Stochastic Analysis

1. 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.

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

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

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

10. 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.

11. 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 $K(x,\xi)$.