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In this paper, we present a Longstaff-Schwartz-type algorithm for optimal stopping time problems based on the Brownian motion filtration. The algorithm is based on Le\~ao, Ohashi and Russo and, in contrast to previous works, our methodology…
We consider an optimal stopping problem where a constraint is placed on the distribution of the stopping time. Reformulating the problem in terms of so-called measure-valued martingales allows us to transform the marginal constraint into an…
We study the height of the maximal particle at time $t$ of a one dimensional branching Brownian motion with a space-dependent branching rate. The branching rate is set to zero in finitely many intervals (obstacles) of order $t$. We obtain…
With a model for two-dimensional (2D) Brownian rotary ratchets being capable of producing a net torque under athermal random forces, its optimization for mean angular momentum ($L$), mean angular velocity ($\omega$), and efficiency ($\eta$)…
The conditional expectation and conditional variance of Brownian motion is considered given the argmax, B(t|argmax), as well as those with additional information: B(t|close, argmax), B(t|max, argmax), B(t|close, max, argmax) where the close…
The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this paper, we consider an extension of the optimal Skorokhod…
We consider a stochastic differential equation involving standard and fractional Brownian motion with unknown drift parameter to be estimated. We investigate the standard maximum likelihood estimate of the drift parameter, two non-standard…
We consider the problem of tracking a target whose dynamics is modeled by a continuous It\=o semi-martingale. The aim is to minimize both deviation from the target and tracking efforts. We establish the existence of asymptotic lower bounds…
We consider an irreducible pair $\mu \leq_c \nu$ of probability measures on $\mathbb{R}^d$ in convex order. In arXiv:2306.11019, Backhoff, Beiglb\"ock, Schachermayer and Tschiderer have shown that the Stretched Brownian Motion from $\mu$ to…
Motivated by a problem posed by Aldous, our goal is to find the maximal-entropy win-martingale: In a sports game between two teams, the chance the home team wins is initially $x_0 \in (0,1)$ and finally 0 or 1. As an idealization we take a…
For martingales with a wide range of integrability, we will quantify the rate of convergence of the central limit theorem via Wasserstein distances of order $r$, $1\le r\le 3$. Our bounds are in terms of Lyapunov's coefficients and the…
The conditional density of Brownian motion is considered given the max, B(t|\max), as well as those with additional information: B(t|close, max), B(t|close, max, min) and B(t|max, min) where the close is the final value: B(t=1)=c and t in…
We consider a reflected backward stochastic differential equations with default time and an optional barrier in a filtration generated by a one-dimensional Brownian motion and a defaultable process. We suppose that the barrier have…
We derive a three-term asymptotic expansion for the expected lifetime of Brownian motion and for the torsional rigidity on thin domains in R^n, and a two-term expansion for the maximum (and corresponding maximizer) of the expected lifetime.…
Motivated by recent results on the dual formulation of optimal stopping problems, we investigate in this short paper how the knowledge of an approximating dual martingale can improve the efficiency of primal methods. In particular, we show…
We consider the Brownian ``spider process'', also known as Walsh Brownian motion, first introduced in the epilogue of Walsh 1978. The paper provides the best constant $C_n$ for the inequality $$ E D_\tau\leq C_n \sqrt{E \tau},$$ where…
We derive closed-form solutions to the optimal stopping problems related to the pricing of perpetual American standard and lookback put and call options in the extensions of the Black-Merton-Scholes model with progressively enlarged…
Stability of linear systems with uncertain bounded time-varying delays is studied under assumption that the nominal delay values are not equal to zero. An input-output approach to stability of such systems is known to be based on the bound…
We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. We give series expansions…
We consider the maximal displacement of one dimensional branching Brownian motion with (macroscopically) time varying profiles. For monotone decreasing variances, we show that the correction from linear displacement is not logarithmic but…