Asymptotic Expansion Prior Initialization

• Asymptotic Expansion Prior Initialization is a method that uses analytic series expansions to derive principled initial values for computational and neural models.
• It computes asymptotic coefficients via binomial convolutions, ensuring enhanced efficiency and accuracy in solving ordinary and backward stochastic differential equations.
• Integrating the AE prior in deep learning architectures accelerates convergence and reduces error rates significantly in high-dimensional financial and PDE/BSDE applications.

Asymptotic Expansion (AE) Prior Initialization refers to a methodology in which analytic asymptotic expansions—typically derived from classical perturbation techniques or power-series analysis—are utilized to provide principled prior values in computational, algorithmic, or statistical workflows. The technique achieves rigorous analytic initialization for both coefficients in series approximations and neural parameter settings, with significant efficiency and accuracy gains when solving ordinary, stochastic, or backward stochastic differential equations (BSDEs), especially in high-dimensional setups. This approach combines explicit recursive expansions with algorithmic injection into learning or computation pipelines, removing the necessity for uninformed, random initialization.

1. Mathematical Foundation of Asymptotic Expansion

Asymptotic expansion in the context of analytic functions addresses the representation of a function $f(x)$, initially defined by its Taylor series around a point $x_0$,

$f(x) = \sum_{n=0}^\infty c_n (x - x_0)^n,$

in terms of negative powers for large $x$:

$$f(x) \sim \sum_{k=0}^\infty q_k (x - x_0 + 1)^{-k}, \qquad x \to +\infty.$$

The coefficients $q_k$ in the asymptotic expansion are given in terms of the derivatives at $t = 1$ of the asymptotically associated function,

$u(t) = \sum_{n=0}^\infty c_n^* (t - 1)^n,$

with the associated coefficients,

$$c_0^* = c_0, \qquad c_n^* = \sum_{k=1}^n (-1)^{n-k} \binom{n-1}{k-1} c_k, \quad n \geq 1.$$

A sufficient condition for the existence of the asymptotic expansion is the analyticity of this associated power series $u(t)$ at $x_0$0; specifically, the radius of convergence $x_0$1 of $x_0$2 must satisfy $x_0$3 about $x_0$4 (Nikitin, 2010).

2. Algorithmic Derivation and Implementation

Computation of the asymptotic coefficients $x_0$5 follows a two-stage convolution:

• Stage 1: Compute the associated coefficients $x_0$6 via binomial sums over the original Taylor coefficients.
• Stage 2: Derive $x_0$7 from $x_0$8 by another binomial transform,

$x_0$9

Alternatively, a single “unitary” formula (a double sum in $f(x) = \sum_{n=0}^\infty c_n (x - x_0)^n,$0) can be used:

$f(x) = \sum_{n=0}^\infty c_n (x - x_0)^n,$1

An explicit pseudocode for the computation is as follows (Nikitin, 2010): $$f(x) \sim \sum_{k=0}^\infty q_k (x - x_0 + 1)^{-k}, \qquad x \to +\infty.$$4 The computational complexity is $f(x) = \sum_{n=0}^\infty c_n (x - x_0)^n,$2, with possible acceleration via FFT-based convolutions for large $f(x) = \sum_{n=0}^\infty c_n (x - x_0)^n,$3.

3. AE Prior Initialization in Deep Learning for BSDEs

Asymptotic expansion as prior knowledge is integrated into the "deep BSDE solver," a neural method for high-dimensional BSDEs (Fujii et al., 2017). The driver $f(x) = \sum_{n=0}^\infty c_n (x - x_0)^n,$4 of the BSDE is expressed in terms of a formal perturbation parameter $f(x) = \sum_{n=0}^\infty c_n (x - x_0)^n,$5,

$f(x) = \sum_{n=0}^\infty c_n (x - x_0)^n,$6

and solutions are expanded as power series in $f(x) = \sum_{n=0}^\infty c_n (x - x_0)^n,$7:

$f(x) = \sum_{n=0}^\infty c_n (x - x_0)^n,$8

Each $f(x) = \sum_{n=0}^\infty c_n (x - x_0)^n,$9 is determined recursively from linear BSDEs, with $x$0 yielding the linearized problem via Feynman–Kac formulas.

In the actual deep solver, the AE-based path $x$1 is precomputed at each time slice and injected into the neural architecture:

$x$2

where $x$3 are residual networks initialized near zero, so that the network learns only corrections to the AE path.

4. Quantitative Impact and Case Studies

Integrating AE priors drastically reduces loss and accelerates convergence in high-dimensional financial PDE/BSDE solvers. Key findings for the deep BSDE solver include (Fujii et al., 2017):

Problem	Without AE	With AE
1D European call (5,000 iter)	Loss ≈ 10, error ∼5%	Loss ≈ 1.7, error ∼1%
30D European call	Slow/unstable, high loss	Stable, error ∼0.5% in ~5,000 steps
50D American basket call (2k iter)	Error >1%, instability	Error <1%, $x$4 ≈ 11.08 vs 11.098 (bench)
50D quadratic-growth BSDE	>10× error after 30,000 iterations	0.1% error in few thousand iterations

In all tested settings—including Bergman’s FVA model, quadratic-growth BSDEs, and American options—AE initialization provided 5–10× faster loss reduction and one to two orders of magnitude improvement in pricing accuracy.

5. Extensions and Theoretical Boundaries

The AE method generalizes to arbitrary shifts $x$5 or to expansions involving more general systems (e.g., powers times logarithms); however, the corresponding unitary formulas must be re-derived for non-power expansions (Nikitin, 2010).

If the analytically associated function $x$6 has radius of convergence $x$7 about $x$8, the formal expansion may exist but lacks asymptotic validity; Padé-type or hyperasymptotic methods must then be considered. Highly oscillatory or branched $x$9 require contour-based rather than direct series-based approaches.

For reflected BSDEs (American options), the AE expansion of the linear part remains unchanged, and the loss function is augmented by lateral-penalty terms. Neural training proceeds via injection of AE paths and residual learning (Fujii et al., 2017).

6. Numerical Stability and Practical Considerations

Cancellation in the alternating binomial sums (for both $$f(x) \sim \sum_{k=0}^\infty q_k (x - x_0 + 1)^{-k}, \qquad x \to +\infty.$$0 and $$f(x) \sim \sum_{k=0}^\infty q_k (x - x_0 + 1)^{-k}, \qquad x \to +\infty.$$1) requires use of compensated summation or extended numerical precision. The radius of convergence of $$f(x) \sim \sum_{k=0}^\infty q_k (x - x_0 + 1)^{-k}, \qquad x \to +\infty.$$2 can be routinely monitored by ratio tests on $$f(x) \sim \sum_{k=0}^\infty q_k (x - x_0 + 1)^{-k}, \qquad x \to +\infty.$$3.

In deep learning applications, network weights for the AE-injected components are initialized close to zero, reflecting the expectation that the residual component is small if the AE prior is accurate. All implementation algorithms for moderate expansion orders are readily executed with standard computational resources; for large-scale expansions, FFT-based acceleration is recommended (Nikitin, 2010).

7. Summary and Current Research Directions

AE prior initialization provides a fully explicit mechanism from Taylor-series data to asymptotic coefficients, avoiding the need for ad hoc or random initializations in both analytic and neural approaches. In deep learning for BSDEs, AE-initialized solvers consistently demonstrate an order of magnitude improvement in convergence rate and solution accuracy across high-dimensional, nonlinear problems. The methodological pipeline—building the asymptotically-associated function, verifying analyticity, computing binomial convolutions, and network initialization with AE—is both theoretically sound and computationally robust (Nikitin, 2010, Fujii et al., 2017).

A plausible implication is that AE prior initialization could extend to other classes of scientific machine learning and numerical analysis frameworks where prior analytic structures admit explicit series representation and efficient binomial transforms.