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Alberto Ohashi

Publications and source records attributed to Alberto Ohashi.

31 records · Page 2Linked to original sources
arXiv2018

Stochastic Near-Optimal Controls for Path-Dependent Systems

In this article, we present a general methodology for control problems driven by the Brownian motion filtration including non-Markovian and non-semimartingale state processes controlled by mutually singular measures. The main result of this paper is the development of a concrete pathwise method for characterizing and computing near-optimal controls for abstract controlled Wiener functionals. The theory does not require ad hoc functional differentiability assumptions on the value process and elipticity conditions on the diffusion components. The analysis is pathwise over suitable finite dimensional spaces and it is based on the weak differential structure introduced by Leão, Ohashi and Simas jointly with measurable selection arguments. The theory is applied to stochastic control problems based on path-dependent SDEs where both drift and possibly degenerated diffusion components are controlled. Optimal control of drifts for path-dependent SDEs driven by fractional Brownian motion is also discussed. We finally provide an application in the context of financial mathematics. Namely, we construct near-optimal controls in a non-Markovian portfolio optimization problem.

math.PR↗
arXiv2017

A weak version of path-dependent functional Itô calculus

We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The main novel idea is to compute the "sensitivities" of processes, namely derivatives of martingale components and a weak notion of infinitesimal generators, via a finite-dimensional approximation procedure based on controlled inter-arrival times and approximating martingales. The theory comes with convergence results that allow to interpret a large class of Wiener functionals beyond semimartingales as limiting objects of differential forms which can be computed path wisely over finite-dimensional spaces. The theory reveals that solutions of BSDEs are minimizers of energy functionals w.r.t Brownian motion driving noise.

math.PR↗
arXiv2017

Weak Functional Itô Calculus and Applications

We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration. The theory provides a differential structure which describes the infinitesimal evolution of Wiener functionals at very small scales. The Markov property is replaced by a finite-dimensional approximation procedure based on controlled inter-arrival times and jumps of approximating martingales. The theory reveals that a large class of adapted processes follow a differential rule which is similar in nature to a fundamental theorem of calculus in the context of Wiener functionals. Null stochastic derivative term turns out to be a non-Markovian version of the classical second order parabolic operator and connections with theory of local-times and $(p,q)$-variation regularity are established. The framework extends the pathwise functional calculus and it opens the way to obtain variational principles for processes. Applications to semi-linear variational equations and stochastic variational inequalities are presented. In particular, we provide a feasible dynamic programming principle for fully non-Markovian/non-semimartingale optimal stopping problems.

math.PR↗
arXiv2016

Principal Components Analysis for Semimartingales and Stochastic PDE

In this work, we develop a novel principal component analysis (PCA) for semimartingales by introducing a suitable spectral analysis for the quadratic variation operator. Motivated by high-dimensional complex systems typically found in interest rate markets, we investigate correlation in high-dimensional high-frequency data generated by continuous semimartingales. In contrast to the traditional PCA methodology, the directions of large variations are not deterministic, but rather they are bounded variation adapted processes which maximize quadratic variation almost surely. This allows us to reduce dimensionality from high-dimensional semimartingale systems in terms of quadratic covariation rather than the usual covariance concept. The proposed methodology allows us to investigate space-time data driven by multi-dimensional latent semimartingale state processes. The theory is applied to discretely-observed stochastic PDEs which admit finite-dimensional realizations. In particular, we provide consistent estimators for finite-dimensional invariant manifolds for Heath-Jarrow-Morton models. More importantly, components of the invariant manifold associated to volatility and drift dynamics are consistently estimated and identified. The proposed methodology is illustrated with both simulated and real data sets.

math.ST↗
arXiv2015

Path-dependent Itô formulas under finite $(p,q)$-variation regularity

In this work, we establish pathwise functional Itô formulas for non-smooth functionals of real-valued continuous semimartingales. Under finite $(p,q)$-variation regularity assumptions in the sense of two-dimensional Young integration theory, we establish a pathwise local-time decomposition $$F_t(X_t) = F_0(X_0)+ \int_0^t\nabla^hF_s(X_s)ds + \int_0^t\nabla^wF_s(X_s)dX(s) - \frac{1}{2}\int_{-\infty}^{+\infty}\int_0^t(\nabla^w_xF_s)(^{x}X_s)d_{(s,x)}\ell^x(s).$$ Here, $X_t= \{X(s); 0\le s\le t\}$ is the continuous semimartingale path up to time $t\in [0,T]$, $\nabla^h$ is the horizontal derivative, $(\nabla^w_x F_s)(^{x}X_s)$ is a weak derivative of $F$ with respect to the terminal value $x$ of the modified path $^{x}X_s$ and $\nabla^w F_s(X_s) = (\nabla^w_x F_s)(^{x} X_s)|_{x=X(s)}$. The double integral is interpreted as a space-time 2D-Young integral with differential $d_{(s,x)}\ell^x(s)$, where $\ell$ is the local-time of $X$. Under less restrictive joint variation assumptions on $(\nabla^w_x F_t)(^{x} X_t)$, functional Itô formulas are established when $X$ is a stable symmetric process. Singular cases when $x\mapsto (\nabla^w_x F_t)(^{x}X_t)$ is smooth off random bounded variation curves are also discussed. The results of this paper extend previous change of variable formulas in Cont and Fournié and also Peskir, Feng and Zhao and Elworhty, Truman and Zhao in the context of path-dependent functionals. In particular, we provide a pathwise path-dependent version of the classical Föllmer-Protter-Shiryaev formula for continuous semimartingales.

math.PR↗
arXiv2014

A Maximal Inequality of the 2D Young Integral based on Bivariations

In this note, we establish a novel maximal inequality of the 2D Young integral $\int_a^b\int_c^d FdG$ in terms of the $(p,q)$-bivariation norms of the section functions $x\mapsto F(x,y)$ and $y\mapsto F(x,y)$ where $G:[a,b]\times [c,d]\rightarrow \mathbb{R}$ is a controlled path satisfying finite $(p,q)$-variation conditions. The proof is reminiscent from the Young's original ideas \cite{young1} in defining two-parameter integrals in terms of $(p,q)$-finite bivariations. Our result complements the standard maximal inequality established by Towghi \cite{towghi1} in terms of joint variations. We apply the maximal inequality to get novel strong approximations for 2D Young integrals w.r.t the Brownian local time in terms of number of upcrossings of a given approximating random walk.

math.FA↗
arXiv2014

A Noisy Principal Component Analysis for Forward Rate Curves

Principal Component Analysis (PCA) is the most common nonparametric method for estimating the volatility structure of Gaussian interest rate models. One major difficulty in the estimation of these models is the fact that forward rate curves are not directly observable from the market so that non-trivial observational errors arise in any statistical analysis. In this work, we point out that the classical PCA analysis is not suitable for estimating factors of forward rate curves due to the presence of measurement errors induced by market microstructure effects and numerical interpolation. Our analysis indicates that the PCA based on the long-run covariance matrix is capable to extract the true covariance structure of the forward rate curves in the presence of observational errors. Moreover, it provides a significant reduction in the pricing errors due to noisy data typically founded in forward rate curves.

q-fin.ST↗
arXiv2014

A Note on the sharp $L^p$-Covergence rate of Upcrossings to the Brownian local time

In this note, we prove a sharp $L^p$-rate of convergence of the number of upcrossings to the local time of the Brownian motion. In particular, it provides novel $p$-variation estimates ($2 < p < \infty$) for the number of upcrossings of the Brownian motion. Our result complements the fundamental work of Koshnevisan \cite{kho} who obtains an almost sure exact rate of convergence in the sup norm.

math.PR↗
arXiv2013

A general Multidimensional Monte Carlo Approach for Dynamic Hedging under stochastic volatility

In this work, we introduce a Monte Carlo method for the dynamic hedging of general European-type contingent claims in a multidimensional Brownian arbitrage-free market. Based on bounded variation martingale approximations for Galtchouk-Kunita-Watanabe decompositions, we propose a feasible and constructive methodology which allows us to compute pure hedging strategies w.r.t arbitrary square-integrable claims in incomplete markets. In particular, the methodology can be applied to quadratic hedging-type strategies for fully path-dependent options with stochastic volatility and discontinuous payoffs. We illustrate the method with numerical examples based on generalized Follmer-Schweizer decompositions, locally-risk minimizing and mean-variance hedging strategies for vanilla and path-dependent options written on local volatility and stochastic volatility models.

q-fin.PR↗
arXiv2013

Weak approximations for Wiener functionals

In this paper we introduce a simple space-filtration discretization scheme on Wiener space which allows us to study weak decompositions and smooth explicit approximations for a large class of Wiener functionals. We show that any Wiener functional has an underlying robust semimartingale skeleton which under mild conditions converges to it. The discretization is given in terms of discrete-jumping filtrations which allow us to approximate nonsmooth processes by means of a stochastic derivative operator on the Wiener space. As a by-product, we provide a robust semimartingale approximation for weak Dirichlet-type processes. The underlying semimartingale skeleton is intrinsically constructed in such way that all the relevant structure is amenable to a robust numerical scheme. In order to illustrate the results, we provide an easily implementable approximation scheme for the classical Clark-Ocone formula in full generality. Unlike in previous works, our methodology does not assume an underlying Markovian structure and does not require Malliavin weights. We conclude by proposing a method that enables us to compute optimal stopping times for possibly non-Markovian systems arising, for example, from the fractional Brownian motion.

math.PR↗
arXiv2012

On the Discrete Cramér-von Mises Statistics under Random Censorship

In this work, nonparametric log-rank-type statistical tests are introduced in order to verify homogeneity of purely discrete variables subject to arbitrary right-censoring for infinitely many categories. In particular, the Cramér-von Mises test statistics for discrete models under censoring is established. In order to introduce the test, we develop the weighted log-rank statistics in a general multivariate discrete setup which complements previous fundamental results of Gill (1980) and Andersen et al. (1982). Due to the presence of persistent jumps over the unbounded set of categories, the asymptotic distribution of the test is not distribution-free. The statistical test for a large class of weighted processes is described as a weighted series of independent chi-squared variables whose weights can be consistently estimated and the associated limiting covariance operator can be infinite-dimensional. The test is consistent to any alternative hypothesis and, in particular, it allows us to deal with crossing hazard functions. We also provide a simulation study in order to illustrate the theoretical results.

math.ST↗
arXiv2009

Fractional term structure models: No-arbitrage and consistency

In this work we introduce Heath-Jarrow-Morton (HJM) interest rate models driven by fractional Brownian motions. By using support arguments we prove that the resulting model is arbitrage free under proportional transaction costs in the same spirit of Guasoni [Math. Finance 16 (2006) 569-582]. In particular, we obtain a drift condition which is similar in nature to the classical HJM no-arbitrage drift restriction. The second part of this paper deals with consistency problems related to the fractional HJM dynamics. We give a fairly complete characterization of finite-dimensional invariant manifolds for HJM models with fractional Brownian motion by means of Nagumo-type conditions. As an application, we investigate consistency of Nelson-Siegel family with respect to Ho-Lee and Hull-White models. It turns out that similar to the Brownian case such a family does not go well with the fractional HJM dynamics with deterministic volatility. In fact, there is no nontrivial fractional interest rate model consistent with the Nelson-Siegel family.

q-fin.PR↗