Stochastic Control and Systemic Stability: Governing AI in Healthcare and Algorithmic Financial Markets

The swift implementation of Artificial Intelligence (AI) and Machine Learning (ML) systems in the healthcare industry and trading infrastructure has always resulted in the development of very high-stakes decision-making which are a complex of adaptive systems, having properties of opacity, feedback, and nonlinear risk propagation. The existing regulatory structures are mainly static and and sectoral fragmented. Therefore, they are not suitable for the regulation of adaptive systems, which are time-varying. The current research work formulates the regulation of AI in the healthcare industry and financial markets as a problem of dynamic risk minimization. The central research problem concerns the design of regulatory architectures that minimize expected social loss: f(x)=a_0+∑_(n=1)^∞▒(a_n cos⁡〖nπx/L〗+b_n sin⁡〖nπx/L〗 ) RminE[L(Ht,Mt∣R)], where (Ht) denotes population health outcomes, (Mt) represents market stability outcomes, and ({R}) is the regulatory regime, subject to constraints on innovation efficiency, information asymmetry, and institutional feasibility. In healthcare, AI-enabled diagnostic and decision-support systems—regulated under adaptive Software as a Medical Device framework like those which are articulated by the U.S. Food and Drug Administration—operating as a probabilistic classifier which are vulnerable to the distributional shift ((Pt \neq P{t+1})) and model drift. In financial markets, algorithmic trading systems governed under frameworks including the Markets in Financial Instruments Directive II (MiFID II) generate endogenous volatility through latency-sensitive strategic interaction and high-frequency feedback dynamics. The current study advances majorly three objectives: (i) construction of a unified stochastic representation of various algorithmic risk across healthcare and financial markets; (ii) evaluating the contemporary regulatory instruments, including the risk-tiered architecture of the Artificial Intelligence Act, through the control-theoretic and principal–agent lenses; and (iii) proposing an adaptive governance model with the integration of ex ante certification with continuous ex post supervisory feedbacks. Methodologically, the research also synthesizes the preliminary stochastic process modelling, network contagion analysis, and dynamic game theory. Healthcare AI systems are modelled as a sequential decision process with cumulative regret: RT=t=1∑T(Lt(πt)-Lt(π*)), The main argument is that both intelligent care systems and high-frequency trading architectures can produce systemic externalities due to correlated error structures and feedback amplification. In the healthcare domain, correlated misclassification errors can always lead to population-level harm, whereas in the financial domain, correlated trading strategies can lead to various flash crashes. These phenomena have a similar structure to the stability thresholds in complex networks, where systemic collapse usually happens if the spectral radius of the interaction matrix is satisfied by the condition (\rho(A) > 1). Thus, regulatory design must incorporate dynamic monitoring, stress testing, auditability constraints, and risk measures which are similar to Conditional Value-at-Risk (CVaR) but adjusted for the clinical harm. By framing AI governance as a problem of stochastic control and systemic stability, the study advances a mathematically grounded framework for safeguarding both clinical integrity and market stability in algorithmically mediated environments.