Conventional wisdom holds that since continuous-time, Black-Scholes hedging is infinitely expensive in a model with proportional transaction costs, there is no continuous-time strategy which hedges a European call option perfectly. Of course, if one is attempting to dominate the European call rather than replicate it, then one can use the trivial strategy of buying one share of the underlying stock and holding to maturity. In this paper we prove that this is, in fact, the least expensive method of dominating a European call in a Black-Scholes model with proportional transaction costs.

This paper confirms a version of a conjecture by Fischer Black regarding consol rate models for the term structure of interest rates. A consol rate model is one in which the stochastic behavior of the short rate is influenced by the consol rate. Since the consol rate is itself determined, via the usual discounted present value formula, by the short rate, such models have an inherent fixed point aspect. Under an equivalent martingale measure, purely technical regularity conditions are given for the stochastic differential equation defining the short rate and the consol rate to be consistent with the definition of the consol rate as the yield on a perpetual annuity. The results are based on an extension of the theory for the forward-backward stochastic differential equations to infinite-horizon settings. Under additional compatibility conditions, we also show that the consol rate is uniquely determined and given as a function of the short rate.

In this paper we present an explicit form of the distribution function of the occupation time of a Brownian motion with a constant drift (if there is no drift, this is the well-known arc-sine law). We also define the $\alpha$-percentile of the stock price and give an explicit form of the distribution function of this random variable. Using this explicit distribution, we calculate the price of a new type of path-dependent option, called the $\alpha$-percentile option. This option was first introduced by Miura and is based on order statistics.

The study of the quantile of a Brownian motion with a drift is undertaken. An explicit formula for its density, as well as a representation of its distribution as the sum of the maximum and the minimum of two rescaled independent Brownian motions with drift, is given. The result is used in the pricing of a financial path-dependent option due to Miura.

This paper concerns the optimal impulse control of piecewise deterministic Markov processes (PDPs). The PDP optimal (full) control problem with dynamic control plus impulse control is transformed to an equivalent dynamic control problem. The existence of an optimal full control and a generalized Bellman-Hamilton-Jacobi necessary and sufficient optimality condition for the PDP full control problem in terms of the value function for the new dynamic control problem are derived. It is shown that the value function of the original PDP optimal full control problem is Lipschitz continuous and satisfies a generalized quasivariational inequality with a boundary condition. A necessary and sufficient optimality condition is given in terms of the value function for the original full control problem.

We consider limits of first passage times to indexed families of nested sets in regenerative processes. The sets are exponentially rare, in the sense that the probability that the process reaches an indexed set in a cycle vanishes exponentially fast in the indexing parameter. Under appropriate formulations of this hypothesis, we prove strong laws, iterated logarithm laws and limits in distribution, both for the index of the rarest set reached in a cycle and for the time to reach a set. An interesting feature of the iterated logarithm laws is an asymmetry in the normalizations for the upper and lower limits. Our results apply to (possibly delayed) wide-sense regenerative processes, as well as those with independent cycles. We illustrate our results with queueing examples.

We consider the problem of estimating first exit distributions for one-dimensional diffusion processes. We provide analytic bounds and an approximation which is shown to be accurate in a range of numerical examples involving Brownian motion.

We study the large time behavior of random fields which are solutions of a nonlinear partial differential equation, called Burgers' equation, under stochastic initial conditions. These are assumed to be of the shot noise type with the Gibbs-Cox process driving the spatial distribution of the "bumps." In certain cases, this work extends an earlier effort by Surgailis and Woyczynski, where only noninteracting "bumps" driven by the traditional doubly stochastic Poisson process were considered. In contrast to the previous work by Bulinski and Molchanov, a non-Gaussian scaling limit of the statistical solutions is discovered. Burgers' equation is known to describe various physical phenomena such as nonlinear and shock waves, distribution of self-gravitating matter in the universe and so forth.

Under suitable conditions, it is proved that limits of sequences of nuclear space-valued solutions of stochastic differential equations driven by Poisson random measures are characterized by diffusion equations. The results are applied to models of environmental pollution and to reversal potential models of neuronal behavior.

We present sharp bounds on the minimal errors of linear estimators for multivariate integration and $L_2$-approximation. This is done for a random field whose covariance kernel is a tensor product of one-dimensional kernels that satisfy the Sacks-Ylvisaker regularity conditions.

Levy bandits are multi-armed bandits driven by Levy processes. As anticipated from existing research, Levy bandits are optimally controlled by an index strategy: One can associate with each arm an index function of its state, and optimal strategies are those that allocate time to arms whose states have the largest index. Furthermore, the index function of an arm is calculated independently of the other arms, and the optimal reward can be expressed in terms of the indices. Somewhat less anticipated, however, is the fact that the index function of an arm, driven by a Levy process, has a representation in terms of the decreasing ladder sets and the exit system of its Levy driver. Moreover, the Wiener-Hopf factorization of the Levy exponents of an arm can be used to obtain the characteristic function of some excursion law, through which the index of the arm is defined. We use this factorization to calculate explicitly index functions and optimal rewards of some interesting Levy bandits, rediscovering along the way that local time naturally quantifies switching in continuous time.

Let $\{Z_n: n \geq 0\}$ be a $p$-type $(p \geq 2)$ supercritical branching process with mean matrix $M$. It is known that for any $l$ in $R^p$, $\big(\frac{l\cdot Z_n}{1\cdot Z_n} - \frac{l\cdot (Z_n M)}{1\cdot Z_n}\big) \text{and} \big(\frac{l\cdot Z_n}{1\cdot Z_n} - \frac{l\cdot v^{(1)}}{1 \cdot v^{(1)}}\big)$ converge to 0 with probability 1 on the set of nonextinction, where $v^{(1)}$ is the left eigenvector of $M$ corresponding to its maximal eigenvalue $\rho$ and 1 is the vector with all components equal to one. In this paper we study the large deviation aspects of this convergence. It is shown that the large deviation probabilities for these two sequences decay geometrically and under appropriate conditioning supergeometrically.