We present a goal programming model for risk minimization of a financial
portfolio managed by an agent subject to different possible criteria. We extend
the classical risk minimization model with scalar risk measures to general case
of set-valued risk measure. The problem we obtain is a set-valued optimization
program and we propose a goal programming-based approach to obtain a solution
which represents the best compromise between goals and the achievement levels.
Numerical examples are provided to illustrate how the method works in practical
situations.

We discuss the use of saddlepoint methods in the analysis of portfolios, with
particular reference to credit portfolios. The objective is to proceed from a
model of the loss distribution, given through probabilities, correlations and
the like, to an analytical approximation of the distribution. Once this is done
we show how to derive the so-called risk contributions which are the
derivatives of risk measures, such as a given quantile (VaR) or expected
shortfall, to the allocations in the underlying assets.

We consider a model of optimal investment and consumption with both
habit-formation and partial observations in incomplete Ito processes markets.
The individual investor develops addictive consumption habits gradually while
he can only observe the market stock prices but not the instantaneous rates of
return, which follow Ornstein-Uhlenbeck processes. Applying the Kalman-Bucy
filtering theorem and Dynamic Programming arguments, we solve the associated
HJB equation fully explicitly for this path dependent stochastic control
problem in the case of power utility preferences.

To any utility maximization problem under transaction costs one can assign a
frictionless model with a price process $S^*$, lying in the bid/ask price
interlval $[\underline S, \bar{S}]$. Such process $S^*$ is called a
\emph{shadow price} if it provides the same optimal utility value as in the
original model with bid-ask spread.

For utility maximization problems under proportional transaction costs, it
has been observed that the original market with transaction costs can sometimes
be replaced by a frictionless "shadow market" that yields the same optimal
strategy and utility. However, the question of whether or not this indeed holds
in generality has remained elusive so far. In this paper we present a
counterexample which shows that shadow prices may fail to exist.

We consider a Black-Scholes market in which a number of stocks and an index
are traded. The simplified Capital Asset Pricing Model is the conjunction of
the usual Capital Asset Pricing Model, or CAPM, and the statement that the
appreciation rate of the index is equal to its squared volatility plus the
interest rate. (The mathematical statement of the conjunction is simpler than
that of the usual CAPM.) Our main result is that either we can outperform the
index or the simplified CAPM holds.

Exchange Traded Funds (ETFs) have been gaining increasing popularity in the
investment community as is evidenced by the high growth both in the number of
ETFs and their net assets since 2000. As ETFs are in nature similar to index
mutual funds, in this paper we examined if this growing demand for ETFs can be
explained through their outperformance as compared to index mutual funds.

We study the consistency of sample mean-variance portfolios of arbitrarily
high dimension that are based on Bayesian or shrinkage estimation of the input
parameters as well as weighted sampling. In an asymptotic setting where the
number of assets remains comparable in magnitude to the sample size, we provide
a characterization of the estimation risk by providing deterministic
equivalents of the portfolio out-of-sample performance in terms of the
underlying investment scenario.

In this paper we study several classical problems of optimal investment with
intermediate consumption and random endowment in incomplete markets. We
establish the key assertions of the utility maximization theory assuming that
both primal and dual value functions are finite in the interiors of their
domains as well as that random endowment at maturity can be dominated by the
terminal value of a self-financing wealth process. In order to facilitate
verification of these conditions, we present alternative, but equivalent
conditions, under which the conclusions of the theory hold.

For an investor with constant absolute risk aversion and a long horizon, who
trades in a market with constant investment opportunities and small
proportional transaction costs, we obtain explicitly the optimal investment
policy, its implied welfare, liquidity premium, and trading volume. We identify
these quantities as the limits of their isoelastic counterparts for high levels
of risk aversion. The results are robust with respect to finite horizons, and
extend to multiple uncorrelated risky assets.

We study the generalized composite pure and randomized hypothesis testing
problems. In addition to characterizing the corresponding optimal tests, we
examine the conditions under which these two hypothesis testing problems are
equivalent, and provide counterexamples when they are not. This analysis is
useful for portfolio optimization problems that maximize some success
probability given a fixed initial capital. The corresponding dual is related to
a pure hypothesis testing problem which may or may not coincide with the
randomized hypothesis testing problem.

Portfolio managers are often constrained by turnover limits, minimum and
maximum stock positions, cardinality, a target market capitalization and
sometimes the need to hew to a style (growth or value). In addition, many
portfolio managers choose stocks based upon fundamental data, e.g.
price-to-earnings and dividend yield in an effort to maximize return. All of
these are typical real-world constraints a portfolio manager faces.

We consider the utility maximization problem of terminal wealth from the
point of view of a portfolio manager paid by an incentive scheme given as a
convex function $g$ of the terminal wealth. The manager's own utility function
$U$ is assumed to be smooth and strictly concave, however the resulting utility
function $U \circ g$ fails to be concave. As a consequence, this problem does
not fit into the classical portfolio optimization theory.

A geometric analysis of the time series of returns has been performed in the
past and it implied that the most of the systematic information of the market
is contained in a space of small dimension. Here we have explored subspaces of
this space to find out the relative performance of portfolios formed from the
companies that have the largest projections in each one of the subspaces. It
was found that the best performance portfolios are associated to some of the
small eigenvalue subspaces and not to the dominant directions in the distances
matrix.

A wealth-process set is abstractly defined to consist of nonnegative cadlag
processes containing a strictly positive semimartingale and satisfying an
intuitive re-balancing property. Under the condition of absence of arbitrage of
the first kind, it is established that all wealth processes are
semimartingales, and that the closure of the wealth-process set in the Emery
topology contains all "optimal" wealth processes.

We consider an illiquid financial market with different regimes modeled by a
continuous-time finite-state Markov chain. The investor can trade a stock only
at the discrete arrival times of a Cox process with intensity depending on the
market regime. Moreover, the risky asset price is subject to liquidity shocks,
which change its rate of return and volatility, and induce jumps on its
dynamics. In this setting, we study the problem of an economic agent optimizing
her expected utility from consumption under a non-bankruptcy constraint.

We investigate the continuity of expected exponential utility maximization
with respect to perturbation of the Sharpe ratio of markets. By focusing only
on continuity, we impose weaker regularity conditions than those found in the
literature. Specifically, for markets of the form $S = M + \int \lambda d<M>$,
we require a uniform bound on the norm of $\lambda \cdot M$ in a suitable $bmo$
space.

We study natural selection in complete financial markets, populated by
heterogeneous agents. We allow for a rich structure of heterogeneity:
Individuals may differ in their beliefs concerning the economy, information and
learning mechanism, risk aversion, impatience (time preference rate) and degree
of habits. We develop new techniques for studying long run behavior of such
economies, based on the Strassen's functional law of iterated logarithm. In
particular, we explicitly determine an agent's survival index and show how the
latter depends on the agent's characteristics.

We provide a detailed characterization of the optimal consumption stream for
the additive habit-forming utility maximization problem, in a framework of
general discrete-time incomplete markets and random endowments. This
characterization allows us to derive the monotonicity and concavity of the
optimal consumption as a function of wealth, for several important classes of
incomplete markets and preferences. These results yield a deeper understanding
of the fine structure of the optimal consumption and provide a further
theoretical support for the classical conjectures of Keynes (1936).

Several portfolio selection models take into account practical limitations on
the number of assets to include and on their weights in the portfolio. We
present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset
Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional
Value-at-Risk (LACVaR) models, where the assets are limited with the
introduction of quantity and cardinality constraints. We propose a completely
new approach for solving the LAM model, based on reformulation as a Standard
Quadratic Program and on some recent theoretical results.

The paper studies problem of continuous time optimal portfolio selection for
a diffusion model of incomplete market. It is shown that, under some mild
conditions, the suboptimal strategies for investors with different performance
criterions can be constructed using a limited number of fixed processes (mutual
funds), for a market with a larger number of available risky stocks. In other
words, a relaxed version of Mutual Fund Theorem is obtained.

We consider a portfolio optimization problem in a defaultable market with
finitely-many economical regimes, where the investor can dynamically allocate
her wealth among a defaultable bond, a stock, and a money market account. The
market coefficients are assumed to depend on the market regime in place, which
is modeled by a finite state continuous time Markov process. We rigorously
deduce the dynamics of the defaultable bond price process in terms of a Markov
modulated stochastic differential equation.

Real Estate Investment Trusts (REITs) are the only truly liquid assets
related to real estate investments. We study the behavior of U.S. REITs over
the past three decades and document their return characteristics. REITs have
somewhat less market risk than equity; their betas against a broad market index
average about .65. Decomposing their covariances into principal components
reveals several strong factors. REIT characteristics differ to some extent from
those of the S&P/Case-Shiller (SCS) residential real estate indexes. This is
partly attributable to methods of index construction.

Consider power utility maximization of terminal wealth in a 1-dimensional
continuous-time exponential Levy model with finite time horizon. We discretize
the model by restricting portfolio adjustments to an equidistant discrete time
grid. Under minimal assumptions we prove convergence of the optimal
discrete-time strategies to the continuous-time counterpart. In addition, we
provide and compare qualitative properties of the discrete-time and
continuous-time optimizers.

We study the problem of super-replication for game options under proportional
transaction costs. We consider a multidimensional model which is an extension
of the usual Black-Scholes (BS) model, in the sense that the volatility is a
progressively measurable function of the stock. For this case we show that the
super-replication price is the cheapest cost of a trivial super-replication
strategy. This result is an extension of previous papers (see [1], [2], [10]
and [11]) in which only European options with Markovian structure were
considered.

In this article we extend earlier work on the jump-diffusion risk-sensitive
asset management problem [SIAM J. Fin. Math. (2011) 22-54] by allowing jumps in
both the factor process and the asset prices, as well as stochastic volatility
and investment constraints. In this case, the HJB equation is a partial
integro-differential equation (PIDE). By combining viscosity solutions with a
change of notation, a policy improvement argument and classical results on
parabolic PDEs we prove that the HJB PIDE admits a unique smooth solution.

We introduce an extension to Merton's famous continuous time model of optimal
consumption and investment, in the spirit of previous works by Pliska and Ye,
to allow for a wage earner to have a random lifetime and to use a portion of
the income to purchase life insurance in order to provide for his estate, while
investing his savings in a financial market comprised of one risk-free security
and an arbitrary number of risky securities driven by multi-dimensional
Brownian motion.

We consider a utility-maximization problem in a general semimartingale
financial market, subject to constraints on the number of shares held in each
risky asset. These constraints are modeled by predictable convex-set-valued
processes whose values do not necessarily contain the origin, i.e., no risky
investment at all may be inadmissible. Such a setup subsumes the classical
constrained utility-maximization problem, as well as the problem where illiquid
assets or a random endowment are present.

Portfolio turnpikes state that, as the investment horizon increases, optimal
portfolios for generic utilities converge to those of isoelastic utilities.
This paper proves three kinds of turnpikes. The abstract turnpike, valid in a
general semimartingale setting, states that final payoffs and portfolios
converge under their myopic probabilities.

This paper is devoted to study the effects arising from imposing a
value-at-risk (VaR) constraint in mean-variance portfolio selection problem for
an investor who receives a stochastic cash flow which he/she must then invest
in a continuous-time financial market.

In this paper we extend the stability results of [4]}. Our utility
maximization problem is defined as an essential supremum of conditional
expectations of the terminal values of wealth processes, conditioned on the
filtration at the stopping time $\tau$. The stability result, in particular,
implies that in the framework of [4], the optimal wealth at any given stopping
time is stable with respect to changes in the Sharpe ratio and initial wealth.
To establish our results, we extend the classical results of convex analysis to
maps from $L^0$ to $L^0$.

A stock loan is a contract whereby a stockholder uses shares as collateral to
borrow money from a bank or financial institution. In Xia and Zhou (2007), this
contract is modeled as a perpetual American option with a time varying strike
and analyzed in detail within a risk--neutral framework. In this paper, we
extend the valuation of such loans to an incomplete market setting, which takes
into account the natural trading restrictions faced by the client.

We revisit the problem of maximizing expected logarithmic utility from
consumption over an infinite horizon in the Black-Scholes model with
proportional transaction costs, as studied in the seminal paper of Davis and
Norman [Math. Operation Research, 15, 1990]. Similarly to Kallsen and
Muhle-Karbe [Ann. Appl. Probab., 20, 2010], we tackle this problem by
determining a shadow price, that is, a frictionless price process with values
in the bid-ask spread which leads to the same optimization problem. However, we
use a different parametrization, which facilitates computation and
verification.

The growth-optimal portfolio optimization strategy pioneered by Kelly is
based on constant portfolio rebalancing which makes it sensitive to transaction
fees. We examine the effect of fees on an example of a risky asset with a
binary return distribution and show that the fees may give rise to an optimal
period of portfolio rebalancing. The optimal period is found analytically in
the case of lognormal returns. This result is consequently generalized and
numerically studied for broad return distributions and returns generated by a
GARCH process.

The typical behavior of optimal solutions to portfolio optimization problems
with absolute deviation and expected shortfall models using replica analysis
was pioneeringly estimated by S. Ciliberti and M. M\'ezard [Eur. Phys. B. 57,
175 (2007)]; however, they have not yet developed an approximate derivation
method for finding the optimal portfolio with respect to a given return set.

In this short report, we discuss how coordinate-wise descent algorithms can
be used to solve minimum variance portfolio (MVP) problems in which the
portfolio weights are constrained by $l_{q}$ norms, where $1\leq q \leq 2$. A
portfolio which weights are regularised by such norms is called a sparse
portfolio (Brodie et al.), since these constraints facilitate sparsity (zero
components) of the weight vector. We first consider a case when the portfolio
weights are regularised by a weighted $l_{1}$ and squared $l_{2}$ norm.

A competing market model with a polyvariant profit function that assumes
"zeitnot" stock behavior of clients is formulated within the banking portfolio
medium and then analyzed from the perspective of devising optimal strategies.
An associated Markov process method for finding an optimal choice strategy for
monovariant and bivariant profit functions is developed.

In this paper an econophysics model for the currency exchange operations with
commission is proposed. With this purpose some analogies and similarities of
the processes that take place in the frame of the electrochemical system made
from electrodes sunk into a solution of electrolytes and the process of the
currency exchange and determination of the international currency purchasing
power have been used.

In this paper we consider dividend problem for an insurance company whose
risk evolves as a spectrally negative L\'{e}vy process (in the absence of
dividend payments) when Parisian delay is applied. The objective function is
given by the cumulative discounted dividends received until the moment of ruin
when so-called barrier strategy is applied. Additionally we will consider two
possibilities of delay. In the first scenario ruin happens when the surplus
process stays below zero longer than fixed amount of time $\zeta>0$.

We present an approach to derivative exposure management based on subjective
and implied probabilities. We suggest to maximize the valuation difference
subject to risk constraints and propose a class of risk measures derived from
the subjective distribution. We illustrate this process with specific examples
for the two and three dimensional case. In these cases the optimization can be
performed graphically.

We assume that an individual invests in a financial market with one riskless
and one risky asset, with the latter's price following a diffusion with
stochastic volatility. In the current financial market especially, it is
important to include stochastic volatility in the risky asset's price process.
Given the rate of consumption, we find the optimal investment strategy for the
individual who wishes to minimize the probability of going bankrupt. To solve
this minimization problem, we use techniques from stochastic optimal control.

This paper considers a portfolio optimization problem in which asset prices
are represented by SDEs driven by Brownian motion and a Poisson random measure,
with drifts that are functions of an auxiliary diffusion 'factor' process.

We consider the optimal investment problem for Black-Scholes type financial
market with bounded VaR measure on the whole investment interval $[0,T]$. The
explicit form for the optimal strategies is found.

We investigate optimal consumption problems for a Black-Scholes market under
uniform restrictions on Value-at-Risk and Expected Shortfall for logarithmic
utility functions. We find the solutions in terms of a dynamic strategy in
explicit form, which can be compared and interpreted. This paper continues our
previous work, where we solved similar problems for power utility functions.

In this article we extend earlier work on the jump-diffusion risk-sensitive
asset management problem by allowing for jumps in both the factor process and
the asset prices as well as stochastic volatility and investment constraints.
In this case, the HJB equation is a PIDE. By combining viscosity solutions with
a change of notation, a policy improvement argument and classical results on
parabolic PDEs we prove that the PIDE admits a unique smooth solution. A
verification theorem concludes the resolutions of this problem.

Kramkov and Sirbu (2006, 2007) have shown that first-order approximations of
power utility-based prices and hedging strategies can be computed by solving a
mean-variance hedging problem under a specific equivalent martingale measure
and relative to a suitable numeraire. In order to avoid the introduction of an
additional state variable necessitated by the change of numeraire, we propose
an alternative representation in terms of the original numeraire.

We study power utility maximization for exponential L\'evy models with
portfolio constraints, where utility is obtained from consumption and/or
terminal wealth. For convex constraints, an explicit solution in terms of the
L\'evy triplet is constructed under minimal assumptions by solving the Bellman
equation. We use a novel transformation of the model to avoid technical
conditions. The consequences for q-optimal martingale measures are discussed as
well as extensions to non-convex constraints.

We study the utility maximization problem for power utility random fields in
a semimartingale financial market, with and without intermediate consumption.
The notion of an opportunity process is introduced as a reduced form of the
value process of the resulting stochastic control problem. We show how the
opportunity process describes the key objects: optimal consumption, value
function, and dual problem. The results are applied to obtain monotonicity
properties of the optimal consumption.

In this paper we derive the optimal execution trajectory for a trader who
wishes to buy or sell a large position of shares which evolve as a geometric
Brownian process in contrast to the arithmetic model which prevails in the
existing literature, and with a general temporary impact $h$. We provide a
couple of examples which illustrate the results. We would like to stress the
fact that in this paper we use understandable user-friendly techniques.

We study the optimal investment problem for a continuous time incomplete
market model such that the risk-free rate, the appreciation rates and the
volatility of the stocks are all random; they are assumed to be independent
from the driving Brownian motion, and they are supposed to be currently
observable.

The paper studies the robust maximization of utility of terminal wealth in
the diffusion financial market model. The underlying model consists with risky
tradable asset, whose price is described by diffusion process with misspecified
trend and volatility coefficients, and non-tradable asset with a known
parameter. The robust utility functional is defined in terms of a HARA utility
function. We give explicit characterization of the solution of the problem by
means of a solution of the HJBI equation.

The optimization of large portfolios displays an inherent instability to
estimation error. This poses a fundamental problem, because solutions that are
not stable under sample fluctuations may look optimal for a given sample, but
are, in effect, very far from optimal with respect to the average risk. In this
paper, we approach the problem from the point of view of statistical learning
theory. The occurrence of the instability is intimately related to over-fitting
which can be avoided using known regularization methods.

The aim of this work is to extend the capital growth theory developed by
Kelly, Breiman, Cover and others to asset market models with transaction costs.
We define a natural generalization of the notion of a numeraire portfolio
proposed by Long and show how such portfolios can be used for constructing
growth-optimal investment strategies. The analysis is based on the classical
von Neumann-Gale model of economic dynamics, a stochastic version of which we
use as a framework for the modelling of financial markets with frictions.

We consider the problem of dynamic buying and selling of shares from a
collection of $N$ stocks with random price fluctuations. To limit investment
risk, we place an upper bound on the total number of shares kept at any time.
Assuming that prices evolve according to an ergodic process with a mild
decaying memory property, and assuming constraints on the total number of
shares that can be bought and sold at any time, we develop a trading policy
that comes arbitrarily close to achieving the profit of an ideal policy that
has perfect knowledge of future events.