Research

26. 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.

27. 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 $R \to \infty$.

28. 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 $0.976$.

29. 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.

30. 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.

31. 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 $x_1=0$, $x_2\to 0$.

32. 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.

33. 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.

34. 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.

35. 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.

36. 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 $\sigma_z$.

37. 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 $\psi(\theta) = q$.

38. A Stochastic Maximum Principle for Volterra SDEs

We establish a stochastic maximum principle for finite-horizon optimal control of stochastic Volterra integral equations with the fractional kernel $k(t) = t^{H-1/2}/\Gamma(H+1/2)$, Hurst index $H \in (0, 1/2]$, Lipschitz coefficients and a convex compact control set. The adjoint process solves a backward stochastic Volterra integral equation (BSVIE) in which the forward kernel is read over later times and the gradient of the terminal cost enters with the singular weight $k(T-t)$. An optimal control satisfies the first-order form of the maximum condition, which is pointwise maximisation of the Hamiltonian when the Hamiltonian is concave in the control; when the terminal cost is convex and the Hamiltonian is jointly concave in state and control, the condition is also sufficient. A linear–quadratic example has a closed-form adjoint in Mittag-Leffler functions, which we check against direct numerical solutions.

39. Universal Laws for the Times of the Maximum and Minimum of Random Walks

Let $t_{\max}$ and $t_{\min}$ be the times of the maximum and the minimum of a random walk of $n$ steps, and $\tau = t_{\min} – t_{\max}$. Mori, Majumdar and Schehr conjectured that $P(\tau = n) = 1/(2n)$ for every symmetric continuous jump law and proved it for $n \le 3$ and for double-exponential jumps. We prove it for every $n$ and every i.i.d. symmetric law without atoms. The proof is the cycle lemma applied to the doubled sequence $(\xi_1, \dots, \xi_n, -\xi_1, \dots, -\xi_n)$, whose cyclic rotations are the doubled sequences of $T^r\xi$, $T(x_1, \dots, x_n) = (x_2, \dots, x_n, -x_1)$: for any law invariant under $T$ without ties, dependent laws included, the highest point of the doubled loop is uniform on its $2n$ positions. These positions are explicit events about $t_{\max}$, $t_{\min}$ and the sign of $M + m – S_n$, each of probability $1/(2n)$. For i.i.d. steps the rest of the joint law depends on the step law through one probability, which gives the universal bound $P(\tau = j) \le 1/(2j)$, the limit $n P(\tau = n – k) \to 1/2$ and, for every stable domain of attraction, the Lévy-flight edge value $1/2$ in averaged form. Brownian motion has a uniform maximizer of the doubled path on $[0, 2)$; on the lattice a uniform split of ties restores the law. Exact enumeration for $n \le 7$ and Monte Carlo over ten step laws, within $4.34$ standard errors, confirm the results.

40. 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.

41. 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.

42. Verification Theorems for HJB Equations with α-Stable Lévy Noise

We establish verification theorems for finite-horizon Hamilton–Jacobi–Bellman equations driven by $\alpha$-stable L\’evy noise, where the classical It\^o calculus is replaced by a non-local infinitesimal generator. For $\alpha \in (1, 2)$ and bounded controls, we prove existence and uniqueness of viscosity solutions, derive optimal feedback laws expressible in terms of the resolvent of the non-local operator, and recover the classical Brownian verification theorem as $\alpha \to 2$. A jump-adapted Monte Carlo scheme for the controlled state is given; on a linear–quadratic subcase with a closed-form value function its statistical error decays as $N^{-1/2}$ at every $\alpha$ (fitted slopes $-0.504$ to $-0.489$ for $\alpha \in \{1.3, 1.5, 1.7, 1.9\}$), as plain Monte Carlo must, and its bias is first order in the time step; extrapolated to $\Delta t \to 0$ it reproduces the closed-form value to $0.11$ % ($1.0$ standard errors).

43. Options Under a Defended Currency Floor

When a central bank defends a currency floor, the exchange rate does not bounce off it: it sits on it. We model the log distance to the floor as a diffusion that is sticky at zero, pushed up at rate $\theta$ while it sits there, and let the bank abandon the floor at hazard $\kappa$ per unit of floor time, the rate jumping by $e^{J}$. The discounted rate is a martingale exactly when the drift above the floor is the carry less half the variance and $\kappa(1 – e^{J}) = \theta – q$, with $q$ the interest differential. Because $\theta$ is a path functional, local time over twice the floor time, it is the same under every equivalent measure, and the integrated break hazard equals the floor’s push net of carry divided by the relative jump. Calls, puts and digitals are explicit Laplace transforms; they agree with finite differences to $1.2 \times 10^{-5}$ and with Monte Carlo within $2.83$ standard errors, and the floor puts a steep downside skew on options. On $858$ ECB fixings of the EUR/CHF floor of 2011 to 2015 the stickiness is weakly identified, $0.20$ to $1.71$ per year across bands, yet the risk-neutral twelve-month probability of a rate below $1.20$ from the floor stays between $16.9$ and $18.7$ per cent with the realized jump to $1.028$. With the realignment to $1.15$ that Hertrich and Zimmermann describe, the same path gives $35.6$ per cent, inside their option-implied range: the path prices the product of break probability and jump size, not either factor alone.

44. Optimal Timing of Carbon-Capture Retrofit on a Mature Offshore Gas Platform under Stochastic EU-ETS Prices and a Stochastic Decommissioning Deadline

We study when the operator of a mature offshore gas platform should retrofit carbon capture and storage when the price of EU Emissions Trading System (EU ETS) allowances is stochastic and a regulatory decommissioning deadline arrives at a random time. The retrofit is irreversible, and once installed it saves the allowances on the avoided emissions until the deadline. We pose the decision as optimal stopping of a geometric Brownian motion killed at an exponential deadline, show that the decommissioning charge cancels from the decision, derive the free-boundary variational inequality, in which the deadline enters with the term $+\lambda\Phi = -\lambda D$, and solve it in closed form for an open-ended retrofit window and numerically for a finite one. The price model is fitted to EU ETS primary-auction clearing prices on 1,343 auction days from January 2018 to 7 May 2024 (EEX auction results as published by the Instrat Foundation; licence and credit in Section 5.1). The monthly volatility is $\hat\sigma = 0.380$; the historical drift, 0.40 per year, exceeds any discount rate and cannot serve as a valuation drift. Investment costs and emissions come from a peer-reviewed study of a North Sea platform; operating costs come from a US DOE/NETL onshore study, used as a stated proxy. With a mean residual life of 12 years the retrofit threshold is 180 €/tCO$_2$, 2.68 times the zero-NPV price and well above the auction price of 69.16 €/tCO$_2$ on 7 May 2024; with a 6-year mean life it rises to 239 €/tCO$_2$. We prove that the threshold rises with the deadline hazard exactly when the annuity on the retrofit capital outweighs the fall in the option premium; at the calibrated costs this holds for every mean residual life below 76 years. The study’s range of capital costs moves the 12-year threshold from 71 to 290 €/tCO$_2$. Each headline number is checked by a grid solution of the variational inequality and by a Monte Carlo simulation of the stopping rule.

45. 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.

46. 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.

47. Dividends When Capital Is Slow: de Finetti’s Problem with a Sticky Boundary

Dividend models give a firm at zero surplus two fates: it is ruined, or capital arrives instantly and reflects it back into business. Neither is what happens when capital must be raised, which takes time, and while it is being raised the firm can neither pay nor operate. This paper models that interval exactly. The surplus is a Brownian motion with drift that is sticky at zero: while frozen there it is pushed off at the speed $\theta$ at which capital arrives, each unit of capital costs $\phi$, and each unit of time frozen costs $\kappa$. The capital raised is the local time of the surplus at zero, and the time frozen is that local time divided by $2\theta$ in the unit case. The equation has a weak but no strong solution, so the control problem is posed on weak solutions.

The optimal policy is a dividend barrier fixed by one explicit equation, the sticky boundary condition $\theta g_b'(0) – q g_b(0) = \phi\theta + \kappa$ applied to the smooth-fit function. Rescue is worth paying for exactly when $\theta(\bar\phi – \phi) > \kappa$, where $\bar\phi = \rho^{(\rho-1)/(\rho+1)}$ is a price ceiling for capital that depends on the model only through the ratio $\rho$ of the two characteristic roots. Below the critical speed $\theta^{\ast} = \kappa/(\bar\phi – \phi)$ the firm should be liquidated at zero and keep de Finetti’s buffer; above it the buffer falls continuously, leaving de Finetti’s level with a kink, and converges to the instant-injection barrier as $b_{LZ} + c_1/\theta$. The value lost at zero to slow capital is $\phi c_1/\theta$ to first order.

At the house parameters the critical speed is $\theta^{\ast} = 0.028906$, the ceiling is $\bar\phi = 8.1189$, the kink has slope $-10.800$ and $c_1 = 1.7351$. A lattice chain that is solved exactly and optimises its own barrier, never touching the closed forms, reproduces the optimal barrier, the value function and the critical speed, with errors of first order in the lattice spacing.

48. 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 $y = 1$.

49. 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.

50. The Distribution of Growing Perpetuities and the Gordon Growth Formula

A cash flow growing at expected rate $g$ with volatility $\sigma$ and discounted at $r > g$ for ever is worth $S_0/(r-g)$ by the Gordon growth formula. That number is the mean of the present value, not its typical size. The present value is an exponential functional of a drifted Brownian motion, and Dufresne’s identity gives its law exactly: $2S_0/(\sigma^2\gamma_\nu)$, with $\gamma_\nu$ a Gamma variable of shape $\nu = 1 + 2(r-g)/\sigma^2$. This one number fixes the shape. The mean is the Gordon value at every volatility; the median, as a share of it, is $0.689$ at $r – g = 3$ per cent and $\sigma = 20$ per cent, and $0.495$ at $\sigma = 30$ per cent. The variance is finite only while $\sigma^2 < 2(r-g)$. At those two settings $70.0$ and $77.4$ per cent of outcomes fall below the Gordon value, a share that exceeds one half for every $\nu$ and tends to one as volatility grows. A ten-year model's terminal value has a median of $0.584$ of its mean, and a simulation free of the identity reproduces its quantiles within $0.43$ per cent.