561 research outputs found
Nonlinear Parabolic Equations arising in Mathematical Finance
This survey paper is focused on qualitative and numerical analyses of fully
nonlinear partial differential equations of parabolic type arising in financial
mathematics. The main purpose is to review various non-linear extensions of the
classical Black-Scholes theory for pricing financial instruments, as well as
models of stochastic dynamic portfolio optimization leading to the
Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both
problems can be represented by solutions to nonlinear parabolic equations.
Qualitative analysis will be focused on issues concerning the existence and
uniqueness of solutions. In the numerical part we discuss a stable
finite-volume and finite difference schemes for solving fully nonlinear
parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387
Expected Utility Maximization and Conditional Value-at-Risk Deviation-based Sharpe Ratio in Dynamic Stochastic Portfolio Optimization
In this paper we investigate the expected terminal utility maximization
approach for a dynamic stochastic portfolio optimization problem. We solve it
numerically by solving an evolutionary Hamilton-Jacobi-Bellman equation which
is transformed by means of the Riccati transformation. We examine the
dependence of the results on the shape of a chosen utility function in regard
to the associated risk aversion level. We define the
Conditional value-at-risk deviation () based Sharpe ratio for
measuring risk-adjusted performance of a dynamic portfolio. We compute optimal
strategies for a portfolio investment problem motivated by the German DAX 30
Index and we evaluate and analyze the dependence of the -based Sharpe
ratio on the utility function and the associated risk aversion level
Jump-Diffusion Risk-Sensitive Asset Management I: Diffusion Factor Model
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. The
criterion, following earlier work by Bielecki, Pliska, Nagai and others, is
risk-sensitive optimization (equivalent to maximizing the expected growth rate
subject to a constraint on variance.) By using a change of measure technique
introduced by Kuroda and Nagai we show that the problem reduces to solving a
certain stochastic control problem in the factor process, which has no jumps.
The main result of the paper is to show that the risk-sensitive jump diffusion
problem can be fully characterized in terms of a parabolic
Hamilton-Jacobi-Bellman PDE rather than a PIDE, and that this PDE admits a
classical C^{1,2} solution.Comment: 33 page
Learning the solution operator of a nonlinear parabolic equation using physics informed deep operator network
This study focuses on addressing the challenges of solving analytically
intractable differential equations that arise in scientific and engineering
fields such as Hamilton-Jacobi-Bellman. Traditional numerical methods and
neural network approaches for solving such equations often require independent
simulation or retraining when the underlying parameters change. To overcome
this, this study employs a physics-informed DeepONet (PI-DeepONet) to
approximate the solution operator of a nonlinear parabolic equation.
PI-DeepONet integrates known physics into a deep neural network, which learns
the solution of the PDE
Controlled diffusion processes
This article gives an overview of the developments in controlled diffusion
processes, emphasizing key results regarding existence of optimal controls and
their characterization via dynamic programming for a variety of cost criteria
and structural assumptions. Stochastic maximum principle and control under
partial observations (equivalently, control of nonlinear filters) are also
discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation
In this paper we propose the notion of continuous-time dynamic spectral
risk-measure (DSR). Adopting a Poisson random measure setting, we define this
class of dynamic coherent risk-measures in terms of certain backward stochastic
differential equations. By establishing a functional limit theorem, we show
that DSRs may be considered to be (strongly) time-consistent continuous-time
extensions of iterated spectral risk-measures, which are obtained by iterating
a given spectral risk-measure (such as Expected Shortfall) along a given
time-grid. Specifically, we demonstrate that any DSR arises in the limit of a
sequence of such iterated spectral risk-measures driven by lattice-random
walks, under suitable scaling and vanishing time- and spatial-mesh sizes. To
illustrate its use in financial optimisation problems, we analyse a dynamic
portfolio optimisation problem under a DSR.Comment: To appear in Finance and Stochastic
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