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An Euler scheme for BSDEs via the Wiener chaos decomposition

This paper proposes a Wiener chaos decomposition-based Euler scheme for solving backward stochastic differential equations (BSDEs) with arbitrary square-integrable terminal conditions, offering a comprehensive convergence analysis and numerical validation as an alternative to traditional forward-backward Markovian approaches.

Original authors: Pere Diaz-Lozano, Giulia Di Nunno

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Pere Diaz-Lozano, Giulia Di Nunno

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the weather for a specific day in the future, but you don't have a perfect crystal ball. Instead, you have a complex set of rules (mathematical equations) that describe how the weather might evolve, starting from today and working backward to that future day. In the world of finance and mathematics, this is called a Backward Stochastic Differential Equation (BSDE). It's like trying to figure out the exact price of a complex financial product today, knowing that its final value tomorrow depends on a chaotic, random market.

The problem is that solving these equations is incredibly hard. You have to calculate "what is the average outcome if this random thing happens?" at every single step of the way. Doing this with standard computer methods is like trying to navigate a maze by checking every single possible path one by one—it gets slow, messy, and often requires you to build a giant, complicated map (a "forward process") just to get to the answer.

The Authors' New Idea: The "Chaos" Recipe

Pere Diaz-Lozano and Giulia Di Nunno propose a new way to solve this puzzle. Instead of building a giant map of the future, they use a technique called Wiener Chaos Decomposition.

Think of a complex, messy sound (like a noisy crowd) as a combination of simple, pure musical notes. Even though the crowd sounds chaotic, it's actually just a mix of specific frequencies. The authors' method treats the random future outcomes of the financial equation like that noisy crowd. They break the randomness down into a "recipe" of simple, standard building blocks (mathematical shapes called Hermite polynomials).

Here is how their method works, step-by-step:

The Recipe Book (The Decomposition): Instead of guessing the future, they write the future outcome as a sum of these simple building blocks. It's like saying, "The final price isn't just a random number; it's 50% of 'Note A', 30% of 'Note B', and 20% of 'Note C'."
The Truncation (Simplifying the Recipe): A perfect recipe might have infinite notes. The authors say, "Let's just use the top 10 most important notes." This makes the math manageable. They call this "truncating the chaos."
The Monte Carlo Tasting (Estimating the Ingredients): To know exactly how much of "Note A" or "Note B" is in the final mix, they run thousands of computer simulations (like tasting the soup many times to get the average flavor). This gives them the coefficients (the numbers) for their recipe.
The Backward Walk: Once they have the recipe, they work backward from the end of the time period to the beginning. Because they have the recipe, they can calculate the "average" and the "randomness" at each step very quickly using formulas, rather than guessing.

Why is this better than the old way?

1. No Need for a "Forward" Map:
Most existing methods require you to simulate a "forward" process first (like simulating the stock price day-by-day) to figure out the backward price. This is like trying to find your way out of a forest by first mapping every tree in the forest.
The authors' method skips the map. It works directly with the final outcome (the "terminal condition"). Whether the final price depends on the stock price today, the price yesterday, or a weird combination of the last 100 days, their method handles it without needing to build a complex, high-dimensional map. It's like solving the puzzle by looking at the picture on the box, not by tracing every tree in the forest.

2. Handling the "Curse of Complexity":
When you have many variables (like 5 different stocks instead of 1), the old methods often break down or become impossibly slow because the "map" gets too big. The authors' method avoids this specific trap of "state augmentation" (making the map bigger to remember the past). However, they do admit that if you have too many notes in your recipe (too many dimensions or too high a complexity), the number of ingredients to calculate still grows fast.

The Results: A Taste Test

The authors tested their new "recipe" against two other popular methods:

The Picard Method: An older, iterative approach that tries to guess the answer, check it, guess again, and check again.
The Deep Learning Method: A modern approach using artificial intelligence to learn the pattern.

In their experiments (which included pricing options on one asset and even five assets at once), their new method:

Was faster: It took about 3 seconds to run a test where the older method took 7 seconds.
Was just as accurate: The results were very close to the "gold standard" benchmark and the deep learning method.
Was more stable: When they changed the settings (like the number of steps or the complexity of the recipe), their method settled on a stable answer more quickly than the older iterative method.

The Bottom Line

This paper presents a new, efficient tool for solving complex financial equations. Instead of building a massive, forward-looking map of every possible future, the authors break the problem down into a simple, manageable "recipe" of random building blocks. They then use computer simulations to figure out the ingredients of that recipe.

It's a clever shortcut that allows mathematicians and financial engineers to solve difficult problems faster and with less computational clutter, especially when the final outcome depends on complicated, messy history rather than just a simple, straight line.

Technical Summary: An Euler Scheme for BSDEs via the Wiener Chaos Decomposition

Problem Statement

The paper addresses the numerical approximation of Backward Stochastic Differential Equations (BSDEs) of the form:
$Y_t = \xi + \int_t^T f(s, Y_s, Z_s)ds - \int_t^T Z_s \cdot dB_s, \quad t \in [0, T]$
where $B$ is a $d$ -dimensional Brownian motion, $\xi$ is an $\mathcal{F}_T$ -measurable square-integrable terminal condition, and $f$ is a measurable generator.

The primary challenge in implementing standard Euler-type discretizations for BSDEs lies in the computation of conditional expectations and martingale terms at each time step. Existing literature predominantly relies on a forward-backward (Markovian) structure, where the solution is represented as deterministic functions of a forward state process $X_t$ . While effective for Markovian problems, this approach requires state augmentation for path-dependent terminal conditions, often leading to high-dimensional systems that are computationally expensive. Furthermore, many existing methods (e.g., least-squares regression, neural networks) lack explicit error quantification regarding the specific approximation tools used (time discretization, basis truncation, Monte Carlo sampling).

Methodology

The authors propose a numerical scheme that bypasses the introduction of a forward process. Instead, the method directly applies the Wiener chaos decomposition to the random variables arising in the backward recursion.

1. Backward Euler Discretization

The scheme begins with the standard backward Euler discretization on a time grid $\pi = \{t_i\}$ . The recursion defines intermediate variables $F^\pi_i$ :
$F^\pi_i = Y^\pi_{t_i} + \Delta_i f(t_i, Y^\pi_{t_i}, \langle Z \rangle^\pi_i)$
where $\langle Z \rangle^\pi_i$ represents the conditional expectation of the martingale term. The solution $Y^\pi_t$ and $Z^\pi_t$ are recovered via conditional expectations and the martingale representation theorem.

2. Wiener Chaos Projection

Rather than approximating deterministic functions of a forward state, the authors approximate the random variables $F^\pi_i$ using a finite-dimensional Wiener chaos expansion. For a random variable $F \in L^2(\mathcal{F}_{t_i})$ , the expansion is truncated to order $P$ and projected onto a basis of size $M$ :
$F^\pi_i \approx \sum_{k=0}^P \sum_{|a|=k, \#a=M} d_a \prod_{j=1}^M H_{a_j} \left( \int_0^{t_i} h^i_j(s) \cdot dB_s \right)$
Here, $H_k$ are Hermite polynomials, and $(h^i_j)$ is a truncated orthonormal basis of $L^2([0, t_i]; \mathbb{R}^d)$ .

3. Closed-Form Implementation

A key feature of the proposed method is the derivation of closed-form formulae for the conditional expectations and martingale terms ( $Y^\pi_t, Z^\pi_t, \langle Z \rangle^\pi_i$ ) based on the chaos coefficients. By selecting a piecewise constant orthonormal basis, the conditional expectations reduce to explicit algebraic operations on the chaos coefficients, avoiding the need for regression or neural network training at each step.

4. Monte Carlo Approximation

Since the chaos coefficients involve expectations that cannot be computed analytically for general generators, the authors employ a Monte Carlo simulation to estimate these coefficients. To ensure independence and avoid bias, the method utilizes independent Brownian motions for the simulation of coefficients at each time step.

Key Contributions

1. General Terminal Conditions

The method accommodates arbitrary square-integrable $\mathcal{F}_T$ -measurable terminal conditions without requiring a forward representation. This eliminates the need for state augmentation when dealing with path-dependent payoffs, thereby avoiding the "curse of dimensionality" associated with high-dimensional Markovian systems derived from path-dependent conditions.

2. Comprehensive Convergence Analysis

The paper provides a rigorous convergence analysis that quantifies errors from all sources:

Time Discretization: Standard Euler error ( $O(|\pi|)$ ).
Chaos Truncation: Error due to limiting the chaos order to $P$ and the basis size to $M$ .
Monte Carlo Error: Error arising from the finite sample size $N$ used to estimate coefficients.

The main error bound (Theorem 4.12) for the fully implementable scheme is given by:
$\text{Error} \leq C \left( |\pi| + \frac{1}{P+1} + \frac{1}{M^{2\beta}} + \frac{C_{P,M}}{N} \right)$
where $\beta$ relates to the Hölder regularity of the chaos kernels, and $C_{P,M}$ depends on the number of chaos terms and the generator's bounds.

3. Comparison with Picard-Type Schemes

The authors compare their Euler-based approach with the Picard-type scheme of [BL14], which also uses Wiener chaos.

Theoretical Advantage: The Euler scheme's analysis includes the error from time integral discretization, whereas the Picard analysis in [BL14] assumes exact time integration.
Practical Advantage: The Euler scheme allows for direct control of the solution at each time step. New samples can be generated and discarded at each step, making it more memory-efficient compared to the Picard scheme, which often requires storing intermediate iterates or rerunning the full iteration loop.

Results and Numerical Experiments

The paper validates the method through three numerical examples:

Linear Driver (1D): Pricing a discrete down-and-out call option. The Euler scheme and the Picard scheme produced similar means and distributions close to the Monte Carlo benchmark. The Euler scheme was faster (3s vs. 7s per run).
Non-linear Driver (1D): A BSDE with $\xi = \max |B_{kT/10}|$ and $f(t,y,z) = \cos(y+z)$ . The study varied parameters $m$ (time steps), $M$ (basis size), and $P$ (chaos order). The Euler scheme generally stabilized faster with respect to $m$ and $M$ and exhibited less variability than the Picard scheme.
Non-linear Driver (5D): Pricing a call on the maximum of five assets with different borrowing and lending rates. The Euler scheme was compared against the Picard scheme and the Deep BSDE (DBDP1) method. All three methods yielded similar approximations for $Y_0$ and $Z_0$ . The Euler scheme completed a run in ~~1.2 minutes, significantly faster than the Picard scheme (~~4.3 minutes).

Significance and Claims

The paper claims that its primary significance lies in providing a direct, implementable numerical scheme for BSDEs that:

Does not rely on a forward-backward Markovian structure, thus handling general terminal conditions naturally.
Offers explicit control over numerical errors through a comprehensive convergence analysis that tracks time, chaos, and Monte Carlo errors simultaneously.
Is computationally efficient and memory-friendly compared to Picard-type chaos methods, particularly for high-dimensional problems where state augmentation is prohibitive.

The authors acknowledge that the method is subject to the curse of dimensionality regarding the Brownian dimension $d$ due to the growth of terms in the truncated chaos expansion. However, they demonstrate that for moderate dimensions and appropriate parameter choices, the method yields accurate and stable results.