Assume that phi(& centerdot;) is a non-negative, continuously differentiable weight function and phi '(& centerdot;) is nondecreasing on [0,infinity), and let 0= 0) of dimension alpha >= 1 and starting at zero such that E[tau(mu)]

We establish explicit estimates for a two-dimensional LQG homing problem with a zero terminal cost under the assumption that the controlled diffusion process is degenerate. The present results extend and complete those in Lefebvre (2013) which are given explicitly for a particular interesting case. Our main tool is a nonlinear integral inequality of the Gronwall type. As an application, we consider a simple example to illustrate the present results.

Under extra conditions, we establish some new bounds for a risk-sensitive homing problem with control only in the diffusion-coefficient of a controlled diffusion process in a given closed interval. Our main results generalize or strengthen those in Lefebvre (2004), and a recent result proved by the present author.

In this paper, we establish a one-sided maximal moment inequality with exact constants for Bessel processes. As a consequence, we obtain an exact constant in the Burkholder-Gundy inequality. The proof of our main result is based on a pure optimal stopping problem of the running maximum process for a Bessel process. The present results extend and complement a number of related results previously known in the literature.

Abstract We prove a one-sided maximal inequality for a randomly stopped Bessel process of dimension For the special case when α = 1, we obtain a sharp Burkholder-Gundy inequality for Brownian motion as a consequence. An application of the present results is also given.

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

In this article, we consider a class of homing problems with controls only in the diffusion-coefficient of a diffusion process in a given closed interval. The controller seeks to minimize an expected cost that accounts for quadratic control costs, and terminal costs until the controlled process hits the end-points of the given interval. Bounds for the value function and the optimal control in question are established under certain conditions. The present results extend and complement an earlier result proved by Lefebvre (2004).

A stochastic Gronwall lemma is proved in Scheutzow (2013) in the case when the exponent $p$ lies in the interval $0 \lt p \lt 1$. In this paper, we extend the lemma to the entire interval $0 \lt p \lt \infty $. We construct simple examples to illustrate t

We prove a stochastic Gronwall lemma of the convolution type. Our results extend that of Scheutzow [A stochastic Gronwall lemma, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2013) 1350019], and the related results established in the non-convolution case. The proofs of the present results are essentially based on the Métivier–Pellaumail inequality for semimartingales.

For a Bessel process X=(Xt)t≥0 with dimension α>0 starting at zero, a result of Dubins, Shepp and Shiryaev (1993) states that there exists a constant γ(α) depending only on α such thatE(max0≤t≤τ⁡Xt)≤γ(α)E(τ) for any stopping time τ of X. In this paper, we give an explicit form of the constant γ(α) in the case 0<α≤1. The present result complements the known case when α>1 treated in Graversen and Peskir (1998).

Let 0 < alpha < infinity be fixed, and let X = (X-t)(t >= 0) be a Bessel process with dimension 0 < theta <= 1 starting at x >= 0. In this paper, it is proved that there are positive constants A and D depending only on theta and alpha such that E-x (exp[alpha max(0 <= t <=tau) X-t]) <= AE(x) (exp[D-tau]) for any stopping time tau of X. This inequality is also shown to be sharp.

A stochastic integral inequality of the Gronwall type is established. The result complements a stochastic Gronwall lemma proved by Scheutzow.(3)

Let X=(Xt)t≥0 be a stock process, assumed a geometric Brownian motion, with drift μ>0 and volatility σ>0 starting at x>0. For any stopping time τ for X, we establish an upper moment bound for Ex(max0≤t≤τ⁡Xt) under certain restrictions. Our method of proof employs the Gronwall inequality, and a comparison principle for a system of first-order nonlinear differential equations. The bound obtained in this paper extends an existing result, and the method of proof employed is quite new.

In this paper, we prove maximal exponential inequalities for a class of diffusion processes under certain conditions. The present results extend the well-known maximal inequalities of the power-type proved in the case of a Brownian motion, and in the geometric Brownian motion case. The method of proof is essentially based on explicit forms of optimal stopping problems related to those treated by Shepp and Shiryaev [Ann. Probab. Appl., 3 (1993), pp. 631-640], but considered here with a multiplicative reward functional. The key tool in the proof of one of our main results is a comparison principle for solutions of a system of first-order nonlinear differential equations. This is one of the novel features in this paper.

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.

Let $X=(X_t)_{t\geq0}$ be a geometric Brownian motion with drift $\mu$ and volatility $\sigma>0$, and let $Y=(Y_t)_{t\geq0}$ be the associated maximum process of $X$. Under certain conditions, we prove a sharp maximal inequality for the geometric Brownian motion. The method of proof is essentially based on explicit forms of the following optimal stopping problem: Find a stopping time $\tau^*$, if it exists, such that \begin{eqnarray*} \Phi(x,y):=\sup_{\tau}\mathbf{E}^{x,y}\left[Y_\tau-c\int_0^\tau X^\theta_sds\right],\quad c,\theta>0 \end{eqnarray*} where the supremum is taken over all stopping times $\tau$ for the process $X$. The present result complements and extends a similar result proved by Graversen and Peskir [1].

This paper treats a class of double optimal stopping problems arising in the pricing of integral options. Under certain conditions, we give an explicit form of the double stopping time for such type of optimal stopping problems. The present results are essentially derived by solving a certain nonlinear bilevel programming problem explicitly.

In this note, using the well-known method of scalarization, we give an explicit characterization of the Pareto optimal stopping time for a vector-valued optimal stopping problem with only two reward functions. The present problem is a natural generalization of the classical McDonald-Siegel optimal stopping problem.

This article concerns a characterization of a vector-valued optimal stopping problem with only two reward functions. Under certain conditions, we give an explicit characterization of the Pareto stopping times for the present problem via a scalar-valued double optimal stopping problem. The latter problem is a natural extension of the classical McDonald-Siegel optimal stopping problem with one stopping rule.

In this note, a vector-valued LQG homing problem subject to a controlled stock-wealth process is formulated. Under certain conditions, the problem is solved explicitly via an auxiliary scalar LQG homing problem.