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Steven E. Shreve Orion Hoch Professor of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213-3890 Wean Hall 6216 (412) 268-8484 Fax: (412) 268-6380 E-mail: shreve at the address andrew.cmu.edu. |
These volumes have replaced the lecture notes
""Steven Shreve: Stochastic Calculus and Finance,
October 1997." The books differ from the lecture
notes in several ways.
Chief among them are:
1. The books include exercises, a bibliography, an index,
and a summary at the end of each chapter,
2. The second volume includes a chapter on jump processes,
3. Explanations have been expanded and topics have been
more fully developed throughout.
Errata for 2004 printing of Volume II, September 2006
More errata for 2004 printing of Volume II, July 2007
More errata for 2004 printing of Volume II, February 2008
Abstract:
This paper presents a heavy-traffic analysis of the behavior of a
single-server queue under an Earliest-Deadline-First (EDF)
scheduling policy, in which customers have deadlines and are served
only until their deadlines elapse. The performance of the
system is measured by the fraction of reneged work (the residual work
lost due to elapsed deadlines), which is shown to be minimized by
the EDF policy. The evolution of the lead time distribution of
customers in queue is described by a measure-valued process.
The heavy traffic limit of this (properly scaled) process
is shown to be a deterministic function of the limit of the scaled
workload process, which, in turn, is identified to be
a doubly reflected Brownian motion. This paper complements
previous work by Doytchinov, Lehoczky and Shreve on the EDF
discipline, in which customers are served to completion
even after their deadlines elapse. The fraction of reneged
work in a heavily loaded system and the fraction of late work
in the corresponding system without reneging are compared using
explicit formulas based on the heavy traffic approximations, which
are validated by simulation results.
Abstract:
A model for optimal consumption and investment is posed whose
solution is provided by the classical Merton analysis when
there is zero transaction cost. A probabilistic argument is
developed to identify the loss in value when a proportional
transaction cost is introduced. There are two sources of this
loss. The first is a loss due to "displacement'' that arises
because one cannot maintain the optimal portfolio of the
zero-transaction-cost problem. The second loss is due to
"transaction,'' a loss in capital that occurs when one adjusts
the portfolio. The first of these increases with increasing tolerance
for departure from the optimal portfolio in the
zero-transaction-cost problem, while the second decreases with
increases in this tolerance. This paper balances the marginal
costs of these two effects. The probabilistic analysis provided
here complements earlier work on a related model that
proceeded from a viscosity solution analysis of the associated
Hamilton-Jacobi-Bellman equation.
Abstract:
An explicit formula for the Skorokhod map $\Gamma_{0,a}$
on $[0,a]$ for $a>0$ is provided and related to similar
formulas in the literature. Specifically, it is shown
that on the space $D[0,\infty)$ of right-continuous
functions with left limits taking values in $\mathbb{R}$,
$$
\Gamma_{0,a}(\psi)(t) = \psi (t)
-\left[\big(\psi(0)-a\big)^+
\wedge\inf_{u\in[0,t]}\psi(u)\right]
\vee
\sup_{s \in [0,t]} \left[ (\psi(s) - a)
\wedge
\inf_{u \in [s,t]} \psi(u)\right]
$$
is the unique function taking values in $[0,a]$ that is
obtained from $\psi$ by minimal ``pushing'' at the
endpoints $0$ and $a$. An application of this result to
real-time queues with reneging is outlined.
Abstract:
The Skorokhod map is a convenient tool for constructing solutions to
stochastic differential equations with reflecting boundary conditions.
In this work, an explicit formula for the Skorokhod map $\Gamma_{0,a}$
on $[0,a]$ for any $a>0$ is derived.
Specifically, it is shown that on the
space $D[0,\infty)$ of right-continuous functions
with left limits taking values in R,
$\Gamma_{0,a} = \Lambda_a \circ \Gamma_0$, where
$\Lambda_a$ mapping $D[0,\infty)$ into itself is defined by
$$
\Lambda_a(\phi)(t)
=\phi(t)-\sup_{s\in[0,t]}[(\phi(s)-a)^+ \wedge \inf_{u\in [s,t]}\phi(u)]
$$
and $\Gamma_0$ mapping $D[0,\infty)$ into itself is the Skorokhod map
on $[0,\infty)$. In addition, properties of $\Lambda_a$ are developed
and comparison properties of $\Gamma_{0,a}$ are established.
Abstract:
A firm issues a convertible bond.
At each subsequent time,
the bondholder must decide
whether to continue to hold the bond,
thereby collecting
coupons, or to convert it to stock.
The bondholder wishes to choose a conversion
strategy to maximize the bond value. Subject to
some restrictions, the bond can be called by the
issuing firm, which presumably acts to maximize the
equity value of the firm by minimizing the bond value.
This creates a two-person game. We show that
if the coupon rate is below the interest rate
times the call price, then conversion should
precede call. On the other hand, if the dividend
rate times the call price is below the coupon rate,
call should precede conversion.
In either case, the game reduces to a problem of optimal
stopping.
"Finite Maturity Convertible Bonds" pdf file.
Abstract:
A random variable, representing the final position
of a trading strategy, is deemed acceptable
if under each of a variety of
probability measures its expectation
dominates a floor associated with the measure.
The set of random variables representing
pre-final positions from which it is possible
to trade to final acceptability is characterized.
In particular, the set of initial capitals
from which one can trade to final acceptability
is shown to be a closed half-line
{x;x\geq a}.
Methods for computing a are provided,
and the application of these ideas to
derivative security pricing is developed.
Abstract:
A firm issues a convertible bond.
At each subsequent time,
the bondholder must decide
whether to continue to hold the bond,
thereby collecting
coupons, or to convert it to stock.
The firm may at any time call the bond.
Because calls and conversions usually occur
far from maturity, we model this situation
with a perpetual convertible bond, i.e,
a convertible coupon-paying bond without maturity.
This model admits a relatively simple solution,
under which the value of the
perpetual convertible bond, as a function of
the value of the underlying
firm, is determined by a nonlinear
ordinary differential equation.
"PerpetualConvertibleBonds" pdf file.
Talk given at AMS meeting in Snowbird, Utah, June 2003, postscript file.
Abstract:
This paper presents a second-order heavy traffic analysis of a single
server queue that processes customers having deadlines using the
earliest-deadline-first scheduling policy. For such systems, referred
to as {\em real-time queueing systems}, performance is measured by the
fraction of customers who meet their
deadline, rather than more traditional performance measures such as
customer delay, queue length, or server utilization. To model such
systems, one must keep track of customer lead times (the time
remaining until a customer deadline elapses) or equivalent
information. This paper reviews the earlier heavy traffic analysis of
such systems that provided approximations to the system's behavior.
The main result of this paper is the development of a second-order
analysis that gives the accuracy of the approximations and the rate of
convergence of the sequence of real-time queueing systems to
its heavy traffic limit.
"Accuracy of State Space Collapse" pdf file.
"A general framework for pricing credit risk," Mathematical Finance 14 (2004), 317-350, with A. Belanger and D. Wong.
"Earliest-deadline-first service in heavy-traffic acyclie networks, Annals of Applied Probability, 14 (2004), 1306-1352 (with L. Kruk, J. Lehoczky and S.-N. Yeung).
"Asymptotic analysis for optimal investment and consumption with transaction costs, Finance and Stochastics 8 (2004), 181-206 (with K. Janecek).
"Multiple-input heavy-traffic real-time queues," Annals of Applied Probability 13 (2003), 54--99, with L. Kruk, J. Lehoczky and S.-N. Yeung.
"Dealing with dangerous digitals," in "Foreign Exchange Risk," Risk Publications, London, 2002, 327--348, with U. Schmock and U. Wystup.
"Real-time queues in heavy traffic with earliest-deadline-first queue discipline," Annals of Applied Probability 11 (2001), 332-378, with B. Doytchinov and J. Lehoczky.
"Valuation of exotic options under shortselling constraints," Finance and Stochastics 6 (2002), 143-172, with U. Schmock and U. Wystup.
"Upgrading your Passport," Risk, July 2000, 81-83.
"Options on a Traded Account: Vacation Calls, Vacation Puts and Passport Options," Finance and Stochastics 4 (2000), 255-274 (with J. Vecer).
"Robustness of the Black and Scholes formula," Mathematical Finance 8 (1998), 93-126 (with N. El Karoui and M. Jeanblanc-Picque).
"Heavy traffic convergence of a controlled, multi-class queueing system," SIAM Journal on Control and Optimization 34 (1996), 2133-2171 (with L. F. Martins and H. M. Soner).
"There is no nontrivial hedging portfolio for option pricing with transaction costs," Annals of Applied Probability 5 (1995), 327-355 (with H. M. Soner and J. Cvitanic).
"Optimal investment and consumption with transaction costs," Annals of Applied Probability 4 (1994), 609-692 (with H. M. Soner).
"Sensitivity of the indirect utility to transaction costs in a consumption-based model," Annals of Applied Probability 4 (1994), 680-689.
"A duality method for optimal consumption and investment under short-selling prohibition, Part I: General market coefficients," Annals of Applied Probability 2 (1992), 87-112 (with G.-L. Xu).
"A duality method for optimal consumption and investment under short-selling prohibition, Part II: Constant market coefficients," Annals of Applied Probability 2 (1992), 314-328 (with G.-L. Xu).
"Equilibrium models with singular asset prices," Mathematical Finance 1 (1991), 11-29 (with I. Karatzas and J. Lehoczky).
"Optimal investment and consumption with two bonds and transaction costs," Mathematical Finance 1 (1991), 53-84 (with H. M. Soner and G.-L. Xu).
"A free boundary problem related to singular stochastic control: the parabolic case," Communications in Partial Differential Equations 16 (1991), 373-424 (with H. M. Soner).
"Martingale and duality methods for utility maximization in an incomplete market," SIAM Journal on Control and Optimization 29 (1991), 707-730 (with I. Karatzas, J. Lehoczky and G.-L. Xu).
"Existence and uniqueness of multi-agent equilibrium in a stochastic, dynamic consumption/investment model," Mathematics of Operations Research 15 (1990), 80-128 (with I. Karatzas and J. Lehoczky).
"Regularity of the value function for a two-dimensional singular stochastic control problem," SIAM Journal on Control and Optimization 27 (1989), 876-907 (with H. M. Soner).
"Optimal portfolio and consumption decisions for a `small investor' on a finite horizon," SIAM Journal on Control and Optimization 25 (1987), 1557-1586 (with I. Karatzas and J. Lehoczky).
"Explicit solution of a general consumption/investment problem," Mathematics of Operations Research 11 (1986), 261-294 (with I. Karatzas, J. Lehoczky and S. Sethi).
"Connections between optimal stopping and singular stochastic control, Part I: Monotone follower problems," SIAM Journal on Control and Optimization 22 (1984), 856-877 (with I. Karatzas).
"Connection between optimal stopping and singular stochastic control, Part II: Reflected follower problems," SIAM Journal on Control and Optimization 23 (1985), 433-451 (with I. Karatzas).
"Trivariate density of Brownian motion, its local and occupation times, with application," Annals of Probability 12 (1984), 819-828 (with I. Karatzas).
"Reflected Brownian motion in the `bang-bang' control of Brownian drift," SIAM Journal on Control and Optimization 19 (1981), 469-478.