161 research outputs found
Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative
In this article, we extend a Milstein finite difference scheme introduced in
[Giles & Reisinger(2011)] for a certain linear stochastic partial differential
equation (SPDE), to semi- and fully implicit timestepping as introduced by
[Szpruch(2010)] for SDEs. We combine standard finite difference Fourier
analysis for PDEs with the linear stability analysis in [Buckwar &
Sickenberger(2011)] for SDEs, to analyse the stability and accuracy. The
results show that Crank-Nicolson timestepping for the principal part of the
drift with a partially implicit but negatively weighted double It\^o integral
gives unconditional stability over all parameter values, and converges with the
expected order in the mean-square sense. This opens up the possibility of local
mesh refinement in the spatial domain, and we show experimentally that this can
be beneficial in the presence of reduced regularity at boundaries
The non-locality of Markov chain approximations to two-dimensional diffusions
In this short paper, we consider discrete-time Markov chains on lattices as
approximations to continuous-time diffusion processes. The approximations can
be interpreted as finite difference schemes for the generator of the process.
We derive conditions on the diffusion coefficients which permit transition
probabilities to match locally first and second moments. We derive a novel
formula which expresses how the matching becomes more difficult for larger
(absolute) correlations and strongly anisotropic processes, such that
instantaneous moves to more distant neighbours on the lattice have to be
allowed. Roughly speaking, for non-zero correlations, the distance covered in
one timestep is proportional to the ratio of volatilities in the two
directions. We discuss the implications to Markov decision processes and the
convergence analysis of approximations to Hamilton-Jacobi-Bellman equations in
the Barles-Souganidis framework.Comment: Corrected two errata from previous and journal version: definition of
R in (5) and summations in (7
On multigrid for anisotropic equations and variational inequalities: pricing multi-dimensional European and American options
Partial differential operators in finance often originate in bounded linear stochastic processes. As a consequence, diffusion over these boundaries is zero and the corresponding coefficients vanish. The choice of parameters and stretched grids lead to additional anisotropies in the discrete equations or inequalities. In this study various block smoothers are tested in numerical experiments for equations of Black–Scholes-type (European options) in several dimensions. For linear complementarity problems, as they arise from optimal stopping time problems (American options), the choice of grid transfer is also crucial to preserve complementarity conditions on all grid levels. We adapt the transfer operators at the free boundary in a suitable way and compare with other strategies including cascadic approaches and full approximation schemes
Piecewise Constant Policy Approximations to Hamilton-Jacobi-Bellman Equations
An advantageous feature of piecewise constant policy timestepping for
Hamilton-Jacobi-Bellman (HJB) equations is that different linear approximation
schemes, and indeed different meshes, can be used for the resulting linear
equations for different control parameters. Standard convergence analysis
suggests that monotone (i.e., linear) interpolation must be used to transfer
data between meshes. Using the equivalence to a switching system and an
adaptation of the usual arguments based on consistency, stability and
monotonicity, we show that if limited, potentially higher order interpolation
is used for the mesh transfer, convergence is guaranteed. We provide numerical
tests for the mean-variance optimal investment problem and the uncertain
volatility option pricing model, and compare the results to published test
cases
Error analysis of truncated expansion solutions to high-dimensional parabolic PDEs
We study an expansion method for high-dimensional parabolic PDEs which
constructs accurate approximate solutions by decomposition into solutions to
lower-dimensional PDEs, and which is particularly effective if there are a low
number of dominant principal components. The focus of the present article is
the derivation of sharp error bounds for the constant coefficient case and a
first and second order approximation. We give a precise characterisation when
these bounds hold for (non-smooth) option pricing applications and provide
numerical results demonstrating that the practically observed convergence speed
is in agreement with the theoretical predictions
Modeling basket credit default swaps with default contagion
The specification of a realistic dependence structure is key to the pricing of multi-name credit derivatives. We value small kth-to-default CDS baskets in the presence of asset correlation and default contagion. Using a first-passage framework, firm values are modeled as correlated geometric Brownian motions with exponential default thresholds. Idiosyncratic links between companies are incorporated through a contagion mechanism whereby a default event leads to jumps in volatility at related entities. Our framework allows for default causality and is extremely flexible, enabling us to evaluate the spread impact of firm value correlations and credit contagion for symmetric and asymmetric baskets
Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems
In this paper, we establish that for a wide class of controlled stochastic
differential equations (SDEs) with stiff coefficients, the value functions of
corresponding zero-sum games can be represented by a deep artificial neural
network (DNN), whose complexity grows at most polynomially in both the
dimension of the state equation and the reciprocal of the required accuracy.
Such nonlinear stiff systems may arise, for example, from Galerkin
approximations of controlled stochastic partial differential equations (SPDEs),
or controlled PDEs with uncertain initial conditions and source terms. This
implies that DNNs can break the curse of dimensionality in numerical
approximations and optimal control of PDEs and SPDEs. The main ingredient of
our proof is to construct a suitable discrete-time system to effectively
approximate the evolution of the underlying stochastic dynamics. Similar ideas
can also be applied to obtain expression rates of DNNs for value functions
induced by stiff systems with regime switching coefficients and driven by
general L\'{e}vy noise.Comment: This revised version has been accepted for publication in Analysis
and Application
Strong convergence rates for Euler approximations to a class of stochastic path-dependent volatility models
We consider a class of stochastic path-dependent volatility models where the
stochastic volatility, whose square follows the Cox-Ingersoll-Ross model, is
multiplied by a (leverage) function of the spot price, its running maximum, and
time. We propose a Monte Carlo simulation scheme which combines a log-Euler
scheme for the spot process with the full truncation Euler scheme or the
backward Euler-Maruyama scheme for the squared stochastic volatility component.
Under some mild regularity assumptions and a condition on the Feller ratio, we
establish the strong convergence with order 1/2 (up to a logarithmic factor) of
the approximation process up to a critical time. The model studied in this
paper contains as special cases Heston-type stochastic-local volatility models,
the state-of-the-art in derivative pricing, and a relatively new class of
path-dependent volatility models. The present paper is the first to prove the
convergence of the popular Euler schemes with a positive rate, which is
moreover consistent with that for Lipschitz coefficients and hence optimal.Comment: 34 pages, 5 figure
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