Heuristic Stochastic Controller for Stabilizing Exchange Rate Dynamics Under Uncertainty

NUMERICAL SIMULATION OF FX RATES IN AN   HESTON STOCHASTIC CORRELATION MODEL

This paper proposes an enhanced approach to modeling and forecasting FX rates. One of the daunting problems in international mathematical finance is the weakness of existing theories to comprehensively model FX rates especially in periods of crisis. In this paper, we develop a new model that incorporates a stochastic volatility-stochastic interest rate-stochastic correlation structure and introduces a self-exciting jump in the variance of the FX process. Based on the $\alpha$-Heston model, the new model introduces an $\alpha$-root term and an $\alpha$-stable Lévy process in the variance of the FX rate. The inclusion of $\alpha$-parameter in the variance structure allows the model to capture both large and small fluctuations and extreme high peaks during crisis periods. We apply the model to the Cedi exchange rate against three major currencies that are exchanged widely in Ghana; the US dollar, the Great British Pounds Sterling and the Euro. The results show that there is significant evidence that the model can predict the Cedi’s performance against all three currencies. We conclude that the model is robust for implementation and forecasting currency rates.

Robust Control under Stationary Ambiguity

Control policies optimized in simulation can perform poorly in the real system when the parameters $x$ of the simulator are estimated from limited data but the resulting parameter uncertainty is not represented inside the simulation. A common way to incorporate such ambiguity is to simulate each trajectory of the system under a randomly drawn value for $x$. Since the policy cannot observe the drawn value, it must initially choose controls that perform well across many possible parameter values. However, if the policy progressively observes the system, it can often gradually infer the value of $x$, so that ambiguity vanishes. Over time, the policy then specializes to its estimate of $x$ and loses its robustness. This is undesirable in many real systems, where latent factors are expected to shift. In financial markets, for example, a policy hedging a derivative payoff should remain robust to changes in the volatility regime. To induce such continual robustness, we propose training policies in simulators where ambiguity varies with the system's state but does not systematically decay over time. We formalize this requirement as stationary ambiguity: the simulator should induce a stationary filter process over the latent state. We show how to construct such simulators and demonstrate, on hedging problems, that policies trained under stationary ambiguity preserve robustness to latent factors over time, leading to strong performance on real market data. As a modeling principle, stationary ambiguity informs many simulator design decisions: which models make realistic simulators, how their parameters should be randomized, and how simulator and policy should be initialized. While our experiments focus on hedging, stationary ambiguity may also be useful for other sequential control problems driven by exogenous stochastic processes with shifting latent structure.

A Predictive Model of Currency Exchange Rates Based on Stochastic Fractional Power-Law Dynamics

This work proposes a stochastic fractional power-law model for currency exchange rate forecasting. The model incorporates nonlocal temporal effects through the Caputo fractional derivative and nonlinear scaling dynamics via a power-law formulation, providing a flexible framework for representing complex temporal behavior in financial time series. Model parameters are estimated by fitting an explicit analytical calibration expression, constructed under the fractional chain-rule framework adopted in this study, to historical currency exchange-rate data using nonlinear optimization techniques. This expression is employed specifically as a tractable parameterization for model calibration and is not claimed as a general exact closed-form solution of the nonlinear problem involving the standard Caputo derivative. The forecasting stage is carried out through numerical simulations using a FORK-2-based stochastic discretization, with stochastic perturbations incorporated via a Wiener process. The proposed methodology is applied to daily EUR/MXN and EUR/CAD exchange rate series, and forecasts are generated through multiple Monte Carlo simulations over different prediction horizons. The results suggest that the fractional formulation can improve forecasting accuracy when longer training periods are employed. In addition, the nonlinear power-law structure increases the model’s flexibility and provides additional structural flexibility. Nevertheless, the integer-order formulation generally exhibits greater predictive stability, largely independent of the training period and the inclusion of the nonlinear power-law extension.

Diverse Private Expectations

Stochastic Optimal Control of Hawkes Jump-Diffusion Systems

This paper is devoted to developing a framework for stochastic growth models with environmental risk, in which rare but catastrophic shocks interact with capital accumulation and pollution. Building on the Poisson point process formulation studied in arXiv:2511.13568, we extend the model to disasters driven by a marked Hawkes process, allowing past disasters to temporarily increase the likelihood of subsequent shocks. Our work focuses on a subcritical Markovian Hawkes specification, in which the state space is augmented by the self-excitation component of disaster risk. We establish the well-posedness and nonexplosion of the resulting controlled Hawkes dynamics. Using the Hamilton-Jacobi-Bellman characterization of the corresponding Poisson control problem, we obtain quantitative estimates for the Hawkes excitation process and prove, under a small-excitation scaling, that the Hawkes value function converges to its Poisson counterpart as the magnitude of self-excitation vanishes. This provides a rigorous Poisson approximation of the stochastic control problem and quantifies the effect of self-excitation on optimal growth under environmental disaster risk.

Stochastic Optimal Control Problem under Inside Information

This paper is concerned with a stochastic optimal control problem under inside information. The control process depends on an $\mathcal{F}_{T_0}$-measurable random variable $Y$, representing the static inside information, and is adapted to the enlarged filtration generated by the underlying Brownian motion and the random variable $Y$. Accordingly, the traditional stochastic integral fails to be well-defined in this non-adapted setting; we adopt forward integrals to formulate the stochastic integral terms in the system. By means of the Donsker delta function, the original controlled system is transformed into a $y$-parameterized system. We further establish the existence of solutions to forward \textit{stochastic differential equations} (SDEs), and prove the uniqueness of solutions via flow transformation techniques. Under a Gaussian assumption on $Y$, we derive both necessary and sufficient optimality conditions for the aforementioned control problem. Subsequently, we formulate the \textit{linear-quadratic} (LQ) optimal control problem under inside information. Through the $y$-parameterized transformation, the original problem is converted into an LQ control problem with random coefficients. A numerical example for the LQ case is provided at the end to validate our theoretical findings.