Strong order 1 adaptive approximation of jump-diffusion SDEs with discontinuous drift

This paper introduces a novel transformation-based, doubly-adaptive quasi-Milstein scheme for jump-diffusion SDEs with discontinuous drift and degenerate diffusion, achieving a strong convergence rate of order 1 in $L^p$ under assumptions weaker than those in existing literature.

The Gist

Here is an explanation of the paper "Strong order 1 adaptive approximation of jump-diffusion SDEs with discontinuous drift" using simple language and creative analogies.

The Big Picture: Navigating a Chaotic River

Imagine you are trying to predict the path of a leaf floating down a river. This isn't a calm, predictable stream; it's a chaotic river with three main features:

1. The Current (Drift): The water pushes the leaf in a certain direction. In this paper, the current is weird. It flows smoothly in some places, but suddenly, at specific points, the direction or speed changes abruptly (like hitting a waterfall or a sudden drop). This is the "discontinuous drift."
2. The Wind (Brownian Motion): Random gusts of wind blow the leaf left and right unpredictably. This is the standard "noise" in math.
3. The Rocks (Jumps): Occasionally, a giant rock falls into the river, instantly knocking the leaf sideways. These are sudden, random "jumps" caused by a Poisson process.

The Problem:
Mathematicians have long been able to simulate rivers with smooth currents or rivers with rocks but smooth currents. But simulating a river with both sudden jumps and sudden changes in the current (discontinuities) is incredibly hard.

If you try to use a standard map (a fixed grid) to track the leaf, you will likely miss the sudden changes. You might step right over a cliff or miss a rock entirely. To get an accurate picture, you usually have to take tiny, tiny steps everywhere, which takes a massive amount of computer time.

The Solution: The "Smart Hiker" Algorithm

The author, Verena Schwarz, has invented a new algorithm called the Transformation-based Doubly-Adaptive Quasi-Milstein Scheme. Let's break down what that mouthful of a name actually means using a hiking analogy.

1. The "Transformation" (Flattening the Cliff)

Imagine the river has a steep cliff where the water suddenly changes speed. It's hard to walk (or calculate) right at the edge of a cliff.

• The Trick: The author uses a mathematical "magic lens" (a transformation function $G$). When you look at the river through this lens, the cliff disappears! The sudden drop is smoothed out into a gentle slope.
• Why it helps: Now, the math thinks the river is smooth and easy to handle, even though the real river is jagged.

2. "Doubly-Adaptive" (The Smart Hiker)

Most algorithms are like a hiker who takes steps of the exact same size (e.g., 1 meter) no matter what. This is wasteful on flat ground and dangerous near cliffs.
This new algorithm is a Smart Hiker that adapts in two ways:

• Adaptivity #1: The "Jump" Radar (Jump-Adapted)
  The hiker knows exactly when the rocks (jumps) will fall. If a rock is scheduled to fall in 3 seconds, the hiker stops exactly at the 3-second mark to see the impact. They never miss a rock.
• Adaptivity #2: The "Cliff" Sensor (Discontinuity-Adapted)
  As the hiker gets closer to the "cliff" (the point where the current changes abruptly), they automatically take smaller and smaller steps.
  • Far from the cliff? Take big, fast steps.
  • Right next to the cliff? Take microscopic steps to ensure you don't fall off.

3. The "Quasi-Milstein" (The High-Tech Compass)

Standard hikers just look at the slope and guess the next step. This algorithm uses a "Quasi-Milstein" compass. It doesn't just look at the current slope; it also checks how the slope is changing (the curvature). This allows it to predict the path much more accurately than a standard guess.

The Result: Why This Matters

Before this paper, the best anyone could do for this specific type of chaotic river was an accuracy of 0.75.

• Analogy: If you wanted to know where the leaf was after 1 hour, the old method might be off by a few meters. To get it closer, you'd have to slow down your computer by a huge factor.

This new method achieves an accuracy of 1.0 (Strong Order 1).

• Analogy: This is the "Gold Standard." It means that if you double your computer power (by taking twice as many steps), you get twice the accuracy. It is the most efficient way possible to solve this specific problem.

Summary in a Nutshell

• The Challenge: Simulating a system that has random jumps and sudden, sharp changes in behavior.
• The Innovation: A computer program that:
  1. Smooths out the sharp corners mathematically.
  2. Stops exactly when a random jump happens.
  3. Slows down automatically when it gets close to a sharp corner.
• The Payoff: It is the first method to solve this problem with maximum efficiency and accuracy, saving massive amounts of computing time compared to older methods.

It's like upgrading from a hiker with a blurry map and a fixed stride to a hiker with a GPS, a cliff-detecting sensor, and the ability to shrink their legs to a millimeter whenever danger is near.

Technical Summary

Here is a detailed technical summary of the paper "Strong order 1 adaptive approximation of jump-diffusion SDEs with discontinuous drift" by Verena Schwarz.

1. Problem Statement

The paper addresses the numerical approximation of jump-diffusion Stochastic Differential Equations (SDEs) where the drift coefficient is discontinuous. The specific SDE considered is:
$$dX_t = \mu(X_t) dt + \sigma(X_t) dW_t + \rho(X_{t-}) dN_t, \quad t \in [0, T]$$
where:

• $W_t$ is a standard Brownian motion.
• $N_t$ is a homogeneous Poisson process with intensity $\lambda$.
• The drift $\mu$ is piecewise Lipschitz continuous with potential discontinuities at points $\zeta_1, \dots, \zeta_m$.
• The diffusion $\sigma$ and jump $\rho$ coefficients are Lipschitz continuous, with $\sigma(\zeta_k) \neq 0$ at the discontinuity points.

The Challenge: Standard numerical schemes (like Euler-Maruyama) typically achieve only a strong convergence rate of $1/2$for SDEs with discontinuous drift. Previous higher-order schemes for jump-diffusions with discontinuous drift (e.g., transformation-based jump-adapted quasi-Milstein) were limited to a strong convergence rate of$3/4$. The goal is to achieve the **optimal strong convergence rate of 1** in terms of the number of evaluations of the driving noise processes ($W$and$N$) in$L^p$ spaces.

2. Methodology

The author proposes a transformation-based doubly-adaptive quasi-Milstein scheme. The methodology combines two distinct types of adaptivity:

A. Transformation Technique

To handle the discontinuous drift, the author applies a transformation $G: \mathbb{R} \to \mathbb{R}$ to the original process $X_t$, defining a new process $Z_t = G(X_t)$.

• The transformation is constructed such that the drift coefficient of the transformed SDE becomes Lipschitz continuous (and differentiable with Lipschitz derivatives between discontinuities).
• The transformed SDE for $Z_t$ retains the jump-diffusion structure but satisfies the regularity conditions required for higher-order schemes.
• The original solution is recovered via the inverse transformation: $X_t \approx G^{-1}(Z_t)$.

B. Doubly-Adaptive Grid Strategy

The numerical scheme for $Z_t$ employs a time grid that adapts in two ways:

1. Jump-Adapted: The grid points explicitly include all jump times of the Poisson process ($N_t$). This ensures the scheme captures jumps exactly, avoiding errors associated with missing jumps between grid points.
2. Discontinuity-Adaptive: The step size $h_\delta(x)$ is a function of the distance of the current approximation to the discontinuity points $\Theta = \{\zeta_1, \dots, \zeta_m\}$.
   • Far from discontinuities: The step size is constant ($\delta$).
   • Near discontinuities: The step size decreases significantly (proportional to the square of the distance or smaller) to resolve the irregularity of the drift.
   • The next grid point is determined by $\tau_{n+1} = (\tau_n + h_\delta(Z_{\tau_n})) \wedge (\text{next jump time}) \wedge T$.

C. Quasi-Milstein Scheme

Between grid points, the scheme uses a quasi-Milstein discretization. This includes a correction term involving the derivative of the diffusion coefficient, which is essential for achieving the strong order 1 rate. The scheme is defined as:
$$Z_{\tau_{n+1}} = Z_{\tau_{n+1}-} + \tilde{\rho}(Z_{\tau_{n+1}-})(N_{\tau_{n+1}} - N_{\tau_n})$$
where $Z_{\tau_{n+1}-}$ is computed using the drift, diffusion, and the Milstein correction term based on the Brownian increment.

3. Key Contributions

1. First Strong Order 1 Scheme: This is the first scheme proven to achieve a strong convergence rate of 1 for jump-diffusion SDEs with discontinuous drift, measured by the number of evaluations of the driving noise processes.
2. Novel Combination of Adaptivity: It successfully combines jump-adaptivity (handling the Poisson noise) with discontinuity-adaptivity (handling the piecewise Lipschitz drift). Previous works could only achieve order $3/4$ when combining these features.
3. Rigorous Cost Analysis: The paper provides a detailed analysis of the computational cost, proving that the expected number of time steps $n(Z_T^{(\delta)})$ scales as $O(\delta^{-1} + E[N_T])$, ensuring the computational effort remains proportional to the inverse of the error tolerance.

4. Main Results

The paper establishes two primary theorems under the assumption that the coefficients satisfy specific piecewise Lipschitz and non-degeneracy conditions ($\sigma(\zeta_k) \neq 0$):

• Theorem 3.16 (Convergence of Transformed Process): For the transformed SDE $Z_t$, the doubly-adaptive quasi-Milstein scheme $Z_t^{(\delta)}$ satisfies:
  $$\left( \mathbb{E} \left[ \sup_{t \in [0,T]} |Z_t - Z_t^{(\delta)}|^p \right] \right)^{1/p} \leq c \delta$$
  for all $p \in [1, \infty)$.

• Theorem 4.3 (Convergence of Original Process): Applying the inverse transformation $G^{-1}$, the approximation for the original SDE $X_t$ satisfies:
  $$\left( \mathbb{E} \left[ \sup_{t \in [0,T]} |X_t - G^{-1}(Z_t^{(\delta)})|^p \right] \right)^{1/p} \leq C \delta$$
  and the computational cost satisfies:
  $$\mathbb{E}[n(G^{-1}(Z_T^{(\delta)}))] \leq \tilde{C}(\delta^{-1} + \mathbb{E}[N_T]).$$

5. Significance

• Optimality: The result recovers the optimal convergence rate of 1, which is known for SDEs with Lipschitz continuous coefficients, extending this optimality to the more challenging class of jump-diffusions with discontinuous drift.
• Theoretical Advancement: The proof requires overcoming significant technical hurdles introduced by the interaction between the jump noise and the adaptive step size near discontinuities. The author introduces series representations of coefficients and careful conditioning on filtrations to handle these terms.
• Practical Application: Such SDEs are used to model energy prices and control actions in energy markets. The proposed scheme offers a more efficient and accurate tool for simulating these systems compared to existing methods, reducing the computational cost required to achieve a specific error tolerance.

In summary, Verena Schwarz presents a breakthrough in numerical SDE theory by constructing a scheme that simultaneously adapts to jump times and drift discontinuities, achieving the theoretically optimal strong convergence rate for this class of problems.