49 research outputs found
Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients
In this work, we generalize the current theory of strong convergence rates for the backward Euler–Maruyama scheme for highly non-linear stochastic differential equations, which appear in both mathematical finance and bio-mathematics. More precisely, we show that under a dissipative condition on the drift coefficient and superlinear growth condition on the diffusion coefficient the BEM scheme converges with strong order of a half. This type of convergence gives theoretical foundations for efficient variance reduction techniques for Monte Carlo simulations. We support our theoretical results with relevant examples, such as stochastic population models and stochastic volatility models
Comparing hitting time behaviour of Markov jump processes and their diffusion approximations
Markov jump processes can provide accurate models in many applications, notably chemical and biochemical kinetics, and population dynamics. Stochastic differential equations offer a computationally efficient way to approximate these processes. It is therefore of interest to establish results that shed light on the extent to which the jump and diffusion models agree. In this work we focus on mean hitting time behavior in a thermodynamic limit. We study three simple types of reactions where analytical results can be derived, and we find that the match between mean hitting time behavior of the two models is vastly different in each case. In particular, for a degradation reaction we find that the relative discrepancy decays extremely slowly, namely, as the inverse of the logarithm of the system size. After giving some further computational results, we conclude by pointing out that studying hitting times allows the Markov jump and stochastic differential equation regimes to be compared in a manner that avoids pitfalls that may invalidate other approaches
Multilevel Monte Carlo methods for applications in finance
Since Giles introduced the multilevel Monte Carlo path simulation method
[18], there has been rapid development of the technique for a variety of
applications in computational finance. This paper surveys the progress so far,
highlights the key features in achieving a high rate of multilevel variance
convergence, and suggests directions for future research.Comment: arXiv admin note: text overlap with arXiv:1202.6283; and with
arXiv:1106.4730 by other author
Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients
We are interested in the strong convergence and almost sure stability of
Euler-Maruyama (EM) type approximations to the solutions of stochastic
differential equations (SDEs) with non-linear and non-Lipschitzian
coefficients. Motivation comes from finance and biology where many widely
applied models do not satisfy the standard assumptions required for the strong
convergence. In addition we examine the globally almost surely asymptotic
stability in this non-linear setting for EM type schemes. In particular, we
present a stochastic counterpart of the discrete LaSalle principle from which
we deduce stability properties for numerical methods
First order strong approximations of scalar SDEs with values in a domain
We are interested in strong approximations of one-dimensional SDEs which have
non-Lipschitz coefficients and which take values in a domain. Under a set of
general assumptions we derive an implicit scheme that preserves the domain of
the SDEs and is strongly convergent with rate one. Moreover, we show that this
general result can be applied to many SDEs we encounter in mathematical finance
and bio-mathematics. We will demonstrate flexibility of our approach by
analysing classical examples of SDEs with sublinear coefficients (CIR, CEV
models and Wright-Fisher diffusion) and also with superlinear coefficients
(3/2-volatility, Ait-Sahalia model).
Our goal is to justify an efficient Multi-Level Monte Carlo (MLMC) method for
a rich family of SDEs, which relies on good strong convergence properties
Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without L\'{e}vy area simulation
In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for
multi-dimensional SDEs driven by Brownian motions. Giles has previously shown
that if we combine a numerical approximation with strong order of convergence
with MLMC we can reduce the computational complexity to estimate
expected values of functionals of SDE solutions with a root-mean-square error
of from to . However, in
general, to obtain a rate of strong convergence higher than
requires simulation, or approximation, of L\'{e}vy areas. In this paper,
through the construction of a suitable antithetic multilevel correction
estimator, we are able to avoid the simulation of L\'{e}vy areas and still
achieve an multilevel correction variance for smooth payoffs,
and almost an variance for piecewise smooth payoffs, even
though there is only strong convergence. This results in an
complexity for estimating the value of European and Asian
put and call options.Comment: Published in at http://dx.doi.org/10.1214/13-AAP957 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Convergence and qualitative properties of modified explicit schemes for BSDEs with polynomial growth
The theory of Forward-Backward Stochastic Differential Equations (FBSDEs)
paves a way to probabilistic numerical methods for nonlinear parabolic PDEs.
The majority of the results on the numerical methods for FBSDEs relies on the
global Lipschitz assumption, which is not satisfied for a number of important
cases such as the Fisher--KPP or the FitzHugh--Nagumo equations. Furthermore,
it has been shown in \cite{LionnetReisSzpruch2015} that for BSDEs with monotone
drivers having polynomial growth in the primary variable , only the
(sufficiently) implicit schemes converge. But these require an additional
computational effort compared to explicit schemes.
This article develops a general framework that allows the analysis, in a
systematic fashion, of the integrability properties, convergence and
qualitative properties (e.g.~comparison theorem) for whole families of modified
explicit schemes. The framework yields the convergence of some modified
explicit scheme with the same rate as implicit schemes and with the
computational cost of the standard explicit scheme.
To illustrate our theory, we present several classes of easily implementable
modified explicit schemes that can computationally outperform the implicit one
and preserve the qualitative properties of the solution to the BSDE. These
classes fit into our developed framework and are tested in computational
experiments.Comment: 49 pages, 3 figure
A limit order book model for latency arbitrage
We consider a single security market based on a limit order book and two
investors, with different speeds of trade execution. If the fast investor can
front-run the slower investor, we show that this allows the fast trader to
obtain risk free profits, but that these profits cannot be scaled. We derive
the fast trader's optimal behaviour when she has only distributional knowledge
of the slow trader's actions, with few restrictions on the possible prior
distributions. We also consider the slower trader's response to the presence of
a fast trader in a market, and the effects of the introduction of a `Tobin tax'
on financial transactions. We show that such a tax can lead to the elimination
of profits from front-running strategies. Consequently, a Tobin tax can both
increase market efficiency and attract traders to a market