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 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.