A note on the stochastic version of the Gronwall lemma

We prove a stochastic version of the Gronwall lemma assuming that the underlying martingale has a terminal random value in Lp, where 1 p < 1: The proof of the present result is mainly based on a sharp martingale inequality of the Doob-type

On maximal inequalities via comparison principle

Under certain conditions, we prove a new class of one-sided, weighted, maximal inequalities for a standard Brownian motion. Our method of proof is mainly based on a comparison principle for solutions of a system of nonlinear first-order differential equations

On the exact constants in one-sided maximal inequalitiesfor Bessel processes

In this paper, we establish a one-sided maximal moment inequalitywith exact constants for Bessel processes. As a consequence, weobtain an exact constant in the Burkholder-Gundy inequality. Theproof of our main result is based on a pure optimal stopping prob-lem of the running maximum process for a Bessel process. The pre-sent results extend and complement a number of related resultspreviously known in the literature

On mean exit time from a curvilinear domain

In this short communication, we consider a mean exit time problem for a non-degenerate, two-dimensional, coupled diffusion process Mt=(xt,yt) in the interior of a curvilinear domain with a C2-boundary, where xt is any arbitrary diffusion process and yt is a geometric Brownian motion evolving under non-explosive conditions, and [psi](.) is a real-valued, positive, increasing, continuous function such that [psi](0)>=0. It is proved that, under certain conditions, the mean exit time is a logarithmic function associated with a certain second-order nonlinear ordinary differential equation. At the end of the note, we shall present several examples to illustrate our main result.

A note on a nonlinear functional equation and its application

AbstractThis paper treats the following type of nonlinear functional equationsφ(x)=mH(x,φ[g(x)])Hφ(x,φ[g(x)]), where m is a real number, H(x,y) and g(x) are given functions, and φ(x) is an unknown function. Under certain conditions, we prove that such type of equations admits a unique continuous solution

Controlling a stopped diffusion process to reach a goal

We consider a problem of optimally controlling a two-dimensional diffusion process initially starting in the interior of a domain until it reaches the line y=[theta][phi](x) at a stopping time [tau] 0 and [theta]>1 are fixed positive constants and [phi](x) is a given positive strictly increasing, twice continuously differentiable function on (0,[infinity]) such that [phi](0)>=0. The goal is to maximize the probability criterion over a class of admissible controls consisting of bounded, Borel measurable functions. Under suitable conditions, it is shown that the maximal probability is given explicitly and the optimal process is determined explicitly byGeometric Brownian motion Optimal stochastic control problem

Exit probability for an integrated geometric Brownian motion

In this note, we present an explicit form for the exit probability of an integrated geometric Brownian motion from a given curved domain. Explicit bounds for the exit probability and one possible application are also given, under certain conditions.

A note on explicit bounds for a stopped Feynman-Kac functional

Let Qt=(xt,yt) be a two-dimensional geometric Brownian motion which is possibly correlated starting at (x,y) in the positive quadrant, and let [tau] be an -stopping time generated by the process Qt. Under certain conditions, we prove that where [Phi] is a bounded Borel function, C>0, [mu]>1, n>1 are constants and g* is an explicit bound for a solution of a certain second order ordinary differential equation. The present result extends and supplements the explicit upper bound in Hu and Øksendal (1998).Geometric Brownian motions Optimal stopping inequality