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.
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.
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.
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.
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.
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 $\psi(\theta) = q$.
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.
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 $K(x,\xi)$.
