Stochastic Optimization with Differential Algebra

• SODA is a methodological framework that employs differential algebra to characterize and manage uncertainty in stochastic optimal control problems.
• It integrates advanced tools such as G-expectation, backward stochastic differential equations, and viscosity solutions for uncertainty propagation and robust decision-making.
• The approach is applicable to diverse fields including finance, astrodynamics, robotics, and systems biology, enabling both theoretical advancements and practical computational strategies.

Stochastic Optimization with Differential Algebra (SODA) refers to a collection of methodologies and theoretical frameworks that leverage concepts from differential algebra to address stochastic optimal control and decision-making problems under uncertainty. These approaches are distinguished by their explicit treatment of uncertainty—often in the system dynamics, control policies, or noise structure—using advanced tools such as differential algebraic equation representations, high-order Taylor expansions, algebraic manipulation of stochastic processes, and integration of robust and chance-constrained optimization strategies. The integration of SODA in practice enables rigorous uncertainty propagation, robust optimization under model ambiguity, and tractable solution schemes for high-dimensional, nonlinear, or constrained stochastic problems.

1. Mathematical Foundations and Core Frameworks

Central to SODA is the application of differential algebra to stochastic dynamic systems, enabling the explicit representation and manipulation of both differential and algebraic relationships under uncertainty. Notable formulations include stochastic differential algebraic equations (SDAEs) and the use of backward stochastic differential equations (BSDEs) driven by G-Brownian motion to encode model and volatility uncertainty. In these settings:

• G-expectation and G-Brownian motion generalize standard stochastic calculus, encoding nonlinear expectations that adapt to volatility and model ambiguity. The G-expectation operator $\hat{\mathbb{E}}$ measures the "worst-case" expected value over a family of probability measures.
• BSDEs in the G-framework take the form:

$$Y_t = \xi + \int_t^T f(s, Y_s, Z_s) ds + \int_t^T g_{ij}(s, Y_s, Z_s) d\langle B^i, B^j \rangle_s - \int_t^T Z_s dB_s - (K_T - K_t)$$

where the tuple $(Y, Z, K)$ captures backward dynamics under non-classical uncertainty and defines the cost functional in optimal control.

• In stochastic optimal control, the system dynamics are modeled as:

$$dX_s = b(s, X_s, v_s)ds + h_{ij}(s, X_s, v_s)d\langle B^i, B^j \rangle_s + \sigma_j(s, X_s, v_s)dB_s$$

with $v(\cdot)$ in an admissible set $\mathcal{U}$. Cost functionals defined by the BSDEs above enable formulation of robust, uncertainty-aware objectives.

2. Dynamic Programming Principle and Viscosity Solutions

A fundamental outcome of the SODA approach is the derivation of a generalized dynamic programming principle (DPP) that accommodates model and noise uncertainty. The DPP in the G-expectation setting is formulated as: $$u(t, x) = \sup_{v \in \mathcal{U}} G_{t, t+\Delta}^{t,x,v}[ u(t+\Delta, X_{t+\Delta}^{t,x,v}) ]$$ where $G_{t, t+\Delta}^{t,x,v}[\cdot]$ denotes the backward semigroup associated with the BSDE over the interval $[t, t+\Delta]$.

The value function $u(t, x)$, representing the optimized cost-to-go, is shown to be a viscosity solution of a fully nonlinear second-order partial differential equation (PDE), specifically a Hamilton-Jacobi-BeLLMan (HJB) type PDE of the form: $$\partial_t u + F(D_x^2u, D_xu, u, x, t) = 0, \quad u(T,x) = \Phi(x)$$ with $F$ expressing supremum over the control set and encoding both drift and diffusion uncertainties. Viscosity solutions provide a robust mathematical tool for ensuring well-posedness of the value function, particularly when classical solutions may not exist due to nonlinearity or degeneracy.

3. Differential Algebraic Methods for Uncertainty Propagation

The algebraic backbone of SODA is realized in the use of differential algebra—most notably, high-order truncated Taylor expansions and moment propagation. Differential algebra enables symbolic or automatic differentiation of system trajectories and constraints, allowing for:

• Analytical propagation of uncertainties: Taylor expansion of state transitions, especially in multi-degree-of-freedom systems (e.g. astrodynamics), allows the input uncertainty (mean, covariance, higher moments) to be mapped analytically onto the system state:

$$\delta x_f^i = \mathcal{P}_i^k (\delta x_0, \delta \mu, \delta T) + O(k)$$

where $\mathcal{P}_i^k$ is computed via differential algebra.

• Efficient computation of expected values and covariances: The moment-generating function $M_X(t) = \exp(\mu^T t + \frac{1}{2} t^T \Sigma t)$ enables straightforward calculation of high-order moments of Gaussian random variables propagated through nonlinear maps.
• Symbolic Poincaré maps: For periodic orbit continuation under uncertainty (e.g., in the CR3BP), the entire map from initial parameters to Poincaré section is analytically represented.

These methods are especially powerful for high-dimensional, nonlinear systems where traditional Monte Carlo approaches become computationally infeasible.

4. Stochastic Control under Model Ambiguity and Robustness

SODA frameworks generalize classical stochastic control by incorporating volatility and model uncertainty at a foundational level:

• G-expectation-driven control is particularly suited for financial applications, enabling robust derivative pricing, risk management, and robust portfolio selection where ambiguity in volatility or drift is prevalent.
• Value functions as viscosity solutions of fully nonlinear PDEs provide rigorous theoretical guarantees even in the non-smooth or non-Markovian regime.
• Extensions to differential games and uncertain environments in engineering and economics are facilitated by the ability of BSDEs and associated PDEs to encode adversarial and cooperative dynamics under uncertainty.

Moreover, the differential algebraic approach ensures that the sensitivity of the value function and optimal control policies to both parametric and stochastic perturbations can be analyzed systematically.

5. Computational Strategies and Numerical Considerations

The translation of SODA theory into computational methods relies on a combination of advanced numerical analysis and symbolic computation. Key aspects include:

• Domain decomposition and parallelization: Piecewise polynomial approximations and the use of local Taylor expansions enable decomposition of high-dimensional domains into tractable subproblems.
• Handling of fully nonlinear PDEs: Viscosity solution frameworks, together with numerical schemes for BSDEs (e.g., Picard iterations, stochastic optimization for residual minimization), make it feasible to compute solutions in regimes where classical solvers fail.
• Algebraic manipulation of candidate libraries: In data-driven identification of DAEs (e.g., SODAs framework), sequential sparse regression and careful conditioning of the candidate function library mitigate issues of multicollinearity and noise amplification.

The integration of these technologies supports robust, scalable solutions with quantifiable error bounds and formal guarantees of convergence where applicable.

6. Applications and Impact

The SODA paradigm is broadly applicable across scientific and engineering domains:

• Finance: Robust derivative pricing, model ambiguity management, and optimal strategy design under volatility uncertainty.
• Space mission design and astrodynamics: Robust trajectory optimization in the presence of non-Gaussian, nonlinear uncertainty for multi-body dynamical environments, including the design of periodic orbits and risk-aware mission planning.
• Optimal control and robotics: Real-time robust control policy development under uncertainty, with applications to navigation, stabilization, and path planning.
• Systems biology and chemical engineering: Structural discovery of conservation laws and dynamic equations directly from noisy experimental time series.

The SODA approach supports both model-based (theoretically driven) and model-free (data-driven, sparse regression) variants, unified by the differential algebraic representation of system structure and the explicit algebraic management of uncertainty.

7. Theoretical and Practical Implications

The SODA framework consolidates advances in differential algebra, stochastic analysis, and robust optimization to provide a unified mathematical and computational toolkit. Its major implications include:

• Theoretical generalization of stochastic control and inverse problems to settings with model ambiguity and weak regularity.
• Enhancement of computational tractability for solving high-dimensional, nonlinear, and/or chance-constrained optimization problems under uncertainty.
• Frameworks for analyzing sensitivity, robustness, and stability in systems governed by both differential and algebraic equations, with or without explicit solutions.

The structure of SODA methodologies opens avenues for the development of new numerical algorithms for nonlinear PDEs, high-dimensional uncertainty propagation, and real-time robust optimization in the presence of unpredictable and complex sources of randomness.