Research

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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 $3.4 \times 10^{-3}$.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. The Feynman–Kac Formula and the Heat Equation with Killing

We study the Feynman–Kac formula in its general form with a killing potential, establishing the probabilistic representation of solutions to the heat equation $\partial_t u = \tfrac{1}{2}\sigma^2 \partial_{xx} u – c(x)\,u$ on a bounded domain with absorbing boundaries. The solution is given by the expectation $u(x,t) = \mathbb{E}\bigl[e^{-\int_0^t c(X_s)\,ds} f(X_t)\,\mathbf{1}_{\{\tau > t\}}\bigr]$, where $\tau$ is the first exit time and the exponential weight is the Feynman path integral with potential $c$. We prove the formula via Itô’s lemma, analyse how the killing rate $c(x)$ suppresses the solution, and establish the connection to the imaginary-time Schrödinger equation. Numerical experiments confirm the probabilistic representation against direct PDE solutions for quadratic, step, and barrier killing potentials.

15. 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.

16. 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.

17. 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 $R \to \infty$.

18. 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.

19. The Controlled Symbol: Pseudo-Differential Operators, HJB Duality, and Spatially Varying Regularity

We develop a rigorous treatment of Feller processes as pseudo-differential operators and apply the resulting symbol calculus to stochastic optimal control. Starting from the Courège–Lévy–Khintchine representation theorem, we define the symbol $q(x,\xi)$ of a Feller generator as the position-dependent analogue of the Lévy exponent, and prove that $\xi \mapsto q(x,\xi)$ is a continuous negative definite function for each $x$. We then introduce controlled Feller processes, in which the full Lévy characteristics $(b(x,u), a(x,u), \nu(x,u,\cdot))$ depend on a control parameter $u$, and establish three principal results: (i) the HJB Hamiltonian $\mathcal{H}$ equals the infimum over $u$ of the controlled symbol evaluated at the gradient of the value function (gradient-symbol identity); (ii) the optimal symbol $q^*(x,\xi) = \inf_{u \in U} q^u(x,\xi)$ preserves the Lévy–Khintchine structure whenever $U$ is convex and the infimum is attained; (iii) a conservativeness criterion for the optimally controlled process stated directly in terms of symbol growth. We conclude by showing that the Blumenthal–Getoor index of $q^*$ governs the local Sobolev regularity of the value function, providing a spatial profile of HJB regularity through the optimal symbol.

20. Forward Rate Dynamics in the Langevin-Zamrik Framework: HJM Structure and Inertial SPDEs

This paper makes four contributions to the Langevin-Zamrik (LZ) interest rate model. (1) HJM identification: we show that LZ is a Gaussian Heath-Jarrow-Morton model with closed-form volatility $\sigma^{\mathrm{HJM}}(\tau) = -(\sigma/m)B_2′(\tau)$, where no-arbitrage is automatic rather than imposed. (2) Three-regime volatility: the overdamped, critically damped, and underdamped regimes produce humped, Nelson-Siegel, and oscillatory volatility shapes respectively; the underdamped volatility changes sign, a structural feature impossible in any existing Gaussian HJM family. The no-arbitrage drift is fully closed-form in all three regimes. (3) Inertial Musiela SPDE: LZ is lifted to an infinite-dimensional second-order SPDE on the space of forward curves; it collapses back to the 2D LZ system via the Björk-Christensen finite-dimensional realisation theorem, with the critically damped case yielding the no-arbitrage Nelson-Siegel family. (4) Forward Rate LZ PDE: the central new result. A change of coordinates from $(r_t,v_t)$ to any two market-observable forward rates $(F_1, F_2)$ transforms the LZ PDE into one whose state variables are directly quoted in swap markets and whose coefficients are exactly the HJM drift and volatility. The original LZ PDE is the special case $x_1=0$, $x_2\to 0$.

21. Optimal Stopping and the Snell Envelope

We study the optimal stopping problem in continuous time, where an agent chooses a stopping time to maximise the expected value of a payoff process, following the classical framework of Snell (1952) and its continuous-time extension via the Doob–Meyer decomposition. The value function is characterised as the Snell envelope — the smallest supermartingale dominating the payoff — whose generator satisfies a Hamilton– Jacobi–Bellman variational inequality of obstacle type. The optimal stopping time is the first entry into the stopping region, where the value function equals the payoff, and the continuation region is determined by the strict inequality V > g. As a canonical application, we solve the perpetual American put option, obtaining the closed-form exercise boundary and value function, and illustrate how the exercise boundary moves with time-to-expiry under the finite-horizon formulation.

22. The Two-Player War of Attrition on a Bivariate Diffusion: A Free Boundary PDE Approach

We study a two-player war of attrition in which each firm’s profitability evolves as an independent geometric Brownian motion, so the state space is two-dimensional. The game is formulated as a Dynkin stopping game whose equilibrium characterises a pair of free boundaries $\Gamma_1$ and $\Gamma_2$ in the $(y_1, y_2)$ plane, each a curve separating the exit region from the continuation region. In the mixed region — where both players randomise — the value functions satisfy a coupled elliptic PDE system of the form $(\mathcal{L}_1 + \mathcal{L}_2 – r)V_i = 0$, which is the two-dimensional analogue of the ODE collapse that characterises the one-dimensional model. We derive a spectral representation for the value functions using the product structure of GBM, show that in the symmetric case the free boundaries reduce to a single curve admitting a closed-form expression via separation of variables, and treat the asymmetric case numerically using a projected successive over-relaxation (SOR) algorithm coupled with a free boundary iteration. As $\sigma_1, \sigma_2 \to 0$, the PDE system degenerates to the ODE system of the one-dimensional model, providing a structural connection between the two frameworks.

23. The Three-Player War of Attrition as a Dynkin Game

We study the three-player war of attrition as a Dynkin game in continuous time, driven by a geometric Brownian motion representing market demand. Each of three symmetric firms controls a stopping time; the last firm to exit captures the entire market. Because the game is not zero-sum, pure strategy Nash equilibria generically fail to exist and equilibrium requires mixed stopping strategies. The model has three regions: an exit region below a common threshold $x^*$, a mixed strategy region in which firms randomise at a state-dependent hazard rate, and a certainty continuation region above the firm-specific break-even level $\bar{x}_n = c/\pi_n$. We show that in the mixed strategy region the value function satisfies $\mathcal{L}V_n – rV_n = 0$ — an ODE collapse that is the mathematical signature of indifference — with the game interaction encoded entirely in the hazard rate and the upper matching condition. The hazard rate $\lambda_n(x)$ is non-negative throughout and vanishes at $\bar{x}_n$, confirming a smooth transition to the certainty region. The recursive structure — each $n$-player problem uses the $(n-1)$-player value as a boundary condition — yields a tractable system solved by backward induction.

24. Bond Pricing Under Inertial Interest Rate Dynamics: The Langevin-Zamrik PDE

We introduce the Langevin-Zamrik partial differential equation, a zero-coupon bond pricing equation derived from the full underdamped Langevin dynamics of the short interest rate. The state space is two-dimensional, comprising the short rate $r_t$ and its velocity $v_t = \dot{r}_t$. For a quadratic (OU-type) potential the equation admits a closed-form affine solution $P = \exp\{A(\tau) + B_1(\tau)r + B_2(\tau)v\}$, where the coefficient pair $(B_1, B_2)$ satisfies a $2\times 2$ linear ODE system solved explicitly via the matrix exponential of the inertia-friction-spring matrix $M$. The three damping regimes — overdamped, critically damped, and underdamped — produce qualitatively distinct yield curve shapes: monotone, single-humped, and oscillatory respectively. The Vasicek formula is recovered exactly in the overdamped limit $m\to 0$. A central result is that the Langevin-Zamrik yield curve in the overdamped regime is structurally equivalent to the Nelson-Siegel-Svensson specification, providing its first no-arbitrage structural derivation. Extensions to nonlinear potentials, state-dependent volatility, multi-factor dynamics, and Lévy noise are discussed.

25. Insider Trading with Random Signal Arrival: A Kyle–Back Model with Poisson Revelation

We study a continuous-time Kyle–Back insider trading model in which the informed agent receives the asset’s true value $V$ at a random Poisson time $\tau \sim \mathrm{Exp}(\mu)$, rather than at inception. The model decomposes into two phases: before signal arrival, the price is uninformative and the insider’s continuation value satisfies a linear ODE; after arrival, a standard Kyle–Back equilibrium operates on the residual horizon. Our main result is a closed-form formula for the insider’s expected profit, $\mathbb{E}[\Pi](\mu) = \tfrac{\sigma_z \sqrt{\Sigma_0}}{2}\!\left[\sqrt{T} – \tfrac{\sqrt{\pi}}{2\sqrt{\mu}} e^{-\mu T} \,\mathrm{erfi}(\sqrt{\mu T})\right]$, involving the imaginary error function. The formula interpolates between zero profit ($\mu \to 0$) and the classical Kyle profit $\tfrac{\sigma_z\sqrt{\Sigma_0 T}}{2}$ ($\mu \to \infty$), and is strictly increasing in $\mu$, $\Sigma_0$, $T$, and $\sigma_z$.

26. Optimal Dividends with a Resurrection Option in the Cramér–Lundberg Model

We study the optimal dividend barrier problem for the Cramér–Lundberg surplus model when the firm’s owner holds a one-shot resurrection option: upon ruin, the owner may pay a fixed cost $R$ to restart operations at a prescribed level $x_0$. The $W^{(q)}$ scale function, characterised by its Laplace transform $\int_0^\infty e^{-\theta x} W^{(q)}(x)\,dx = 1/(\psi(\theta)-q)$, serves as the fundamental building block of the analysis. We prove that the optimal dividend barrier $b_1^*$ in the presence of the resurrection option satisfies $b_1^* \leq b_0^*$, where $b_0^*$ is the standard de Finetti barrier, with strict inequality when the option has positive value. For exponential claim sizes, every quantity — scale function, value functions, and optimal barriers — is given in fully explicit closed form via the two roots of the quadratic $\psi(\theta) = q$.

27. A Stationary Mean Field Congestion Game on a Torus

We study a stationary discounted mean field game on the one-dimensional torus $\mathbb{T}=[0,1]$, in which a continuum of identical agents controls a Brownian drift subject to a local congestion cost and a sinusoidal external potential $f(x)=A\sin(2\pi x)$. The equilibrium pair $(V^*,m^*)$ satisfies a coupled Hamilton–Jacobi–Bellman and Kolmogorov–Fokker–Planck system. Lasry–Lions monotonicity guarantees a unique Nash mean field equilibrium. Numerically, Howard policy iteration solves the HJB at each Picard step, while the stationary KFP is handled by a direct adjoint-matrix solve; the outer loop converges in 33 iterations to an equilibrium featuring a bang-bang optimal policy and a unimodal density concentrated in the low-cost region.

28. Brownian Local Time, Tanaka’s Formula, and the Quantum Delta Potential

We construct Brownian local time $L_t^x$ as the density of the occupation measure of standard Brownian motion and establish three foundational results: the occupation time formula, Tanaka’s formula extending Itô’s lemma to $|B_t – a|$, and Lévy’s representation theorem identifying $L_t^0$ in distribution with $|B_t|$. The entire development is motivated by a single problem in quantum mechanics: the Schrödinger operator $H = -\tfrac{1}{2}\partial_{xx} + \alpha\delta$ requires, via the Feynman–Kac formula, a rigorous interpretation of $\int_0^\tau \delta(B_s)\,ds$ — which is precisely the local time $L_\tau^0$. The quantum consequences follow as direct corollaries: the Feynman–Kac weight for the delta potential is $e^{-\alpha L_\tau^0}$, and the bound state energy $E_0 = -\alpha^2/2$ (for $\alpha < 0$) is derived from the Laplace transform of $L_t^0$ established via Lévy's theorem.

29. Russian Options: HJB and Reflected BSDE — Two Methods, One Price Surface

The Russian option, introduced by Shepp and Shiryaev (1993), is a perpetual American lookback contract whose payoff is the running maximum of the underlying asset. We price it via two independent frameworks: a Hamilton-Jacobi-Bellman (HJB) variational inequality and a singly reflected backward stochastic differential equation (BSDE) with barrier $M_t$. The HJB approach yields a closed-form solution through a scalar Euler ODE, while the BSDE Bellman iteration — solved by Gauss-Hermite quadrature — converges to the same value function with mean relative error $0.10\%$. A third result ties the two together: the expected stopping time $u(x) = \mathbb{E}[\tau^* \mid X_0 = x]$ satisfies a companion Poisson equation $\mathcal{L}u = -1$ under the same operator and boundary conditions, and its closed form reveals that the option premium equals, to leading order, the discount rate times the expected wait times the current value.

30. Robust Israeli Options with Finite Maturity: A Hamilton-Jacobi-Isaacs Approach under Drift Ambiguity

We study finite-maturity Israeli put options in the Black-Scholes framework when the writer faces Knightian uncertainty about the risk-neutral drift. Modelling this ambiguity as a bounded drift distortion controlled adversarially by the writer, the valuation problem becomes a three-player zero-sum game whose value satisfies a Hamilton-Jacobi-Isaacs (HJI) equation coupled with a double obstacle. The bang-bang structure of the optimal drift control is established in closed form, and the resulting nonlinear PDE is solved numerically via an explicit upwind finite-difference scheme. We find that drift ambiguity strictly lowers the option value, contracts the early-exercise boundary, and generates a positive ambiguity discount $V^0 – V^\kappa$ that peaks near the exercise boundary and vanishes throughout the cancellation interval centered at $s = K$. The cancellation region is a bounded interval $[s_l(\tau), s_r(\tau)]$ around $K$ that is absent near expiry and grows with time-to-maturity, shrinking and eventually disappearing as $\kappa$ increases.

31. The Russian Option: Optimal Stopping on the Running Maximum and Peskir’s Maximality Principle

The Russian option, introduced by Shepp and Shiryaev (1993), is a perpetual American option whose payoff is the running maximum $M_\tau = \sup_{s \leq \tau} X_s$ of the underlying stock price, discounted to the stopping time $\tau$. We analyse the perpetual problem for a geometric Brownian motion with drift $\mu < r$, and exploit the homogeneity of the payoff to reduce the two-dimensional state space $(X_t, M_t)$ to the one-dimensional ratio process $Y_t = X_t/M_t \in (0,1]$. In the continuation region $(y^*, 1)$ the reduced value function $v(y)$ solves an Euler--Cauchy ODE whose general solution is $v(y) = Ay^{\beta_+} + By^{\beta_-}$, and the four constants $(A, B, y^*)$ are determined by a Neumann reflection condition at $y = 1$ and smooth-pasting conditions at $y^*$. We present Peskir's maximality principle, which characterises $y^*$ as the largest candidate threshold for which the ODE solution dominates the obstacle, and we identify the two Doob--Meyer compensators of the discounted value process as the local times of $Y$ at $y^*$ and at the reflecting barrier $y = 1$.

32. The Israeli Option: Doob-Meyer Decomposition and the Double Obstacle PDE

The Israeli option, introduced by Kifer (2000), is a financial contract in which both the holder and the writer possess the right to terminate the contract at any time, making it a zero-sum Dynkin game. We analyse the perpetual Israeli put on a geometric Brownian motion under the risk-neutral measure, and show that its value function satisfies a double obstacle variational inequality with two free boundaries $x_1^* < x_2^*$. In the continuation region $(x_1^*, x_2^*)$ the value solves the Black–Scholes ODE, yielding the explicit form $V(x) = Ax + Bx^{-\alpha}$ with $\alpha = 2r/\sigma^2$. We derive the smooth-pasting system that determines all four unknowns $(A, B, x_1^*, x_2^*)$, recover the perpetual American put as the limit $\delta \to \infty$, and identify the two Doob–Meyer compensators of the discounted value process as the local times of the stock price at each free boundary.

33. The Doob h-Transform: Harmonic Functions, Conditioned Brownian Motion, and the Martin Boundary

The Doob $h$-transform is a fundamental technique for conditioning Markov processes on rare events. Given a strictly positive harmonic function $h$ for the generator $\mathcal{A}$ of a Markov process $X$, the $h$-transform reweights the original measure via the local martingale $M_t = h(X_t)/h(X_0)$, producing a new Markov process whose generator is $\mathcal{A}^h f = h^{-1}\mathcal{A}(hf)$. We develop the theory systematically: harmonic functions and Dynkin’s formula, the measure-change construction, conditioning standard Brownian motion to stay positive (yielding the three-dimensional Bessel process $\mathrm{BES}(3)$), conditioning to hit a fixed point (yielding the Brownian bridge), and Doob’s general theorem connecting $h$-transforms to conditional distributions. We conclude with Martin boundary theory, which classifies all positive harmonic functions via minimal harmonic functions and provides the canonical integral representation against the Martin kernel $K(x,\xi)$.