Abstract : Stochastic numerical analysis becomes so important in probability theory, statistics and applied mathematics especially in machine learning and data science and achieves a great success in computational finance. The aim of the minisymposium is to highlight recent developments in stochastic numerical analysis and computational finance, and to interact with researchers working on the fields. Topics will include deep learning methods for stochastic differential equations and PDEs, new computational methods for pricing derivatives, portfolio optimization and risk management, and their theoretical analysis.
[03032] Policy improvement algorithm for an optimal consumption and investment problem under general stochastic factor models
Format : Talk at Waseda University
Author(s) :
Kazuhiro Yasuda (Hosei university)
Hiroaki Hata (Hitotsubashi university)
Abstract : In this talk, we propose a policy improvement algorithm for a consumption and investment problem on a finite time horizon to optimize a discounted expected power utility of consumption and terminal wealth. We employ a general stochastic factor model which means that the returns and volatilities of assets are random and affected by some economic factors, modeled as diffusion processes. We establish an iteration procedure converging to the value function and the optimal strategies obtained in Hata, Nagai and Sheu (2018). Some numerical results are shown to understand convergence behaviors of the algorithm.
[03381] Growth in Fund Models
Format : Talk at Waseda University
Author(s) :
Hyeng Keun Koo (Ajou University)
Constantinos Kardaras (London School of Economics)
Johannes Ruf (London School of Economics)
Abstract : We study estimation of growth in fund models, i.e., statistical descriptions of markets where all asset returns are spanned by the returns of a lower-dimensional collection of funds, modulo orthogonal noise. The loss of growth due to estimation error in fund models under local frequentist estimation is determined entirely by the number of funds. A shrinkage method that targets maximal growth with the least amount of deviation is proposed.
[03015] Carbon Emissions Pricing by Forward and Double Barrier Backward SDE approach
Format : Talk at Waseda University
Author(s) :
Tadashi Hayashi (Mitsubishi UFJ Trust and Banking Corporation)
Abstract : Under the circumstances of global warming caused by increasing in greenhouse gases, there are many theoretical and empirical studies in carbon emissions to control and reduce the gases. Our study is focused on the carbon emissions pricing via Forward and Double Barrier BSDE as another pricing approach. This modelling would be a new approach of carbon emissions pricing and therefore lead to a new chance of empirical simulation.
[03307] Irreversible consumption habit under ambiguity: singular control and optimal G-stopping time
Format : Talk at Waseda University
Author(s) :
Kyunghyun Park (Nanyang Technological University)
HOI YING WONG (The Chinese University of Hong Kong )
Kexin Chen (The Hong Kong Polytechnic University)
Abstract : Consider robust utility maximization with an irreversible consumption habit, where an agent concerned about model ambiguity is unwilling to decrease consumption and must simultaneously contend with a disutility (i.e., an adjustment cost) due to a consumption increase. While the optimization is a robust analog of singular control problems over a class of consumption-investment strategies and a set of probability measures, it is a new formulation that involves non-dominated probability measures of the diffusion process for the underlying assets in addition to singular controls with an adjustment cost. This paper provides a novel connection between the singular controls in the optimization and the optimal $G$-stopping times in a $G$-expectation space, using a duality theory. This connection enables to derive the robust consumption strategy as a running maximum of the stochastic boundary, which is characterized by a free boundary arising from the optimal $G$-stopping times. The duality, which relies on arguments based on reflected $G$-BSDEs, is achieved by verifying the first-order optimality conditions for the singular control, the budget constraint equation for the robust strategies, and the worst-case realization under the non-dominated measures.
[03674] New deep NN architecture using higher-order weak approximation
Format : Talk at Waseda University
Author(s) :
Syoiti Ninomiya (Tokyo Institute of Technology)
Yuming MA (Tokyo Institute of Technology)
Abstract : New deep learning neural networks based on high-order weak approximation algorithms for stochastic differential equations are proposed. The behavior of these new algorithms when applied to the problem of pricing financial derivatives is also reported. The architectural key to the deep learning neural network proposed here is a high-order discretization method of Runge-Kutta type, in which the weak approximation of stochastic differential equations is realized by iterative substitutions and their linear summation.
[03488] A higher order discretization scheme for backward stochastic differential equations combined with a non-linear discrete Clark-Ocone formula
Format : Talk at Waseda University
Author(s) :
Kaori Okuma (Ritsumeikan University)
Abstract : In this talk, the author first introduces a discretization scheme of arbitrary order for backward stochastic differential equations. Then, by establishing a mathematical algorithm based on a non-linear discrete Clark-Ocone formula, which was previously established by the author and her collaborators, the author claims that the scheme is potentially implementable in a “deep solver” type numerical algorithm —- a scheme using approximation by deep neural networks and stochastic gradient descent — for a semi-linear partial differential equation.
[03622] New deep learning-based algorithms for high-dimensional Bermudan option pricing
Format : Talk at Waseda University
Author(s) :
Riu Naito (Hitotsubashi University)
Toshihiro Yamada (Hitotsubashi University)
Abstract : In this talk, we introduce efficient algorithms for pricing high-dimensional Bermudan options. The proposed methods provide an accurate approximation for Bermudan options by discretizing the interval of early-exercise dates with weak approximation schemes for stochastic differential equations. The deep learning-based approximation for conditional expectations at each exercise date works well for high-dimensional problems compared to the least squares Monte Carlo method. Numerical experiments confirm the validity of the methods.
[05412] On-Policy and Off-Policy q-Learning in Continuous Time
Format : Talk at Waseda University
Author(s) :
Yanwei Jia (Chinese University of Hong Kong )
Xunyu Zhou (Columbia University)
Abstract : We study the continuous-time counterpart of Q-learning for reinforcement learning (RL) under the entropy-regularized, exploratory formulation introduced by Wang et al (2020). As the conventional (big) Q-function collapses in continuous
time, we consider its first-order approximation and coin the term ``(little) q-function". This function is related to the instantaneous advantage rate function as well as the Hamiltonian. We develop a ``q-learning" theory around the
q-function that is independent of time discretization. Given a stochastic policy, we jointly characterize the associated q-function and value function by martingale conditions of certain stochastic processes, in both on-policy and off-policy
settings. We then apply the theory to devise different actor-critic algorithms for solving underlying RL problems,
depending on whether or not the density function of the Gibbs measure generated from the q-function can be
computed explicitly.
[03150] An Approximation Scheme for Path-Dependent BSDEs
Format : Talk at Waseda University
Author(s) :
Hyungbin Park (Seoul National University)
Ji-Uk Jang (Seoul National University)
Abstract : In this work, we study an approximation scheme for solutions to forward-backward stochastic differential equations (FBSDEs) with non-anticipative coefficients. When the non-anticipative coefficients have Fréchet derivatives or can be approximated by non-anticipative functionals having Fréchet derivatives, we show the Picard-type iteration converges to the FBDSE solution and provide its convergence rate. Using this result, we establish a numerical method for solutions of second-order parabolic path-dependent partial differential equations. To achieve this, weak approximation of martingale representation theorem (Cont, Rama, and Yi Lu. “Weak approximation of martingale representations." Stochastic Processes and their Applications 2016) is employed. Our results generalize the scheme for Markovian cases in (Bender, Christian, and Robert Denk. “A forward scheme for backward SDEs." Stochastic processes and their applications, 2007)
[03659] Practical high-order recombination algorithms for weak approximation of stochastic differential equations : Recursive patch dividing and its effects to singularities of terminal conditions
Format : Talk at Waseda University
Author(s) :
Syoiti Ninomiya (Tokyo Institute of Technology)
Yuji Shinozaki (Bank of Japan)
Abstract : This study proposes practically feasible implementation algorithms of the high-order recombination to apply to the weak approximation problem of SDEs, by extending and refining the work of Lyons and Litterer(2012). Specifically, new recursive patch dividing algorithms, which are based on the refined patch radius criteria, are proposed. Our numerical experiments demonstrate that the new recursive patch dividing algorithms are still efficient even when the terminal condition $f$ becomes more singular.
[03645] Extended Milstein scheme for hypoelliptic diffusions
Format : Talk at Waseda University
Author(s) :
Yuga Iguchi (University College London)
Toshihiro Yamada (Hitotsubashi University)
Abstract : For a wide class of diffusion processes, precisely hypoelliptic diffusions, we propose an effective and simple numerical scheme as an extension of Milstein scheme that outperforms Euler-Maruyama (EM) scheme (and standard Milstein scheme), though they share the same convergence rate in a weak sense. Analytic error term for the new scheme is derived and compared with that for EM scheme under non-smooth test functions. The effectiveness of the proposed scheme is also shown through numerical experiments for hypoelliptic diffusions appearing in finance.
[03489] Wong-Zakai approximation for stochastic PDEs and HJM model
Format : Talk at Waseda University
Author(s) :
TOSHIYUKI NAKAYAMA (MUFG Bank, Ltd.)
Abstract : We talk about semi-linear stochastic differential equation (SPDE) driven by a finite dimensional Brownian motion.
$$dX(t)=(AX(t)+b(X(t)))dt+\sum_{j=1}^r\sigma_j(X(t))dB^j(t),\quad X(0)=x_0.$$
Our goal is to establish a convergence rate with the generator $A$ which is allowed to be the infinitesimal generator of an arbitrary strongly continuous semigroup.
Finally, we will introduce an application example for SPDE called HJMM that appears in mathematical finance.
This talk is based on a co-authored paper with Stefan Tappe.
