Embedding laws in diffusions by functions of time

We present a constructive probabilistic proof of the fact that if $B=(B_t)_{t\ge0}$ is standard Brownian motion started at $0$, and $\mu$ is a given probability measure on $\mathbb{R}$ such that $\mu(\{0\})=0$, then there exists a unique left-continuous increasing function $b:(0,\infty)\rightarrow\mathbb{R}\cup\{+\infty\}$ and a unique left-continuous decreasing function $c:(0,\infty)\rightarrow\mathbb{R}\cup\{-\infty\}$ such that $B$ stopped at $\tau_{b,c}=\inf\{t>0\vert B_t\ge b(t)$ or $B_t\le c(t)\}$ has the law $\mu$. The method of proof relies upon weak convergence arguments arising from Helly's selection theorem and makes use of the L\'{e}vy metric which appears to be novel in the context of embedding theorems. We show that $\tau_{b,c}$ is minimal in the sense of Monroe so that the stopped process $B^{\tau_{b,c}}=(B_{t\wedge\tau_{b,c}})_{t\ge0}$ satisfies natural uniform integrability conditions expressed in terms of $\mu$. We also show that $\tau_{b,c}$ has the smallest truncated expectation among all stopping times that embed $\mu$ into $B$. The main results extend from standard Brownian motion to all recurrent diffusion processes on the real line.Comment: Published at http://dx.doi.org/10.1214/14-AOP941 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A note on the continuity of free-boundaries in finite-horizon optimal stopping problems for one dimensional diffusions.

We provide sufficient conditions for the continuity of the free-boundary in a general class of finite-horizon optimal stopping problems arising, for instance, in finance and economics. The underlying process is a strong solution of a one-dimensional, time-homogeneous stochastic differential equation (SDE). The proof relies on both analytic and probabilistic arguments and is based on a contradiction scheme inspired by the maximum principle in partial differential equations theory. Mild, local regularity of the coefficients of the SDE and smoothness of the gain function locally at the boundary are required

Global C¹ regularity of the value function in optimal stopping problems

We show that if either the process is strong Feller and the boundary point is probabilistically regular for the stopping set, or the process is strong Markov and the boundary point is probabilistically regular for the interior of the stopping set, then the boundary point is Green regular for the stopping set. Combining this implication with the existence of a continuously differentiable flow of the process we show that the value function is continuously differentiable at the optimal stopping boundary whenever the gain function is so. The derived fact holds both in the parabolic and elliptic case of the boundary value problem under the sole hypothesis of probabilistic regularity of the optimal stopping boundary, thus improving upon known analytic results in the PDE literature, and establishing the fact for the first time in the case of integro-differential equations. The method of proof is purely probabilistic and conceptually simple. Examples of application include the first known probabilistic proof of the fact that the time derivative of the value function in the American put problem is continuous across the optimal stopping boundary

Time-randomized stopping problems for a family of utility functions

This paper studies stopping problems of the form $V=\inf_{0 \leq \tau \leq T} \mathbb{E}[U(\frac{\max_{0\le s \le T} Z_s }{Z_\tau})]$ for strictly concave or convex utility functions U in a family of increasing functions satisfying certain conditions, where Z is a geometric Brownian motion and T is the time of the nth jump of a Poisson process independent of Z. We obtain some properties of $V$ and offer solutions for the optimal strategies to follow. This provides us with a technique to build numerical approximations of stopping boundaries for the fixed terminal time optimal stopping problem presented in [J. Du Toit and G. Peskir, Ann. Appl. Probab., 19 (2009), pp. 983--1014]

Optimal prediction of resistance and support levels

Assuming that the asset price X follows a geometric Brownian motion we study the optimal prediction problem in

Precautionary Measures for Credit Risk Management in Jump Models

Sustaining efficiency and stability by properly controlling the equity to asset ratio is one of the most important and difficult challenges in bank management. Due to unexpected and abrupt decline of asset values, a bank must closely monitor its net worth as well as market conditions, and one of its important concerns is when to raise more capital so as not to violate capital adequacy requirements. In this paper, we model the tradeoff between avoiding costs of delay and premature capital raising, and solve the corresponding optimal stopping problem. In order to model defaults in a bank's loan/credit business portfolios, we represent its net worth by Levy processes, and solve explicitly for the double exponential jump diffusion process and for a general spectrally negative Levy process.Comment: 31 pages, 4 figure

Optimization in task--completion networks

We discuss the collective behavior of a network of individuals that receive, process and forward to each other tasks. Given costs they store those tasks in buffers, choosing optimally the frequency at which to check and process the buffer. The individual optimizing strategy of each node determines the aggregate behavior of the network. We find that, under general assumptions, the whole system exhibits coexistence of equilibria and hysteresis.Comment: 18 pages, 3 figures, submitted to JSTA