Geodesic walks converge to Brownian motion on Finsler manifolds.
problem Understanding random walks on Finsler manifolds.
method Analyzing convergence of geodesic random walks to diffusion processes.
result The Brownian motion on a Riemannian metric is a key result.
New method calculates geometric Brownian motion with affine drift and its integral.
problem Calculating the distribution of geometric Brownian motion with affine drift and its integral.
method Laplace transform approach and Heun differential equation.
result Joint distribution of geometric Brownian motion with affine drift and its integral can be determined.
Two insurance companies collaborate to maximize the probability of none going bankrupt.
problem Maximizing the probability of no company bankruptcy in a correlated Brownian motion model.
method Analyzing optimal strategies and deriving explicit formulas for minimal ruin probability.
result Maximizing collaboration benefits when Brownian motions are positively correlated.
Unified geometric framework for Brownian motion on various manifolds.
problem Modeling Brownian motion on complex Riemannian manifolds.
method Constructing stochastic differential equations with noise and drift terms aligned with Laplace-Beltrami operators.
result Geometrically transparent and mathematically consistent foundation for diffusion processes.
In the present paper, an expansion of the transition density of Hyperbolic Brownian motion with drift is given, which is potentially useful for pricing and hedging of options under stochastic volatility models. We work on a condition on the drift which dramatically simplifies the proof.
Study refracted skew Brownian motion, find densities and asymptotics.
problem Modeling and analyzing refracted skew Brownian motion.
method Perturbation approach to find potential densities, transition density, and asymptotic behaviors.
result Expressions and asymptotic behaviors of refracted skew Brownian motion.
The paper models term structures under volatility uncertainty using G-Brownian motion.
problem Modeling term structures with volatility uncertainty.
method Modeling instantaneous forward rates as a diffusion process driven by G-Brownian motion.
result Derives a sufficient condition for the absence of arbitrage under volatility uncertainty.
Solves optimal liquidation problem for stock price following geometric Brownian motion.
problem Optimal liquidation problem for stock price process following geometric Brownian motion.
method Functional analysis tools; working in terms of cash.
result Explicit solution to the problem, extending to stochastic drift.
In the paper "On Truncated Variation of Brownian Motion with Drift" (Bull. Pol. Acad. Sci. Math. 56 (2008), no.4, 267 - 281) we defined truncated variation of Brownian motion with drift, Wt=Bt+μt,t≥0, where (Bt) is a standard Brownian motion. Truncated variation differs from regular variation by neglect…
This paper develops the first method for the exact simulation of reflected Brownian motion (RBM) with non-stationary drift and infinitesimal variance. The running time of generating exact samples of non-stationary RBM at any time t is uniformly bounded by O(1/γˉ2) where γˉ is the average drift of…
Proposes a virtual bidding strategy for electricity markets using stochastic control.
problem Optimizing electricity prices in day-ahead and real-time markets.
method Modeling price differences as Brownian motion with meteorological variables, transforming into portfolio management problem.
result Developed a strategy to manage electricity prices efficiently.
Optimizes quickest detection of drift in Brownian motion with false negatives.
problem Quickest detection of drift in Brownian motion with false negatives.
method Formulated as an optimal multiple stopping problem, then equivalent to a recursive optimal stopping problem, solved using free boundary methods.
result Explicit formulae for expected cost and optimal strategy found.
Investing in declining tech boosts profits, study finds.
problem Optimal decision-making in declining profit streams.
method Modeling profit stream as Brownian motion with negative drift, analyzing thresholds for investment and exit.
result Investment threshold decreases in volatility when profit boost is large.
A new process generalizes geometric Brownian motion with asymmetry.
problem Creating a positive process with asymmetry parameter.
method Introducing asymmetry parameter α to describe volatility at new lows.
result Preserves GBM properties while expressing volatility as weighted mean.
Paper offers a fast method to assess DeFi liquidation risk.
problem Assessing liquidation risk in DeFi stablecoin lending.
method Modeling collateral exchange rate as zero-drift geometric Brownian motion.
result Derives an exact formula for liquidation probability.
The Lie group Sol(p,q) is the semidirect product induced by the action of the real numbers R on the plane R^2 which is given by (x,y) --> (exp{p z} x, exp{-q z} y), where z is in R. Viewing Sol(p,q) as a 3-dimensional manifold, it carries a natural Riemannian metric and Laplace-Beltrami operator. We add a linear drift …
Unified framework for Brownian motion distances on specific geometric manifolds.
problem Understanding Brownian motion distances on radially isoparametric manifolds.
method Developed a geometric framework and derived drift-window inequalities.
result Unified framework for coadapted Brownian couplings on RIM.
Optimal dividend payout strategy found for Brownian risk model with ratcheting constraint.
problem Optimal dividend payout from a surplus process governed by Brownian motion with drift under ratcheting constraint.
method Solved a two-dimensional optimal control problem using viscosity solutions of Hamilton-Jacobi-Bellman equations.
result Threshold and curve strategies identified as optimal for different dividend rate sets.
We consider a general class of continuous asset price models where the drift and the volatility functions, as well as the driving Brownian motions, change at a random time τ. Under minimal assumptions on the random time and on the driving Brownian motions, we study the behavior of the model in all the filtrations whi…
Analyzes first exit times in a modified Barndorff-Nielsen and Shephard model.
problem Analyzing first exit times in a modified Barndorff-Nielsen and Shephard model.
method Formulated an approximate model driven by Brownian motion and Lévy subordinator, analyzed first exit times of log-return process.
result First exit time process decomposes into Brownian motion and Lévy subordinator components.
Optimizes spending by adjusting a discount factor modelled as an exponential CIR process.
problem Maximizing discounted spendings/dividend payments given an exponential CIR discounting factor.
method Analytical and numerical methods for deterministic and stochastic surplus processes.
result Explicit expressions for optimal strategies in deterministic cases, and constant-barrier strategies for small volatility in stochastic cases.
Unified treatment of eigenvalue processes using Riemannian geometry.
problem Eigenvalue processes in various settings.
method Riemannian submersion and gradient flow of isospectral orbits.
result Eigenvalue processes are projections of Brownian motion through Riemannian submersions.
In this work we introduce Heath-Jarrow-Morton (HJM) interest rate models driven by fractional Brownian motions. By using support arguments we prove that the resulting model is arbitrage free under proportional transaction costs in the same spirit of Guasoni [Math. Finance 16 (2006) 569-582]. In particular, we obtain a …
Introduces Neural-Brownian Motion for modeling dynamics under learned uncertainty.
problem Modeling dynamics under uncertainty with learned parameters.
method Defines NBM using a neural network to replace classical martingale property with a non-linear expectation operator.
result Proves existence and uniqueness of canonical NBM as a continuous εθ-martingale. Bayesian investor learns unknown asset drift, trades mean-variance optimal portfolio, but policy is robust to observation model distortion.
problem Bayesian portfolio selection with observation model distortion
method Robust Bayesian portfolio selection
result Robust policy and its price are closed form, with price of robustness half the variance of the non-robust investor's loss.
Financial contracts with options that allow the holder to extend the contract maturity by paying an additional fixed amount found many applications in finance. Closed-form solutions for the price of these options have appeared in the literature for the case when the contract underlying asset follows a geometric Brownia…
We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold …
New unbiased methods for generating stochastic bridges with given extrema.
problem Generating unbiased stochastic bridges with a specified extremum.
method Comparison and generalization of two algorithms for Brownian bridges to other diffusions, and application to Ornstein-Uhlenbeck and unconstrained processes.
result Generalization of unbiased generation methods to other diffusions and application to various processes.
Develops a method to estimate the shadow riskless rate from empirical data.
problem No risky asset in market, need for a shadow riskless rate.
method PCA, SVD, regularization to estimate SRR from correlated geometric Brownian motion.
result Estimates the shadow riskless rate from empirical datasets.
Estimates roughness of volatility from discrete variance data.
problem Estimating roughness exponent of stochastic volatility from discrete observations of integrated variance.
method Pathwise estimator based on fractional Brownian motion with drift.
result Strong consistency theorems for rough volatility models.
An optimal extraction strategy is found for a price-maker company selling an exhaustible commodity.
problem Maximizing profits from selling an extractable commodity with price impact.
method Two-dimensional degenerate singular stochastic control problem with finite fuel. Explicit solution to Hamilton-Jacobi-Bellman equation verified.
result Optimal extraction rules differ based on the type of price process (drifted Brownian motion vs. Ornstein-Uhlenbeck process).
This paper solves a Bayes sequential impulse control problem for a diffusion, whose drift has an unobservable parameter with a change point. The partially-observed problem is reformulated into one with full observations, via a change of probability measure which removes the drift. The optimal impulse controls can be ex…
The study extends GBM to include stable nonzero prices and finds a pronounced potential well.
problem The standard GBM model cannot describe stable nonzero prices in financial dynamics.
method Generalized GBM with polynomial drift of order q, model selection, and Markov chain Monte Carlo ensembles of potential functions.
result The optimal model for financial data is q=2, indicating the existence of a stable price.
We present the collaborative Kalman filter (CKF), a dynamic model for collaborative filtering and related factorization models. Using the matrix factorization approach to collaborative filtering, the CKF accounts for time evolution by modeling each low-dimensional latent embedding as a multidimensional Brownian motion.…
We find a simple expression for the probability density of ∫exp(Bs−s/2)ds in terms of its distribution function and the distribution function for the time integral of exp(Bs+s/2). The relation is obtained with a change of measure argument where expectations over events determined by the time integral…
Study of a generalized geometric Brownian motion with varying entry and exit rates.
problem Understanding the long-run behavior of economic systems with growth, volatility, entry, and exit.
method Generalized geometric Brownian motion framework with varying entry and exit rates, analyzing moments and survival probability.
result Optimal exit rate minimizes mean first-passage time, influencing system outcome.
This paper provides guarantees for DFM models using KL divergence.
problem Ensuring generative models match target distributions efficiently.
method Using KL divergence and Brownian motion bridge for generative models.
result Non-asymptotic guarantees for DFM models under specific conditions.
Optimal probability measure found for constrained stochastic processes.
problem Finding optimal probability measure with constraints for stochastic processes.
method Existence and uniqueness proof, explicit measure change, optimal drift and compensator adjustments.
result Explicit form of the optimal measure change and characterisation of adjustments.
In this note we find a formula for the supremum distribution of spectrally positive or negative Lévy processes with a broken linear drift. This gives formulas for ruin probabilities in the case when two insurance companies (or two branches of the same company) divide between them both claims and premia in some specifie…
We discuss a simple extension of the Ho and Lee model with generic time-dependent drift in which: 1) we compute bond prices analytically; 2) the yield curve is sensible and the asymptotic yield is positive; and 3) our analytical solution provides a clean and simple way of separating volatility from the drift in the sho…
We use drifted Brownian motion in warped product model spaces as comparison constructions to show p-hyperbolicity of a large class of submanifolds for p≥2. The condition for p-hyperbolicity is expressed in terms of upper support functions for the radial sectional curvatures of the ambient space and for the rad…
Study on determinants of unitary Brownian motion and their asymptotic laws.
problem Understanding determinants of unitary Brownian motion and their behavior over time.
method Using Stiefel fibration and skew-product decomposition of the Stiefel Brownian motion.
result Prove asymptotic laws for determinants of block entries of unitary Brownian motion.
Extends Local Variance Gamma model with geometric Brownian motion and piecewise linear local variance.
problem Modeling volatility dynamics in financial markets.
method Develops a geometric version of the Local Variance Gamma model with drift and piecewise linear local variance functions.
result Derives an ordinary differential equation for option prices and solves it in closed form.
Solves inventory control with unknown demand trend using singular control.
problem Optimally managing inventory with an unknown demand trend.
method Formulates as a stochastic control problem under partial observation, solves equivalent separated problem using transition between formulations, and applies viscosity theory.
result Constructs an optimal control rule and shows bounded Lipschitz continuity of free boundaries.
Researchers created a continuous Markov martingale that mimics Brownian motion but lacks the strong Markov property.
problem Constructing a continuous Markov martingale with Brownian marginals that misses the strong Markov property.
method Developed a new approach to create a continuous Markov martingale that differs from Brownian motion in terms of the strong Markov property.
result A continuous Markov martingale with Brownian marginals that lacks the strong Markov property was successfully constructed.
We develop a variational framework for SDEs driven by fractional noise.
problem Capturing long-term dependencies in SDEs driven by fractional noise.
method Markov approximation of fractional Brownian motion, variational inference, neural networks.
result Efficient variational inference of posterior path measures for neural-SDEs.
We consider the problem of utility maximization for investors with power utility functions. Building on the earlier work Larsen et al. (2016), we prove that the value of the problem is a Frechet-differentiable function of the drift of the price process, provided that this drift lies in a suitable Banach space. We then …
In this paper, we study a risk process modeled by a Brownian motion with drift (the diffusion approximation model). The insurance entity can purchase reinsurance to lower its risk and receive cash injections at discrete times to avoid ruin. Proportional reinsurance and excess-of-loss reinsurance are considered. The obj…