This paper deals with moduli of continuity for paths of random processes indexed by a general metric space
Θ
with values in a general metric space
X
. Adapting the moment condition on the increments ...from the classical Kolmogorov–Chentsov theorem, the obtained result on the modulus of continuity allows for Hölder-continuous modifications if the metric space
X
is complete. This result is universal in the sense that its applicability depends only on the geometry of the space
Θ
. In particular, it is always applicable if
Θ
is a bounded subset of a Euclidean space or a relatively compact subset of a connected Riemannian manifold. The derivation is based on refined chaining techniques developed by Talagrand. As a consequence of the main result, a criterion is presented to guarantee uniform tightness of random processes with continuous paths. This is applied to find central limit theorems for Banach-valued random processes.
We analyse an optimal trade execution problem in a financial market with stochastic liquidity. To this end, we set up a limit order book model in continuous time. Both order book depth and resilience ...are allowed to evolve randomly in time. We allow trading in both directions and for càdlàg semimartingales as execution strategies. We derive a quadratic BSDE that under appropriate assumptions characterises minimal execution costs, and we identify conditions under which an optimal execution strategy exists. We also investigate qualitative aspects of optimal strategies such as e.g. appearance of strategies with infinite variation or existence of block trades, and we discuss connections with the discrete-time formulation of the problem. Our findings are illustrated in several examples.
The stochastic exponential
of a continuous local martingale
M
is itself a continuous local martingale. We give a necessary and sufficient condition for the process
Z
to be a true martingale in the ...case where
and
Y
is a one-dimensional diffusion driven by a Brownian motion
W
. Furthermore, we provide a necessary and sufficient condition for
Z
to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function
b
and the drift and diffusion coefficients of
Y
. As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting.
We prove several properties of the EMCEL scheme, which is capable of approximating one-dimensional continuous strong Markov processes in distribution on the path space (the scheme is briefly ...recalled). Special cases include irregular stochastic differential equations and processes with sticky features. In particular, we highlight differences from the Euler scheme in the case of stochastic differential equations and discuss a certain “stabilizing” behavior of the EMCEL scheme like “smoothing and tempered growth behavior”.
We provide a new algorithm for approximating the law of a one-dimensional diffusion M solving a stochastic differential equation with possibly irregular coefficients. The algorithm is based on the ...construction of Markov chains whose laws can be embedded into the diffusion M with a sequence of stopping times. The algorithm does not require any regularity or growth assumption; in particular it applies to SDEs with coefficients that are nowhere continuous and that grow superlinearly. We show that if the diffusion coefficient is bounded and bounded away from 0, then our algorithm has a weak convergence rate of order 1/4. Finally, we illustrate the algorithm's performance with several examples.
In this paper we present a novel approach towards variance reduction for discretised diffusion processes. The proposed approach involves specially constructed control variates and allows for a ...significant reduction in the variance for the terminal functionals. In this way the complexity order of the standard Monte Carlo algorithm (ε−3 in the case of a first order scheme and ε−2.5 in the case of a second order scheme) can be reduced down to ε−2+δ for any δ∈0,0.25) with ε being the precision to be achieved. These theoretical results are illustrated by several numerical examples.
Most of the existing literature on optimal trade execution in limit order book models assumes that resilience is positive. But negative resilience also has a natural interpretation, as it models ...self-exciting behaviour of the price impact, where trading activities of the large investor stimulate other market participants to trade in the same direction. In the paper we discuss several new qualitative effects on optimal trade execution that arise when we allow resilience to take negative values. We do this in a framework where both market depth and resilience are stochastic processes.
In this paper, we introduce a modification of the free boundary problem related to optimal stopping problems for diffusion processes. This modification allows the application of this PDE method in ...cases where the usual regularity assumptions on the coefficients and on the gain function are not satisfied. We apply this method to the optimal stopping of integral functionals with exponential discount of the form $E_{x}\int_{0}^{\tau }e^{-\lambda s}f(X_{s})\ ds$, λ ≥ 0 for one-dimensional diffusions X. We prove a general verification theorem which justifies the modified version of the free boundary problem. In the case of no drift and discount, the free boundary problem allows to give a complete and explicit discussion of the stopping problem.
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