Research
51. Saddle-Point Structures for Two-Player Zero-Sum SDEs with Lévy Noise
Existence, coincidence of values, and a non-local first-order condition
Technical Note · Stochastic Optimal Control
02-MAY-2023 · 18 pages · TN-2023-58552860
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.
52. Resolvent Operators for One-Dimensional Lévy Processes: A Unified Treatment
Wiener–Hopf factorisation and rational families
Preprint · Stochastic Analysis
01-APR-2023 · 34 pages · PR-2023-68980035
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.
53. Axial Dispersion as Elastic Diffusion: Micromixing in Tubular Reactors
Danckwerts Boundary Conditions, Zwietering’s Bounds and the Method of Lines
Working Paper · Miscellaneous
29-JAN-2023 · 28 pages · WP-2023-86760774
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 $\mathrm{Da} \le 10$.
54. The Price of a Yaw: Optimal Deadbands and Filters for Wind-Turbine Yaw Control, Calibrated on SCADA Data
A Random-Walk Wind with Fast Turbulence, the Fourth-Root Deadband, and an Average Measured in Minutes
Working Paper · Miscellaneous
09-NOV-2022 · 29 pages · WP-2022-76705281
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.
55. Projected SOR, Penalty and Policy Iteration for Real-Options Variational Inequalities
A Comparison on Two-Factor Investment Problems
Technical Report · Stochastic Optimal Control
08-MAY-2022 · 30 pages · TR-2022-08169923
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.