Research

26. 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.

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

28. 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.

29. 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.

30. 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.

31. 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.

32. 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 $\sigma_z$.

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

34. 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.

35. 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.

36. 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.

37. 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).

38. 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.

39. 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.

40. 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.

41. 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.

42. 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 $y = 1$.

43. 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.

44. 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.

45. Saddle-Point Structures for Two-Player Zero-Sum SDEs with Lévy Noise

We extend the zero-sum stochastic differential game to a state process driven by Lévy noise. With the generator and standing assumptions of the resolvent paper, the lower and upper values have at most linear growth and are Lipschitz in the state, and under the Isaacs condition they coincide as the unique viscosity solution of at most linear growth of a Hamilton–Jacobi–Isaacs integro-differential equation. Saddle strategies satisfy a first-order condition that is non-local in the attacker’s control: it integrates the value against the jump measure the attacker controls. A central-bank-versus-speculator game on a Kou state has a constant saddle pair and an affine value in closed form, which a numerical evaluation of the Isaacs equation and an exact Monte Carlo at an exponential time both confirm.

46. Resolvent Operators for One-Dimensional Lévy Processes: A Unified Treatment

We give a unified treatment of the resolvent operator $\mathscr{R}_q$ of a one-dimensional Lévy process, organised around its Wiener–Hopf factorisation $\mathscr{R}_q = q\,\mathscr{R}_q^+ \mathscr{R}_q^-$ into the resolvents of the running supremum and the running infimum. For three rational families — Kou double-exponential, hyperexponential and meromorphic — the factors act on exponentials through the Wiener–Hopf factors $\Phi_q^\pm$, finite or convergent infinite products over the roots of $\psi(r) = q$ and the jump rates, and the supremum and infimum at an exponential time are mixtures of exponential laws. We prove a comparison principle for the optimal-stopping variational inequality, identify its solution with the stopping value, and show that the perpetual put is governed by the infimum: the optimal threshold is $b^{\ast} = K + \mathbb{E}[\underline X_{\mathbf{e}(q)}]$ for the payoff $(K – x)^+$ and $e^{b^{\ast}} = K\,\mathbb{E}[e^{\underline X_{\mathbf{e}(q)}}]$ for $(K – e^x)^+$. Under Kou jumps both close in the two negative roots of $\psi(r) = q$; at the benchmark $b^{\ast} = -0.0897$. Every closed form is checked by an independent method — exact Monte Carlo of the process killed at an exponential time, and a finite-difference solve of the integro-differential equations — and agrees within the Monte Carlo error and at the grid’s second-order rate.

47. Axial Dispersion as Elastic Diffusion: Micromixing in Tubular Reactors

The axial dispersion model of a tubular reactor is closed by Danckwerts’ boundary conditions, argued over as physics since 1953. Read probabilistically they are a boundary behaviour: the closed vessel is the exit law of a drifted Brownian particle reflected at the inlet and removed at the outlet at rate $\mathrm{Pe}$ per unit of its local time there, Feller’s elastic boundary. First-order conversion is then a Feynman–Kac expectation, Wehner–Wilhelm’s formula follows, and an absorbing outlet is another vessel, off by up to 0.64 in outlet concentration. For a second-order reaction, the dispersion model lies between Zwietering’s complete segregation and maximum mixedness for the same residence-time distribution. The three agree through $\mathrm{Da}^2$; at $\mathrm{Da}^3$ they differ in one integral, the mean square of the age averaged over nothing, over position or over life expectancy. Near plug flow the dispersion model’s position in the band tends to $\xi_\infty(\mathrm{Da}) = (1+\mathrm{Da})(\mathrm{Da} – \ln(1+\mathrm{Da}))/\mathrm{Da}^2$, from one half toward one (0.58 at $\mathrm{Pe} = 1000$, $\mathrm{Da} = 1$, against 0.61); at low conversion it rises from 0 near the stirred tank to 0.23 at $\mathrm{Pe} = 10$ and 0.46 at $\mathrm{Pe} = 300$. The method of lines, collocation, Laplace inversion and a Monte Carlo compute every number at least twice. Above $\mathrm{Pe} \approx 20$ the average of the two limits gives the dispersion model’s conversion to within 0.1 percentage points for $\mathrm{Da} \le 10$.

48. The Price of a Yaw: Optimal Deadbands and Filters for Wind-Turbine Yaw Control, Calibrated on SCADA Data

A wind turbine must keep its rotor facing a wind whose direction never stops moving, and every yaw that realigns it wears the yaw system. Using a year of 10-minute records from the six turbines of the Kelmarsh wind farm, this paper measures what sets the right policy: below rated power the wind direction wanders like a random walk, at 11.2 degrees per square-root hour on average but at a rate that changes with the weather, with 8.2 degrees of fast turbulence on top, and misalignment costs power as $\cos^k$ with $k = 2.59$. With the turbulence idealised as white noise, the optimal controller separates into a Kalman-Bucy filter whose time constant is $\eta\sqrt{2\tau_u}/s$, 3 to 8 minutes for turbulence correlation times of 10 to 60 seconds, and a deadband of half-width $(6Ks^2/a)^{1/4}$ on the filtered misalignment. An ideal band of 8.9 degrees run on the site’s own varying wander reproduces the turbines’ bursts of yaws, 40.9% of them back to back against 41.1% in the records and 12.4% under a steady wander, so their controller behaves like that band. It reveals a price of 2.3 kWh per yaw, at which the cost of misalignment and wear together is 1.0% of below-rated energy; a band that widens with the fourth root of the current wander rate lowers it by 18%, and a 30-second average instead of a filter holds about three times the misalignment for the same number of yaws.

49. Projected SOR, Penalty and Policy Iteration for Real-Options Variational Inequalities

An option to invest whose payoff depends on two random factors is valued by a variational inequality; after a monotone discretisation it is a linear complementarity problem with an M-matrix, which has one solution. This report compares projected SOR, the penalty method, policy iteration and, for an option that expires, Ikonen–Toivanen splitting, on three problems: the McDonald–Siegel option, checked against its closed form on the full two-factor grid; an option on a mean-reverting price, checked by simulating the rule its boundary defines; and an option with a finite life. All methods reach the same discrete solution, to $2.1 \times 10^{-9}$, and the boundary converges to the closed form $X^{\ast} = 2.8560$ at second order, to $2.4 \times 10^{-4}$ on $512 \times 512$ cells. With a sparse direct solver on one thread, projected SOR was the fastest on the perpetual problems at the larger grids, 2.6 s against 26 s and 27 s at $512 \times 512$. Policy iteration from the payoff moved the boundary about one cell per iteration, so its count grew with the grid, from 4 to 23. The penalty error fell as one over the penalty, which must be set by a rule. The nine-point stencil, never an M-matrix, converged and was the most accurate at strong correlation. For the expiring option, splitting with one factorisation matched the exact step solvers at a fraction of their time. The recommendations hold for a direct solver, one thread and problems of this size.