is just a fancy word for a process where events happen randomly over time, like raindrops hitting a roof or phone calls coming into a call center.

Why is this better?

1. It ignores the "bad times": If the wind is chaotic at 12:00:00, the standard method must check it. The random method might check at 12:00:03 or 11:59:58, avoiding the chaos.
2. It handles "Infinite" spikes: If the wind speed theoretically goes to infinity for a split second, a fixed grid might get stuck trying to calculate it. The random method just skips over it naturally because the probability of landing exactly on that split second is zero.
3. It works for "Memory" problems: Some systems (like Volterra equations) depend not just on the current moment, but on the entire history of the journey. The authors showed that even with these complex "memory" effects, their random clock method works better than the old grid method when the history is messy.

The Results
The authors proved mathematically that this random-jump method converges to the true answer. They also ran computer simulations (experiments) where they tested the method on equations with "singular" (infinite) spikes.

• The Old Way (Euler–Maruyama): The simulation got noisy and inaccurate near the spikes.
• The New Way (Compound Poisson): The simulation stayed smooth and accurate, even when the math got crazy.

The Analogy Summary

• The Old Method: Trying to walk across a field of landmines by stepping exactly on the grid lines. If a mine is on a grid line, you blow up.
• The New Method: Walking across the same field, but taking random, slightly unpredictable steps. You might step near a mine, but you are unlikely to step on the same mine twice, and you can navigate around the danger zones much more safely.

In short, this paper introduces a robust, "fuzzy" way to solve complex math problems that are too messy for the rigid, "sharp" methods we usually use. It turns a rigid schedule into a flexible, random dance that somehow ends up in the right place.

Technical Summary

Here is a detailed technical summary of the paper "Strong Approximation for Stochastic Volterra Equations by Compound Poisson Processes" by Xicheng Zhang and Yuanlong Zhao.

1. Problem Statement

The paper addresses the numerical approximation of Stochastic Differential Equations (SDEs) and Stochastic Volterra Equations (SVEs) where the coefficients exhibit irregular time dependence. Specifically, the authors consider scenarios where:

• The drift coefficient $b(t, x)$ is merely measurable in time, potentially containing jump discontinuities (e.g., regime switching) or integrable singularities of the form $|t-t_0|^{-\alpha}$.
• The diffusion coefficient $\sigma(t, x)$ may lack time continuity.
• For SVEs, the kernel may be singular (e.g., fractional Brownian motion kernels).

The Challenge: Classical time-stepping methods, such as the Euler–Maruyama (EM) scheme, rely on evaluating coefficients on a deterministic grid. When coefficients are irregular in time, fixed grids may repeatedly sample "bad" points (discontinuities or spikes), leading to instability or a failure of strong convergence. Standard strong error analysis for EM typically requires Hölder continuity in time, which these models violate.

2. Methodology: Compound Poisson Time Randomization

The authors propose a novel discretization principle based on time randomization using a Compound Poisson Process.

• The Poisson Clock: Instead of a deterministic grid, time is driven by a Poisson process $N_t$ with intensity $1/\varepsilon$. The jump times are$S_k^\varepsilon = \varepsilon \sum_{i=1}^k T_i$, where$T_i \sim \text{Exp}(1)$.
• Approximation of Brownian Motion: The Brownian motion $W_t$ is approximated by a time-changed process $W_{N_t^\varepsilon}$. The increments of this process are Gaussian with variance proportional to the jump size $\varepsilon$.
• The Scheme:
  • For SDEs: The equation $dX_t = \sigma(t, X_t)dW_t + b(t, X_t)dt$ is approximated by:
    $$X_t^\varepsilon = X_0 + \int_0^t \sigma(s, X_{s-}^\varepsilon) dW_{N_s^\varepsilon} + \int_0^t b(s, X_{s-}^\varepsilon) dN_s^\varepsilon$$
    This leads to a fully discrete jump scheme where updates occur at random times $S_k^\varepsilon$.
  • For SVEs: The scheme is adapted to the two-time structure of the kernel $K(t, s)$, evaluating the kernel at the random jump times while maintaining the dependence on the terminal time $t$.

Key Distinction from EM: The method does not require pointwise evaluation of coefficients on a fixed grid. Instead, it relies on the probabilistic properties of the random clock. The error is controlled via moment bounds of the clock fluctuation ($N_t^\varepsilon - t$) rather than pointwise time continuity of the drift.

3. Key Contributions

1. Strong Convergence with Explicit Rates: The authors prove strong convergence (in $L^{2p}$) for both SDEs and SVEs under minimal regularity assumptions.
   • SDEs: Convergence rate is $O(\varepsilon^{\beta p/2})$, where $\beta$ reflects the temporal regularity of the diffusion coefficient. If the drift is discontinuous, the scheme remains stable.
   • SVEs: Convergence rate is $O(\varepsilon^{\gamma/(2(2+\gamma))})$, where $\gamma$ depends on the temporal regularity of the kernel and the coefficients.
2. Handling Irregular Drifts: Unlike EM, the proposed scheme does not require the drift $b(t, x)$ to be continuous in time. It accommodates jump discontinuities and integrable singularities ($|t-t_0|^{-\alpha}$) naturally.
3. Adaptation to Volterra Structure: The authors develop a specific discretization argument for SVEs to handle the non-Markovian nature and the two-time dependence $(t, s)$ of the kernel, which prevents a direct application of SDE proof techniques.
4. Numerical Validation: The theory is applied to equations driven by Fractional Brownian Motion (fBm) and singular kernels, with numerical experiments demonstrating superior performance over EM in singular regimes.

4. Main Results

Theorem 1.1 (SDEs)

Under global Lipschitz/linear growth conditions in space and a specific temporal regularity condition on the diffusion coefficient $\sigma$ (Hölder-like with exponent $\beta$), the compound Poisson approximation $X_t^\varepsilon$ satisfies:
$$\mathbb{E}|X_t^\varepsilon - X_t|^{2p} \leq C \varepsilon^{\frac{\beta p}{2}}$$

• Significance: If $\sigma$ is Hölder continuous in time with exponent $\alpha$ (via a transformation $t^\alpha$), the rate is optimal ($\varepsilon^{p/2}$). Crucially, no continuity is required for the drift $b$.

Theorem 1.3 (SVEs)

For SVEs with singular kernels satisfying specific integrability conditions ($H^\gamma_1, H^\gamma_2, H^\gamma_3$), the approximation $Y_t^\varepsilon$ satisfies:
$$\mathbb{E}|Y_t^\varepsilon - Y_t|^2 \leq C \varepsilon^{\frac{\gamma}{2(2+\gamma)}}$$

• Significance: The rate reflects a trade-off between the singularity of the kernel (parameter $\gamma$) and the intrinsic fluctuation of the Poisson clock.

Application to Fractional Brownian Motion (Theorem 4.1)

The authors verify the assumptions for SVEs driven by fBm with Hurst parameter $H \in (0, 1) \setminus \{1/2\}$.

• The convergence rate is determined by $\gamma = \beta \wedge (H \wedge |1-2H| \wedge (2-2H))$.
• This provides a rigorous convergence guarantee for fBm-driven equations, a class of problems where standard Itô calculus fails.

5. Numerical Experiments

The paper presents two numerical experiments comparing the Compound Poisson scheme against the Euler–Maruyama method:

1. Linear SDE with Singular Drift: A model with drift $\mu(t)$ containing integrable singularities ($|t-s_0|^{-\alpha}$) and jump discontinuities.
   • Result: The Poisson scheme accurately tracks the analytical mean and second moment, while the EM scheme shows significant deviation and instability near the singularities.
2. Linear SVE with Singular Kernel: A model with a kernel $(t-s)^{-\alpha}|s-s_0|^{-\beta}$.
   • Result: The Poisson approximation achieves visibly higher accuracy. The EM scheme struggles with the singular kernel, whereas the random time discretization of the Poisson scheme averages out the singularities effectively.

6. Significance and Impact

• Robustness: The method offers a robust alternative for models with "rough" or "spiky" time dependence, common in finance (regime switching), physics (memory effects), and biology.
• Theoretical Advancement: It bridges the gap between stochastic analysis and numerical methods for non-smooth coefficients, proving that strong convergence is possible without time-regularity assumptions on the drift.
• Practical Implementation: The algorithm is simple to implement (Algorithm 1 & 2 in the paper), requiring only the generation of exponential random variables and Gaussian jumps, making it a viable tool for complex simulations where standard grids fail.

In summary, this work establishes a new paradigm for simulating stochastic equations with irregular time dependence by replacing deterministic time grids with a random Poisson clock, thereby bypassing the stability issues inherent in classical Euler-type methods.