Papers

Variance swaps are a convenient instrument for trading vega and convexity, and a listed contract now trades on Cboe. We ask when a trader should put such a position on and when she should take it off, and for a perpetual, continuously settled contract we answer both in closed form: each threshold is the unique root of a smooth-pasting equation in confluent hypergeometric functions. Under the pricing measure the question has no content, the mark-to-market being a martingale. Under the physical measure with a variance risk premium it becomes meaningful, and then reduces: the accrued variance separates exactly, the maturity, strike and costs are absorbed into a single forcing term whose sign fixes the geometry of the exercise region, and what is left on the perpetual is an affine reward on a CIR process, which the optimal-stopping literature already solves. Entering the position and exiting it are not mirror images. An exit rule follows from the premium and the trading spread, both observable. For the short — the only side worth opening under the empirical sign of the premium — an entry rule exists only for an interval of carrying charges, and even there triggers only deep in the upper tail of the physical law: a trader who is out of the market pays nothing to stay out, so an operational entry rule needs a cost of idle capital that the exit rule does not.

We study robust bond valuation with endogenous short-rate feedback under volatility uncertainty. Within the $G$-expectation framework, the dependence of the short rate on the bond price yields a nonlinear fixed-point problem, represented by a quadratic $G$-BSDE for the logarithmic price. Under suitable assumptions, we establish existence, uniqueness, comparison, and stability for bounded finite-horizon solutions. An additional strict monotonicity condition yields a unique bounded infinite-horizon solution and exponential convergence of finite-horizon approximations on compact time intervals. We apply these results to inverse short-rate design, constructing discount-rate coefficients that reproduce admissible smooth bond-price targets at a fixed maturity. For long maturities, we construct feedback rules under which the compensated logarithmic price converges exponentially to a prescribed bounded state-dependent profile, while the asymptotic yield equals a specified target.

This paper studies the regularity of the value function arising from a multidimensional continuous-time principal-agent model with separable, nonquadratic effort costs. The associated stochastic control problem has the output and the agent's continuation utility as state variables, and its Hamilton-Jacobi-Bellman equation is fully nonlinear and degenerate, with potentially unbounded coefficients. We address these difficulties by adding an independent regularization noise and bounding the effort. For the resulting problem, we establish classical regularity of the value function and show that the optimal effort is unique, positive and remains in a fixed compact subset, uniformly with respect to both the control restriction and the regularization parameter. These estimates allow us first to remove the control restriction and then to let the additional noise vanish. Consequently, we prove that the regularized value function converges to the original value function and conclude that the latter belongs locally to the Sobolev space $W^{2,1}_{\infty, \mathrm{loc}}$, thereby extending the regularity analysis to separable nonquadratic effort costs, for which the arguments yielding classical solutions in the quadratic-cost setting no longer apply.

Alòs, Gatheral and Radoičić derived the Bergomi-Guyon expansion of the implied variance smile from the forest expansion of the cumulant generating function. Its coefficients are sums of products of diamond trees, with prefactors that are polynomials in the log-strike $k$. Matching moments order by order produces, at order $\epsilon^\ell$, terms of degree greater than $\ell$ in $k$ that mysteriously cancel. We show that, in suitable variables, the matching condition can be formulated as a nonlinear heat equation. The resulting recursion computes the prefactor of each product of trees from those of products of fewer trees, without generating the higher-degree terms; its only model-independent input is a cumulant series, which we compute in closed form. Consequently, at order $\epsilon^\ell$ every prefactor has degree exactly $\ell$ in $k$. The prefactors are universal and need only be computed once. Code and coefficients are provided.

We present SAiFE_gym, a Python module that provides a collection of simulation environments for studying trading problems in Constant Product Markets (CPMs) with Concentrated Liquidity (CL). These markets give Liquidity Providers (LPs) granular control over how their capital is allocated and enable them to adjust their range of liquidity provision dynamically based on market conditions, which in turn, dictates how they earn fees. We decompose the microstructure of CPMs with CL in interactive components that allow researchers and practitioners to capture various economic settings. We employ a vectorized approach to optimize our environments, making them scalable for high dimensional Reinforcement Learning (RL) workflows that best describe sequential decision problems. We demonstrate the benefits of our environments by evaluating the performance of RL agents in CPMs with CL under uncertainty in market parameters.

We study strategic trading around index reconstitution in a continuous-time, multiasset game with transient cross-asset price impact and heterogeneous beliefs about future index membership. Opportunistic traders position before a public announcement, adjust to the revealed composition, and trade around an indexer following a prescribed execution schedule. Under a no-price-manipulation condition, we construct a subgame-perfect Nash equilibrium on every finite horizon. The affine feedback policies are computed from a non-standard matrix Riccati equation and linear ordinary differential equations whose number and dimensions do not grow with the number of traders. Aggregate inventories and price impact depend on beliefs only through the population-average belief, while differences in beliefs affect individual inventory positions. Under mean-field scaling, we construct a mean-field equilibrium and obtain quantitative convergence and approximate-Nash bounds. Numerical illustrations show how competition and impact decay determine the balance between adverse price displacement from anticipatory trading and savings in the indexer's execution costs when opportunists trade against its orders during implementation.

Return models for risk-neutral financial valuation based on skew Brownian motions (SBMs) have been introduced about twenty years ago, and have recently enjoying growing popularity. Unfortunately, the story behind their development is one of mistakes and erroneous interpretations, beginning from the foundational misrepresentations that the prevalent financial model is based on the Itô-McKean SBM – which, in fact, it is not. Besides, and more seriously, the clarification of \cite{rossello2012arbitrage} that price models with a local time in their returns, such as the SBM, are arbitrageable has been, by and large, ignored. In this paper, we try to clear the field from the confusions and misconceptions lurking in the standing option pricing literature on SBM, by exposing all the errors we could trace in the treatment so far. Recognizing however the potential of the SBM skew-normal marginals for risk-neutral valuation, as a positive contribution, we reformulate the putative SBM call pricing formula and show that, even if the SBM return model admits arbitrage, its option pricing formula does not. The correct Markovian SDE with skew-normal marginals is then identified, its strong well-posedness shown, and by exploiting the availability of closed formulae, an asymptotic analysis of the implied volatility surface is offered. En route to our conclusions, we obtain a novel normal/skew-normal stochastic dominance property of independent interest.

Normalized implied-volatility coordinates reveal no-arbitrage structure that is less transparent in strike space. This paper centers on a single quantity that gives those coordinates a direct probabilistic interpretation.

We derive a first-order representation of Black-Scholes implied variance in a continuous local martingale model. Total implied variance is the conditional expectation of the quadratic variation of the log price given its terminal value, up to a smaller-order term, for bounded standardized log-strikes. The framework incorporates small volatility-of-volatility, fast mean-reverting, and short-maturity asymptotics.

We consider two canonical market-microstructure regularities: the long-memory of trade signs and the square-root law of meta-order impact. The point is not to propose new empirical laws, but to separate the clocks on which existing laws are defined. The sign-memory law is an event-time statement about the ordering and fragmentation of hidden orders. The square-root impact law is an operational-time statement about front motion in a locally linear latent order book. Starting from a discrete-time random-walk bid/ask reaction–diffusion order book with a separate event clock, we identify an event-time imbalance reduction, derive the operational-time front and impact equations in the locally linear regime, and then subordinate both sign and impact observables to calendar time. Fractional or tempered clock effects enter through this event-to-calendar projection, not through a different operational-time impact mechanism. This gives a compact clock-aware framework in which event-time sign persistence, operational-time square-root impact, and anomalous calendar-time effects appear as distinct but compatible consequences of a common order-book representation.

In this note, we propose a general structural approach to model a default time $\tau$ as the first-passage time (FPT) of a (“firm-value”) process $S$ below a (“debt”) barrier $K$ that comply with a pre-specified survival probability curve $G(t)=\Pr(\tau>t)$. Following an idea of Mbaye and Vrins (Mathematical Finance, 2022) applied to reduced-form models, our approach consists in two steps: choose a latent FPT model driven by a barrier $\tilde{K}$ and process $\tilde{S}$, and time-change those using a deterministic clock $\Theta$ to get $K_t=\tilde{K}_{\Theta(t)}$ and $S_t:=\tilde{S}_{\Theta(t)}$, leading to the final FTP model $(K,S,\Theta)$. As the market curve $G$ and the latent model $(\tilde{K},\tilde{S})$ are assumed to be given, the calibration step simply consists in finding the clock $\Theta$ such that the distribution of the FPT of $S$ below $K$ coincides with the survival curve $G$. We show that this is achievable for a broad class of specified curves $G$ and latent FTP models. The calibration amounts to a simple inversion of a function, which is almost immediate provided that the latent model is tractable enough. In particular, we show that the AT1P model of Brigo, Morini and Tarenghi - which is able to reproduce a broad range of CDS term-structures - can be regarded as the FPT of a time-changed drifted Brownian motion to a constant barrier: $\tilde{V}_t=\mu t+W_t$ and $\tilde{K}_t=k<0$. This connection offers an elegant interpretation for the instantaneous volatility function featured in AT1P and yields an immediate calibration of the latter to perfectly match a target survival curve.

We give a two-period counterexample to the absence of arbitrage for the posterior-weighted pricing rule in Estimate nothing by Duembgen and Rogers. Both physical models have strictly positive transition densities, and each model is equipped with an equivalent martingale measure. Nevertheless, the mixed price of a single derivative falls deterministically from $5/2$ to $2$ between two trading dates. If these prices are tradable, shorting the derivative and closing the position one period later yields a certain profit. A finite-state appendix also illustrates the failure of recursive consistency.

We prove that the skew stickiness ratio converges to two at short maturity under local volatility models. This appears to be the first rigorous proof of this limit for a general time-dependent local volatility function. As a by-product, we strengthen the one-half rule of the implied volatility skew by removing uniform ellipticity and global bounds on spatial derivatives of order at least two. The proof uses a first-order Watanabe expansion.

We study how to combine diffusion models that disagree about drift and covariance. Candidate-first relative-entropy minimization gives geometric pooling, whereas expert-first minimization gives the arithmetic mixture associated with weighted logarithmic wealth. Different quadratic variations can make path-space entropy infinite, and the arithmetic mixture need not be a Markov diffusion. We therefore specify a local criterion combining drift information, normalized by the second argument's covariance, with quadratic transport between Gaussian shocks in a fixed state metric. A Gaussian identity and an Euler convergence estimate justify this chosen criterion. The expert-first projection has posterior-mean drift and an inverse-covariance penalty for drift dispersion; in one dimension this penalty increases volatility. For Ornstein–Uhlenbeck experts with a common mean-reversion rate, coefficient regularity holds on the full horizon for common volatility and away from the initial time for heterogeneous volatilities. The candidate-first problem has a Hamilton–Jacobi–Bellman characterization. Its matrix covariance selector reduces by congruence to a Bures–Wasserstein barycenter. The condition $H+\lambda M\succ0$, with value Hessian $H$ and state metric $M$, is sharp for finiteness of the unrestricted local covariance problem; compact constraints keep that problem finite. A covariance-disagreement budget interprets the penalty parameter. Linear–quadratic, exact-transition, and financial examples distinguish dynamic volatility reduction, drift-dispersion inflation, and martingale restrictions.

We define the variance swap delta as the sensitivity of the price of variance to a change in underlying price. We use Carr-Madan spanning formulas to analyze this sensitivity when the implied volatility smile curve may depend on the underlying price. We show that the variance swap total delta is zero for the class of smile curves that are pure functions of (log) moneyness, which goes against the empirical observation that variance is up when the market is down. We propose a simple modification of the smile to correct this issue.

Let \(P\) be a fixed physical law and let \(Q\) be an equivalent martingale measure selected from the martingale-measure set associated with a chosen numeraire. A change of numeraire maps \(Q\) to \(T_LQ\), where \(d(T_LQ)=L\,dQ\) and \(L\) is the terminal likelihood ratio. The forward relative-entropy projection minimizing \(D_{\mathrm{KL}}(P\Vert Q)\) commutes with this transform because its objective changes only by the constant \(-E_P\log L\). The minimal entropy martingale measure (MEMM) orientation \(D_{\mathrm{KL}}(Q\Vert P)\) does not have this property, and a trinomial counterexample shows that independently recomputed MEMMs need not be likelihood compatible. We make two economic consequences explicit. First, the two entropy orientations are precisely the \(Q\)-dependent terms in the classical convex-dual objectives for logarithmic and exponential utility, respectively. Second, likelihood compatibility is equivalent to equality of the pricing functionals obtained in the two numeraires. Hence the forward selectors value every integrable claim consistently across numeraires, whereas the two MEMMs in the counterexample assign different prices to a nonreplicable digital claim. We also prove a finite-state class-level characterization: uniform invariance over the elementary one-period likelihood-ratio families forces a smooth convex \(f\)-divergence to be logarithmic, up to scaling and affine equivalence. Finally, in finite-state markets, the forward projection exists under the usual strictly positive feasible-point condition; its density \(dP/dQ^*\) is attainable log-optimal terminal wealth, and the minimum forward entropy equals maximal expected log growth.

We study variance-optimal stock hedging and the convergence of approximate strategies in the rough Hawkes–Heston model. Starting from the model's affine conditional transform and the affine Volterra jump framework, we obtain semi-explicit hedges for European calls and a representation of the minimum quadratic error through the Galtchouk–Kunita–Watanabe projection. Our main approximation result keeps the original stock, variance driver, and information flow fixed while regularizing the kernel used to evaluate the hedge. To handle singular memory and common marked jumps, we construct the approximate holdings from histories available before trading and preserve the conditional transform's random modulus envelope. Riccati–Volterra stability and weighted truncation then yield convergence in the original stock's trading norm on compact Fourier intervals. For calls, a joint choice of kernel regularization and Fourier cutoff gives convergence of the initial capitals and strategies, uniform-in-time square-mean convergence of continuous-time gains, and convergence of the terminal mean-square error to the variance-optimal value. A numerical experiment with shifted fractional kernels illustrates the construction on common original-market paths.

Financial markets alternate between periods of relative stability and instability, with structural breaks marking the transitions between these regimes. Identifying such breaks in real time is a central requirement for any trading or risk system operating at high frequency. This report studies Bayesian Online Changepoint Detection (BOCPD) and two extensions proposed in the literature, and applies them to the signed order flow of NASDAQ-listed equities.

We establish existence, uniqueness, stability, and convergence results for one-dimensional $q$-Bass martingales, characterized as the martingales with prescribed initial and terminal marginals whose transition kernels are closest to a reference measure $q$. Their existence is equivalent to the solvability of a fixed-point problem for probability distributions. Building on Acciaio and Marini (2026), that requires the first marginal to be supported on finitely many points, we study the case of general marginals in convex order. Under the assumption that $q\ll\lambda$, we prove existence, uniqueness and stability of fixed-point distributions, $\mathcal{W}_\infty$-convergence of the fixed-point iteration, and support-diameter estimates. We also extend the martingale Benamou-Brenier formula from Brownian motion to any additive reference process $X$ and show that the corresponding $X$-Bass martingale is optimal whenever it exists, with an interpretation as an adapted Wasserstein projection of $X$.

In this note, we introduce a new comonotonic approximation for sums of lognormal random variables based on the famous Perron-Frobenius theorem.

We consider the Heston model with perfect negative spot–variance correlation and its one-dimensional local-volatility projection. Let $I_T^{\mathrm H}$ and $I_T^{\mathrm{LV}}$ denote their respective integrated variances over $[0,T]$. We establish the inequality \[ \mathbb{E}\bigl[(I_T^{\mathrm H}-K)^+\bigr] < \mathbb{E}\bigl[(I_T^{\mathrm{LV}}-K)^+\bigr] \] for every maturity $T>0$ and every strike $K>0$. Consequently, Heston integrated variance is strictly smaller in convex order than the integrated variance of the calibrated local-volatility model. This strict ordering gives a Heston-model counterexample to the convex-order inequality conjectured by J. Gatheral.

This paper develops a framework to assess loan portfolio budget in credit portfolios exposed to climate risk over a 3-5 year horizon. Using short-term scenarios from the NGFS, the approach translates transition and physical risk pathways into changes in obligor default risk. Firm performance is modeled within a structural credit risk framework where scenario-dependent climate shocks adjust the drift of firm value dynamics. Defaults occur when firm value falls below a liability threshold. Joint defaults are simulated using Monte Carlo methods with dependence captured via principal components. Resulting default-probability estimates inform a constrained optimization used to examine lending allocation across exposures. Climate scenario choice affects model-implied default probabilities and the resulting risk-return tradeoff across obligors and sectors. The results show scenario-specific differences in default-probability distributions, upper-tail risk, and lending allocation patterns. Under the benchmark synthetic portfolio, HWTP produces the highest mean and upper-tail PDs, while SWUC produces the lowest mean and median PDs. These rankings are model-implied and calibration-dependent rather than empirical rankings of climate-scenario severity. The framework remains transparent and can be refined as more granular climate and firm-level data become available. The framework helps banks link NGFS scenarios to sector-level credit risk and portfolio loss outcomes. It can support ICAAP and climate stress testing by highlighting exposures most sensitive to climate risk, adding a forward-looking overlay to existing models.

We develop a global martingale optimal transport framework for joint S&P 500 (SPX)-VIX smile calibration across multiple maturities, without the conditional independence that Markovian stitching imposes between months. We prove that exact local and global feasibility are equivalent, yet the stitched laws, which carry earlier months only through the current SPX level, can form a strict subset of the global ones, so adjacent smiles cannot identify the dependence across months. We show that relative entropy against the standard Markov reference selects the stitched law, so this dependence must be supplied deliberately. We supply it through a finite-memory reference, in which a latent autoregressive state drives the VIX levels while every VIX marginal stays exact, and calibrate the resulting law on a junction tree whose size grows linearly with the number of maturities. On market data with up to nine SPX maturities, a law with memory and a law without it fit the same quotes but price VIX-conditioned SPX Asian claims differently: on every calendar, memory makes a claim on the maximum VIX cheaper and a claim on the minimum VIX dearer. Varying the memory strength on one calendar moves these differences steadily from zero, so the vanilla quotes, which fix the law of each date, do not determine prices that depend on the joint law across dates.

Robust portfolio rules that reconstruct confidence sets after learning need not preserve the evaluator obtained by prior-by-prior Bayesian transport. In the Gaussian model, this discrepancy is summarized by natural-coordinate displacement: inherited transport preserves it whereas fresh reconstruction can replace it. We price evaluator replacement and trace the resulting optimized curvature through endogenous research. Optimized robust value represents protocol regret as a functional Bregman divergence, while a within-vintage rectangular Gaussian benchmark with constant absolute risk aversion (CARA) yields a stopped recalibration tax. In a versioned model-release economy, validated history propagates through a strictly causal network and a same-cycle share of current optimized marginal value feeds back into research supply. The capacity-constrained equilibrium reduces to a scalar equation with protocol-indexed gain \(\mathfrak g_I^P=\lambda\beta_{R,I}(W_R^P)''\). Purely causal validation cannot create a same-cycle unit mode; provenance changes criticality through optimized curvature. For a scalar primitive supplier-score shock \(z\) in direction \(h_I\) and financial outcome \(\mathcal O\), sensitivity factors as \(\omega_{\mathcal O,h,I}/(1-\mathfrak g_I^P)\). Conditional on a smooth equilibrium state and active cell, completion-time information sharply bounds this multiplier when all compatible timing laws are subcritical; no finite uniform bound exists when the timing set reaches the pole.

Although market participants generally have access to a common information set, they make decisions based on forecasts formed over heterogeneous horizons. Because market impact depends on aggregate positions rather than trader identities, these decisions feed back into prices through their collective effect. We introduce a linear mean-field model of this interaction. The observed price is decomposed into a martingale component, a common predictable signal represented by a Volterra process, and the market impact generated by aggregate positions. Agents take positions according to conditional forecasts of future signal increments and a fraction of anticipated aggregate impact over their respective horizons. Within a Gaussian-Volterra framework, we characterize equilibrium through a linear fixed-point equation for aggregate positions and establish existence and uniqueness under explicit conditions. At equilibrium, we identify a balance condition that cancels the direct transmission of the common signal to the observed price. We then study the limit in which agents fully account for market impact. Along a suitably scaled family of equilibria satisfying explicit conditions, the contributions of the predictable signal and the market impact cancel in the limit, and the observed price converges to its martingale component. For fractional-type signals and Gamma-distributed horizons, we further derive local Hölder bounds and identify the horizon distributions for which the observed price has local regularity compatible with that of Brownian motion.

We study a parabolic obstacle partial integro-differential equation (PIDE) with a dynamically moving bilateral free boundary. This type of problem arises in the mathematical modeling of speculative asset bubbles with Lévy jump processes. We investigate the existence of viscosity supersolution barriers within the class of functions exhibiting linear asymptotic growth ($O(|g|)$ at infinity) across three distinct parametric regimes. Our intention is to determine when such a barrier can be constructed by analyzing the balance between the stabilizing local drift, defined by the discount rate $r$ and mean-reversion $\rho$, and the non-local jump dispersion, characterized by the large-jump intensity $\lambda$ and Lipschitz constant $L_\gamma$. First, when $r+\rho > \sqrt{\lambda}L_\gamma$, we prove the global existence of non-negative viscosity supersolutions. Second, in the deficit regime ($r+\rho < \sqrt{\lambda}L_\gamma$), we construct non-negative supersolutions for every finite horizon $T>0$. However, by utilizing an asymptotic slope envelope, we prove that these barriers cannot be bounded by a fixed, pre-determined linear growth ceiling $C_{\max}$ across arbitrarily large horizons; rather, the required linear growth constant must inflate exponentially as the horizon length increases. Finally, at the exact critical boundary ($r+\rho = \sqrt{\lambda}L_\gamma$), we show the existence of a supersolution with a uniform spatial growth bound, provided an additional spatial no-crossing condition holds on the negative tail.

Multinomial probit choice probabilities over n alternatives are Gaussian orthant integrals, computed by simulation for thirty years, one expensive integral per alternative. Inversion, which is to say determining item attractiveness consistent with a prescribed choice probability vector, is even more difficult and has been considered impractical for correlated contests when n is large. Yet here, for families lying within a grammar including factor, block and hierarchical covariance structures, we exhibit a calibration tested at n = 1,000,000 reproducing probabilities to very high accuracy, even in the extreme tail. We must return to much smaller problems for any performance comparison to be possible due to limitations of the prior art. The Geweke-Hajivassiliou-Keane simulator is the standard (and still appropriate for high rank) but is two hundred times slower already at n = 200, and its measured cost grows roughly as n^2.8 while ours is linear. Furthermore our approach applies to any continuous performance distributions within reason: the Thurstone-Mosteller model families thereby become a practical alternative to logit at modern scale.

We consider a discrete-time financial market model where, in addition to finitely many dynamically traded assets, there are also (possibly infinitely many) static options to choose from. We introduce the concept of small cones of random variables and present a sufficient condition for the attainable positions in the market to be closed in probability. A fundamental theorem of asset pricing is shown in the present context. Utility maximization will also be considered. We will provide economically meaningful examples of infinite dimensional small cones to demonstrate the pertinence of our approach.

Loss-versus-Rebalancing (LVR) is the dominant adverse-selection cost borne by liquidity providers on automated market makers. Under geometric Brownian motion, arbitrage profit scales with the probability of a profitable block, which vanishes as the block time $\Delta t \to 0$; this is the standing argument for ever-shorter blocks. Modeling the reference price instead as a jump-diffusion, I show that the constant-product LVR rate splits into a diffusion channel carrying the known multiplier $F(\gamma/(\sigma\sqrt{\Delta t}))$ and a jump channel $\lambda V \cdot G(\gamma;m,\delta^2)$ carrying no $\Delta t$, the two interacting only through an explicitly bounded remainder. The block schedule therefore governs only one channel. For symmetric jump laws the jump channel is moreover an exact lower bound, $\ell(\Delta t) \ge \lambda V G > 0$, so the rate does not vanish as $\Delta t \to 0$, and descends slowly, as $\sqrt{\Delta t}$. At Ethereum's calibrated 12-second slot the rate is 471 bp/yr against a floor of 125, so only three quarters of LP loss is schedule-addressable. At Solana's 400 ms slot the jump channel already dominates. Netting the rate against per-block consensus cost, the LP-side optimal block time is invariant in pool size and in every jump parameter $(\lambda,m,\delta)$: jumps shift the level of LP loss but not the planner's marginal tradeoff. Volatility, the fee tier, and consensus cost set the optimum, near 8 s. However, LVR is only one input to block-time welfare, so this bounds the LP-side contribution rather than settling the design question.

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.