Research

1. What Does a Nonlinear Fokker–Planck Equation Buy a Mean Field Game?

Mean field games almost always pair the agents’ Hamilton–Jacobi–Bellman equation with a Fokker–Planck equation linear in the density: the agents’ noises are independent, every interaction sits in the cost, and the agents never touch. This discussion paper asks what each kind of nonlinearity in the forward equation buys. In a stationary game in one dimension the crowd is an explicit statistics of the value function; the linear equation forces Boltzmann statistics, and without a crowd cost the game decouples. Noise, capacity and congestion each select another statistics and couple the game through the dynamics alone. Noise that grows with the density gives Tsallis statistics with edges and, past an exponent of 1.146, an empty gap; noise that falls with it gathers the crowd into one cluster until the branch folds at 0.219; a capacity gives a Fermi–Dirac plateau once the linear crowd would overflow the cap 1.7-fold; congestion gives the noise family in another game, whose optimum, 0.5 to 3 per cent better, needs a fee set by the crowd’s flow. A gradient pull buys nothing: it is a crowd cost in disguise. The open questions are where the computation sees what the theory cannot yet prove.

2. Littlewood’s Rule for a Home Battery

Ontario’s Ultra-Low Overnight electricity plan charges 3.9 cents a kilowatt-hour overnight and 39.1 cents between 4 and 9 PM on weekdays, and home batteries are sold on that tenfold spread. This paper asks how such a battery should be run when the price is set by the clock, the household’s own evening usage is uncertain and stored energy cannot be sold back, and when, if ever, a household should buy one. Running the battery is a revenue-management problem: the morning and evening windows are two fare classes for the same stored energy. The optimal policy fills the battery every night and during the morning mid-peak discharges only down to a reserve set by Littlewood’s rule, extended for a second mid-peak window after 9 PM and for the cost of refilling overnight. On 462 days of one house’s hourly usage, reconciled to its twelve bills, the reserve for a 13.5 kWh battery is 11.9 kWh in summer, 7.4 in spring and autumn and 11.4 in winter. It earns $757 a year off the bill, 94% of what a battery that knew each day’s usage could earn and 92% when each month’s reserve is set without that month, against $567 for discharging whenever the house draws. Buying the battery is McDonald and Siegel’s problem of investing when both the project’s value and its cost are random. Its savings over a ten-year warranted life are worth about $5,500 against an installed cost of $19,000 to $24,000, and on the regulator’s prices since 2006 and the market’s storage prices since 2021 the HJB variational inequality puts the installation boundary at an installed cost of $430 to $2,700, depending on the drifts one believes, decades away on present trends.

3. Queue-Reactive Dynamics on the Full Order-Book Profile

Queue-reactive models of the limit order book make order arrival and cancellation rates functions of the current queue size, and reproduce the mean reversion of depth that constant-rate models miss. They are, however, formulated at the best bid and best ask only: they carry no depth, and therefore no coupling between adjacent price levels. The reflected-SPDE family carries both, but specifies rates that depend on distance from the mid alone and are linear in volume. We give a model that holds both properties at once — rates depending jointly on position and on local queue size, adjacent levels coupled by a discrete Laplacian, volumes kept non-negative without a reflection measure, and the mid-price moving only when a best queue is depleted. We prove the model contains each parent family as a parameter restriction, and that a specification whose rates do not depend on the state has a stationary law of a single shape: its coefficient of variation is pinned at unity whatever its parameters, so the stationary mean and variance of depth cannot be matched independently. This is a structural account of the over-estimation of profile volatility reported for scaling-limit reductions in which the state dependence is lost. We further show that the sine eigenbasis usually assumed for the linearised generator is a property of uniform cancellation alone: once cancellation varies with position, the modes localise. Every claim is verified by simulation, and an acceptance test against order-by-order data is stated in advance of any such data being held.

4. The Contact Field: Ray-Knight Laws and Reaction in a Distributed Reactive Medium

A particle diffusing through a reactive medium reacts not on first contact but when the contact it has accumulated crosses a threshold, so the quantity that decides its fate is a random field: the contact accumulated at every level it can react at. For one-dimensional Brownian motion that field has an exact law — the Ray-Knight theorems identify it, read in the space variable, as a squared Bessel process — and this paper applies that law to the functional physical reaction theory measures. The result is that the survival probability in an arbitrary reactivity profile $\hat v$ is the reciprocal of $W(a)$, where $W” = 2\hat v W$ with $W(0) = 1$ and $W'(0) = 0$: one linear equation with two initial conditions, the reactivity entering as a potential, with no eigenvalue problem and no geometry remaining.

Two cases fix the result. A medium of finite width $w$ and reduced reactivity $\hat v_0$ has the closed form $S = [\cosh kw + k\sinh kw\,(a-w)]^{-1}$ with $k = \sqrt{2\hat v_0}$; a reactive point of the same total strength has $S = (1 + 2\hat\kappa a)^{-1}$, which recovers Collins-Kimball kinetics and the Robin boundary condition. Between them, a medium of finite width behaves as a point whose reactivity is reduced by the factor $1 – w/2a$ to leading order, a deficit set by the geometry of the search and independent of the strength of the chemistry. At the parameters used throughout, simulation gives $0.67569 \pm 0.00113$ against an exact $0.676195$ for the medium and $0.66681 \pm 0.00117$ against an exact $2/3$ for the point, both scored on one batch of paths so that the gap between them is resolvable.

Because the same equation returns the transform at every argument, the whole law of the reacted contact is available, and with it the kinetics of a surface whose reaction threshold is not exponential — which no local boundary condition can express.

5. Skew Brownian Motion: One Biased Site and the Asymmetry It Buys

Skew Brownian motion is ordinary Brownian motion everywhere except one point: at the origin each excursion chooses the positive side with probability $\alpha$ and the negative side with probability $1-\alpha$. Nothing else is altered — no drift, no varying diffusivity, no boundary. This note sets out how completely that single site fixes the law. The transition density from the origin is the normal density doubled and split, $2\alpha\varphi_t$ to the right and $2(1-\alpha)\varphi_t$ to the left, so the two halves keep the Brownian shape and differ only in the mass they carry. The probability of being positive is exactly $\alpha$ at every time, the expected occupation of the positive half-line is $\alpha t$ with no transient, and the mean displacement is $(2\alpha-1)\sqrt{2t/\pi}$.

What the parameter does not touch is as sharp as what it does. Every even moment equals that of ordinary Brownian motion, for every $\alpha$: the interface moves mass from one side to the other without changing how far the particle travels. An experiment that reports only a width or a dispersion coefficient carries no information about the interface at all.

A lattice walk that is fair at every site but the origin recovers all of it. At $\alpha = 0.75$ it returns the probability of being positive as $0.75023 \pm 0.00098$, the mean as $0.39858 \pm 0.00205$ against $0.398942$, the second moment as $1.00067 \pm 0.00317$ against $1$, and the occupation as $0.74995 \pm 0.00068$ — each within a quarter of a standard error. The one genuine difference between lattice and continuum is the atom the walk places at the origin: it is of order the lattice spacing, it belongs to neither side, and apportioning it by $\alpha$ is a derived correction that tracks across four halvings of the spacing.

6. Making a Market: the HJB a Naive Quoter Is Ignoring

A market maker who posts a fixed spread around the mid earns that spread on every fill and carries whatever inventory the fills leave behind. We treat the two halves of that sentence as two problems. The naive symmetric quoter is solved in closed form: its expected profit is $2AT\delta e^{-k\delta}$, maximised at a half-spread of $1/k$, and its inventory is a symmetric random walk whose variance grows linearly in the horizon, so that inventory risk outweighs spread-capture risk by a factor $\sigma^2 T / 2\delta^2$ that grows without bound. The optimal quoter is the solution of a Hamilton–Jacobi–Bellman equation which, under an exponential-utility ansatz, collapses to a linear system of ordinary differential equations. We solve that system exactly rather than asymptotically, and two things follow. The optimal quotes are stationary: they settle within a fraction of a unit of time and do not depend on the horizon thereafter. And the closed form in general use is the short-horizon expansion of this solution, wrong by twenty-one per cent in the spread and eighty-seven per cent in the inventory skew at parameters where it is routinely applied. Simulation of all three quoters on the same order flow, scored by certainty equivalent, confirms the ordering and measures the cost: the approximation gives up fifty-six units of certainty equivalent at a horizon of four, and the naive quoter turns negative.

7. A Leverage Cap on an Inconsistent Investor

The mean–variance criterion does not satisfy Bellman’s principle, so “optimal” splits into three incompatible strategies: a pre-commitment plan that requires a commitment device, a naive agent who re-solves continually and follows none of his plans, and an equilibrium from which no future self wishes to deviate. We compute all three for a market whose Sharpe ratio is an Ornstein–Uhlenbeck factor, under a leverage constraint of the kind every mandate carries.

Unconstrained, the equilibrium value is affine in wealth and the policy splits into a myopic term and a hedging term that exists only because the problem is inconsistent. The correlation between the asset and the factor enters through that term and nowhere else: an uncorrelated factor leaves the policy unchanged to within solver noise, however volatile it is. The term’s share of the policy is governed by $\kappa T$, the number of times the opportunity set turns over inside the horizon, falling from thirty-six per cent to six across a factor of twenty-four — time-inconsistency costs something only when there is something to hedge over one’s horizon.

The leverage cap destroys that structure. The binding set has two components and occupies a third of the state space throughout the horizon, while the value it destroys falls by an order of magnitude; a mandate is therefore expensive in proportion to the runway it removes rather than to how often it binds. It also changes the policy in regions it never touches, by nearly four per cent at wealths well clear of the boundary. Simulated on common noise under the cap, the three strategies rank naive, pre-commitment, equilibrium — the reverse of what unconstrained optimality suggests, separated by tens of standard errors and by under one per cent of objective. A constraint charges a strategy in proportion to the leverage it wanted, and the most modest rule survives it best.

8. Diffusions with Stochastic Resetting

We study diffusions subject to Poissonian resetting: at rate $r$ the process is returned instantaneously to a fixed point $x_0$. Two consequences separate this from the diffusions of classical stochastic analysis. First, the reset generator $\mathcal{L}_r f = \mathcal{L}f + r\,[f(x_0) – f(x)]$ is non-local and, even when $\mathcal{L}$ is self-adjoint in its natural weight, $\mathcal{L}_r$ is not; the stationary state is consequently a genuine nonequilibrium steady state carrying a nonzero probability current, and cannot be written as a Gibbs measure. For Brownian motion we obtain the stationary density in closed form — a cusped exponential $p_{\mathrm{ss}}(x) = \tfrac{1}{2}\sqrt{r/D}\,e^{-\sqrt{r/D}\,|x – x_0|}$ — and show that its current jumps by exactly $r$ across the reset point, the teleported flux closing the balance. Second, resetting changes first-passage times qualitatively. A renewal argument gives the mean first-passage time under restart as $\langle T_r\rangle = (1 – \tilde{T}(r))/(r\,\tilde{T}(r))$, where $\tilde{T}$ is the Laplace transform of the un-restarted passage time, and we prove the resulting sharp criterion: restart reduces the mean first-passage time if and only if the un-restarted passage time has coefficient of variation exceeding one, and at the optimal rate the restarted passage time has coefficient of variation exactly one. Numerical experiments — using exact stationary sampling and an exact renewal simulation, so that no time-discretisation bias enters — confirm the criterion, locate the optimum, and verify the CV $=1$ signature to within $3.4 \times 10^{-3}$.

9. The Derived Time Stop

A swing trade is a hypothesis with a deadline. The trader enters a setup believing it may carry an edge, the price then either drifts the way the setup promised or it does not, and the setup expires at a known date. We model the drift as one of two constants, unknown to the trader, and ask when the position should be closed and what the usual rules of thumb cost against the answer.

The trader’s belief that the edge is real is a filtered diffusion and an explicit function of time and return. Holding on is worth the expected drift under that belief, accumulated until exit, and the problem becomes a finite-horizon optimal stopping problem in the belief alone, with two parameters in information units: the break-even belief $\theta$ and the deadline $S$. Two partial differential equations answer it. A Hamilton–Jacobi–Bellman equation, in the form of a variational inequality solved backward from the deadline, gives the value of holding on, and its free boundary is the exit rule, which says where the trader should get out. A pair of Fokker–Planck equations, one for a real edge and one for no edge, solved forward from entry with that boundary as an absorbing wall, gives the law of the exit time, which says when trades are actually cut. The HJB boundary rises continuously to $\theta$ exactly at the deadline, so patience is largest at entry and vanishes at expiry.

The boundary is universal. In log-odds it is $\operatorname{logit}\theta+g(S-s)$ for a single function $g$ of the time left, the same for every break-even belief, because the value of any exit rule is a combination of two expected holding times that do not depend on $\theta$. The function is tabulated once: $g(u)/\sqrt{u}$ stays between $-0.63$ and $-0.58$ for time left $u$ between $0.01$ and $4$. On the return chart the boundary is a stop line that tightens as the deadline nears whenever the average of the two drifts is non-negative, and can first loosen when it is negative.

The Fokker–Planck densities show that the rule cuts $13.8$ per cent of real edges and $44.7$ per cent of dead trades before the deadline, most of them late, and that a trade cut at time $s$ had a real edge with probability exactly the boundary $b(s)$. A time stop that ignores the price is worth exactly $(p_0-\theta)$ per unit of time held and can never beat holding to the deadline. The trader’s conditional stop, which exits at a fixed day unless the trade has worked, cuts $31.0$ per cent of real edges at half the deadline, gives up between $3.3$ and $20.6$ per cent of the attainable value across the parameters studied, and is worse than simply holding in thirty of the forty-five cases computed. Every number is computed by finite differences for both equations and checked against closed forms, a no-look and a one-look bound, and a million-path simulation.

10. The Optimal Carbon-Tax Trajectory: A Finite-Horizon HJB Approach

We derive the optimal time-varying carbon-tax trajectory as the solution of a finite-horizon Hamilton–Jacobi–Bellman problem for a planner who balances the output cost of the tax against quadratic climate damage on the cumulative-emissions state. For the linear–quadratic specification the value function and the optimal tax admit a closed form: a stationary, Pigouvian term plus a finite-horizon correction that decays at an explicit rate. With physical anchors from the IPCC Sixth Assessment Report (cumulative emissions of $2\,390$ GtCO$_2$ by 2019, $0.45$ °C of warming per $1\,000$ GtCO$_2$) and an illustrative, DICE-style damage and cost calibration, the 25-year planner’s initial tax is $2.61/tCO$_2$ against a stationary benchmark of $5.80, 55 % lower. The dollar levels follow the illustrative anchors; the size of the gap is set mainly by how the terminal condition values the stock left at the horizon. Every headline number is computed twice, by the closed form and by an independent method: an ODE solve, a direct optimisation that does not use the HJB equation, Monte Carlo, or finite differences.

11. Bridging Technical Analysis and Order-Flow Microstructure: A Quantitative Framework

Technical analysis (TA) and order-flow microstructure are two cultures that rarely converse. TA operates on bar-level price and volume aggregates and enjoys near-universal use among retail and discretionary traders; microstructure operates on tick-level limit-order-book (LOB) events and forms the standard quantitative vocabulary at sell-side desks and in the academic literature. We argue that several common TA features admit natural interpretations as coarse-grained observers of underlying microstructure dynamics, and we offer a framework for stating such interpretations as falsifiable hypotheses. We do not claim a unified theory bridging the two domains; rather, we propose a research programme of pre-registered empirical studies — one TA feature at a time — to identify which patterns are microstructure-substrate-grounded and which are folklore. Four bridging hypotheses are developed in detail. No implementation is reported in this paper; the empirical programme is left as future work.

12. Five-Type Queue-Reactive Hawkes Simulation for Futures Price Discovery

We present a simulation framework for limit order book (LOB) dynamics based on the Five-Type Queue-Reactive (FTQR) model coupled with multivariate Hawkes processes. The five event types — limit bid arrivals ($\lambda_b$), limit ask arrivals ($\lambda_a$), bid cancellations ($\theta_b$), ask cancellations ($\theta_a$), and market order arrivals ($\mu$) — are modelled with state-dependent intensities $a_i \, q^{\beta}$ that scale with queue size $q_b$ or $q_a$, with short-term temporal clustering captured by exponential Hawkes kernels. Price is emergent: the mid-price moves only when a best-queue is depleted below a depletion threshold, never by direct prescription. We demonstrate via simulation that the model reproduces key microstructural stylized facts including queue size mean-reversion, bid-ask asymmetry, and clustered price moves, providing a tractable foundation for reversal signal detection in liquid futures markets.

13. Wishart Stochastic Volatility and the FTQR-Hawkes Bridge

We develop a Wishart stochastic volatility framework for the two-sided futures order book and establish its role as the correct continuum limit of the multivariate FTQR-Hawkes microstructure model. The variance state $V(t)$ evolves as a $2 \times 2$ positive-definite matrix CIR process, capturing the full covariation $v_{12}(t)$ between bid-side and ask-side volatility — a quantity that the Double Heston model is structurally forced to set to zero. We prove that the Wishart model retains an exponential-affine characteristic function, with the matrix Riccati ODE solved via a $4 \times 4$ Hamiltonian matrix exponential, enabling Carr-Madan FFT calibration to the joint call-put option surface. The off-diagonal entry $v_{12}(0)$ recovered from this calibration initialises the Hawkes cross-excitation matrix $\alpha_{12}$, and three mean-field game systems yield rational intensity benchmarks $\lambda^*_b$, $\lambda^*_a$ whose signed divergence from empirical intensities generates high-conviction intraday trading signals.

14. Market Making Under Inventory Risk with Adverse Selection

We study optimal market making in a limit order book where the dealer faces inventory risk and adverse selection from informed traders, following the Avellaneda–Stoikov (2008) framework extended to include asymmetric information costs. The dealer controls bid and ask quote depths to maximise expected terminal wealth minus a quadratic inventory penalty, leading to a Hamilton–Jacobi–Bellman equation whose solution yields closed-form reservation prices and optimal spreads. The reservation price shifts linearly with inventory and remaining horizon, while the optimal spread decomposes into a risk-aversion compo
nent proportional to γσ 2 τ and an adverse-selection component γ2 ln 1 + κγ . Simulated inventory paths under the optimal policy exhibit mean-reverting behaviour governed by the asymmetric quoting rule, with the dealer widening quotes on the side that would exacerbate a large inventory position.

15. Hawkes Process Models for High-Frequency Trade Arrivals

We study multivariate Hawkes processes as models for high-frequency trade arrivals, capturing the self-exciting and cross-exciting structure of order flow across asset classes. The intensity of each arrival stream depends on the full history of all streams via a matrix of exponential kernels, whose parameters are estimated by maximum likelihood using an expectation-maximisation algorithm. Calibrated on simulated tick data, the model reproduces the clustering of trades, the elevated autocorrelation of inter-arrival times, and the asymmetric cross-excitation patterns characteristic of correlated liquid assets.

16. Order Book Dynamics and Price Impact in Limit Order Markets

We study the dynamics of a continuous-time limit order book model in which market orders arrive as Poisson processes and the mid-price evolves as a diffusion driven by order-flow imbalance. Price impact is decomposed into a temporary component and a permanent component that shifts the fundamental value. Using a Hamilton–Jacobi–Bellman framework we derive the optimal liquidation strategy for a large trader minimising expected execution cost subject to a terminal inventory constraint, obtaining a closed-form feedback control in the linear–quadratic case and a numerical solution via backward Euler for nonlinear impact functions.

17. Lee’s Moment Formula in the Heston Model

Lee’s moment formula fixes the slopes of the implied total variance at extreme strikes through the largest finite moments of the price. In the Heston model these follow from the moment-explosion time, a closed form with three branches that we write in one convention, so the wing slopes $\beta_R$ and $\beta_L$ are explicit at every maturity. As the maturity falls the critical moments grow like $1/T$ and the ratio of the two slopes tends to $\arccos\rho/(\pi – \arccos\rho)$; as it grows they tend to the roots of a quadratic at the rate $T^{-2}$, and the slopes to closed forms in the ratio $(1-\rho)/(1+\rho)$. For an equity-like set ($v_0 = \theta = 0.04$, $\kappa = 1.5$, $\xi = 0.5$, $\rho = -0.7$, $T = 1$), $\beta_R = 0.024222$ and $\beta_L = 0.085741$. Damped Fourier prices down to $4.9 \times 10^{-70}$, agreeing with 34-digit arithmetic to $8.5 \times 10^{-14}$, show how slowly the smile gets there: at a strike $2981$ times the forward the call wing is still $9.16$ per cent above its slope, and at $1/2981$ of the forward the put wing is $16.2$ per cent above. Lee’s slope bounds the wings; it describes no quoted strike.

18. Last Passage Times, the Azéma–Yor Martingale, and Optimal Prediction of the Maximum

We study last passage times of standard Brownian motion and their role in the optimal prediction of the running maximum. Using explicit distributional formulas from Borodin and Salminen’s Handbook of Brownian Motion, we characterise the law of the last passage time $g_a = \sup\{t \leq 1 : B_t = a\}$ and connect it to progressive enlargement of filtrations and the theory of honest times. The Azéma–Yor martingale $M_t = \bar{B}_t – B_t$ is shown to be the key object linking last passage times to optimal stopping. We then solve Shiryaev’s problem of predicting the time at which a Brownian motion achieves its maximum on $[0,1]$, deriving the free boundary $b(t) = z^*\sqrt{1-t}$ (with $z^* \approx 0.84$) explicitly via a parabolic variational inequality, and validating the boundary numerically.

19. The Square-Root Wall

Stop a Brownian path the first time it reaches $c$ standard deviations of its own age, $|B_t| = c\sqrt{1+t}$. In the clock $s = \log(1+t)$ the wall becomes the fixed interval $(-c, c)$ for an Ornstein–Uhlenbeck process, and every moment of the wait follows: $\mathbb{E}[(1+\tau_c)^p] = 1/M(-p, \tfrac12, c^2/2)$, with $M$ Kummer’s function. The mean wait $c^2/(1-c^2)$ is finite exactly when $c < 1$, the variance exactly when $c < 0.741964$, and the tail is a power law whose exponent is the first zero of Kummer's function in $p$. A one-sided wall has an exponent below one half at every height, so its mean wait is always infinite. A bracket that widens with $\sqrt{1+t}$, stop at one standard deviation and target at two, wins $20.17$ per cent of the time, not a third, and its exponent $0.3633$ is below one half, so its outcome has no mean. A bridge-corrected log-time simulator confirms every formula whose estimator has a finite variance, within $1.9$ standard errors; a once-a-day check lengthens the mean wait up to $7.7$ times, and the tail exponent recovers only slowly.

20. A Huggett-Moll Mean Field Game with Mean-Reverting Kou Jump-Diffusion Productivity

We formulate a Huggett-Moll mean field game in which agents’ labor productivity follows a Kou double-exponential jump-diffusion, capturing asymmetric income shocks: sudden collapses and rare windfalls superimposed on geometric Brownian motion. Working in log-productivity space $x = \log z$, the multiplicative jump structure becomes additive, and both the Hamilton-Jacobi-Bellman and Fokker-Planck partial integro-differential equations admit a complete reduction to coupled PDE systems via auxiliary first-order equations derived from the exponential jump kernel. General equilibrium is closed by a Cobb-Douglas production sector, with aggregate capital $K[m] = \iint a \, m(a,x) \, da \, dx$ determining wages and returns endogenously. We state the MFG fixed-point problem, derive the auxiliary PDE systems explicitly, and document a Method of Lines numerical algorithm combining upwind finite differences in wealth with centred fourth-order stencils in log-productivity and a stiff implicit integrator.

21. Fredholm Methods for Optimal Stopping under Lévy Noise

We develop a Fredholm-operator framework for optimal stopping of Lévy processes with exponential tails, built on the Wiener–Hopf factorisation of the resolvent. The resolvent symbol factorises as $q – \psi(\xi) = \phi^{+}_{q}(\xi)\,\phi^{-}_{q}(\xi)$, where $\phi^{\pm}_{q}$ are the reciprocal transforms of the supremum and the infimum of the process at an independent exponential time. For payoffs that increase with the state, the variational inequality is solved by a two-level cascade. A Fredholm convolution equation of the first kind, whose kernel is the law of the supremum, gives an averaging function. The stopping region is the half-line above the root of that function, and the value is the supremum-average of its positive part. Under Kou double-exponential jumps both levels are explicit, and the investment threshold for the payoff $(P – I)^+$ is $P^\ast = I\,\mathbb{E}[e^{S}] = I\,\alpha_1\alpha_2(\eta_+ – 1)/(\eta_+(\alpha_1 – 1)(\alpha_2 – 1))$, where $\alpha_1, \alpha_2$ are the two positive roots of $\kappa(r) = q$. We fit the Kou model by exact maximum likelihood to 1,567 EU ETS primary-auction clearing prices from 2018-01-08 to 2025-05-20 (EEX auction results as published by the Instrat Foundation). A probability-integral-transform test does not reject the Kou fit (Kolmogorov–Smirnov $p = 0.98$) and rejects a Gaussian fit to the same prices ($p = 0.005$). With the price expected to grow at the euro risk-free rate of 2.73 %, the threshold multiple $P^\ast/I$ falls from 4.53 at a 7 % hurdle rate to 2.29 at 18 %. The McDonald–Siegel threshold of a geometric Brownian motion fitted to the same prices lies 0.2–0.3 % above the Kou threshold at every hurdle rate. On these data the jump-aware threshold is therefore not stable across hurdle rates: the jumps move it by less than half a per cent, and the hurdle rate moves it by a factor of 1.98. Each threshold is checked by a grid solution of the variational inequality and by Monte Carlo simulation of the stopping rule.

22. Wealth Stratification, Norm Contagion, and Institutional Capture: An Agent-Based Model of Endogenous Accountability Collapse

We introduce an agent-based model of $N$ interacting agents, each carrying wealth $R_i(t)$ evolving under Kou pure-jump dynamics and an accountability level $\Omega_i(t) \in [0,1]$. Three interlocking feedback loops drive the system: Loop 1 links wealth rank $r_i(t)$ to an erosion rate $\alpha(r_i) = \alpha_0 + \alpha_1 r_i^\gamma$, accelerating accountability decay at the top of the distribution; Loop 2 is an asymmetric norm-contagion process on a weighted social network, where observed impunity pulls accountability downward faster than virtue restores it; Loop 3 couples the Gini coefficient $G(t)$ to an institutional capture variable $\kappa(t)$, which weakens the social restoration mechanism and directly erodes $\Omega_i$. The full accountability dynamics are $\dot{\Omega}_i = -\alpha(r_i)\Omega_i + (1-\kappa)\,c\sum_j w_{ij}f(\Omega_j – \Omega_i) – \kappa\,\Omega_i$. Simulations reveal a phase transition at a critical Gini threshold $G^*$: below $G^*$ accountability recovers, above it the system locks into a collapse regime where the top tier reaches $\Omega \to 0$ and norm erosion propagates downward through the wealth distribution.

23. Accountability Collapse under Wealth Growth: A Stochastic Control Problem with Integral Equations

We formulate the erosion of social accountability $\Omega(t)$ under growing private wealth $R(t)$ as a stochastic optimal control problem on $\mathbb{R}_+$. The wealth process follows a pure Kou double-exponential jump model with no diffusion and no drift: two independent compound Poisson processes with rates $\lambda_1, \lambda_2$ and exponential jump sizes $\mathrm{Exp}(\eta_1)$, $\mathrm{Exp}(\eta_2)$, capturing large upward and small downward jumps respectively. We derive the Hamilton-Jacobi-Bellman integro-differential equation for the value function $V(R,\Omega)$, convert it to a Fredholm integral equation of the second kind with bilateral exponential kernel $K(R,x)$, and identify the Wiener-Hopf structure on the half-line via the rational symbol $\Phi(\xi) = 1 – \hat{k}(\xi)$. The Kou model’s quadratic numerator yields an analytic factorisation $\Phi = \Phi^+ \Phi^-$, reducing the problem to a second-order ODE whose characteristic roots $\beta_1, \beta_2$ are the Cramér-Lundberg exponents, giving the explicit solution $V(R) = A\,e^{-\zeta R} + V_p(R)$ with $\zeta = |\beta_2|$. The optimal control is bang-bang: a justice curve $\mathcal{J} = \{R^{\gamma}\eta = \Omega P\}$ separates prosocial from antisocial behaviour, and above $\mathcal{J}$ accountability collapse $\Omega \to 0$ is the rational optimum as $R \to \infty$.

24. Stochastic Stackelberg Games with a Jump-Diffusion Follower: Existence, Verification, and a Fixed-Point Algorithm

We study a continuous-time Stackelberg game in which a leader sets the drift of a one-dimensional jump-diffusion through a feedback rule and a follower, who observes the rule, chooses when to exit. In the feedback sense, each rule a best response to the other, the equilibria are the solutions of a coupled system: the leader’s Hamilton–Jacobi–Bellman (HJB) equation off the follower’s exit region, and the follower’s obstacle problem under the leader’s drift. We prove existence of an equilibrium by a Schauder–Tychonoff argument, a half-line structure of the exit region for a put-type exit payoff under a non-increasing drift, and a comparison theorem for the leader’s equation when the exit region is a half-line. For the alternating fixed-point algorithm we show that each half-step, computed by splitting off the jump term, is a sup-norm contraction with rate at most $\lambda/(q+\lambda)$, and that in the half-line case the outer iteration is a scalar map on the follower’s threshold whose rate is not fixed by $q$ and $\lambda$. For double-exponential (Kou) jumps the follower’s best response to a constant drift is in closed form through the Wiener–Hopf factors of the resolvent paper of the Stochastic Analysis Desk: the threshold is $K – 1/\beta_1 – 1/\beta_2 + 1/\eta_-$. A grid implementation of the two half-steps reproduces this closed form at second order, and the follower sweeps contract at measured rates $0.899$, $0.947$ and $0.973$ against the bounds $0.909$, $0.952$ and $0.976$.

25. Forward Rate Derivatives Pricing under the Langevin-Zamrik Model

We price interest rate derivatives — caplets, floorlets, bond options, and swaptions — under the Langevin-Zamrik (LZ) model, using its representation as a Gaussian Heath-Jarrow-Morton (HJM) model in forward-rate space. Since the LZ forward rate is normally distributed (not lognormal), caplets are priced exactly as put options on zero-coupon bonds — whose log-prices are Gaussian — giving a Black formula on the lognormal bond price with variance $\sigma_P^2(\tau_0, \delta)$. Swaptions use the Bachelier (normal) formula on the approximately-Gaussian swap rate. We derive closed-form expressions for the bond option variance $\sigma_P^2(\tau_0, \tau_S)$, the HJM variance profile $V(\tau) = \int_0^\tau [\sigma^{\mathrm{HJM}}(s)]^2\,ds$, and the swaption cross-covariance $C_{ij}(\tau_0)$ in all three LZ damping regimes and in 1FHW. A key structural finding is that in the underdamped regime the bond option volatility is non-monotone in bond tenor $\tau_S$ — it can decrease as tenor increases near resonance periods $n\pi/\omega$ — a phenomenon impossible in any Gaussian HJM model with positive $\sigma^{\mathrm{HJM}}$. A unified LZ-HJM pricer algorithm computes all instruments in a single pass.