FIRST ORDER BSPDEs IN HIGHER DIMENSION FOR OPTIMAL CONTROL PROBLEMS

This paper studies the first order backward stochastic partial differential equations suggested earlier for a case of multidimensional state domain with a boundary. These equations represent analogues of Hamilton--Jacobi--Bellman equations and allow one to construct the value function for stochastic optimal control problems with unspecified dynamics where the underlying processes do not necessarily satisfy stochastic differential equations of a known kind with a given structure. The problems considered arise in financial modeling

Dynkin Game of Stochastic Differential Equations with Random Coefficients, and Associated Backward Stochastic Partial Differential Variational Inequality

A Dynkin game is considered for stochastic differential equations with random coefficients. We first apply Qiu and Tang's maximum principle for backward stochastic partial differential equations to generalize Krylov estimate for the distribution of a Markov process to that of a non-Markov process, and establish a generalized It\^o-Kunita-Wentzell's formula allowing the test function to be a random field of It\^o's type which takes values in a suitable Sobolev space. We then prove the verification theorem that the Nash equilibrium point and the value of the Dynkin game are characterized by the strong solution of the associated Hamilton-Jacobi-Bellman-Isaacs equation, which is currently a backward stochastic partial differential variational inequality (BSPDVI, for short) with two obstacles. We obtain the existence and uniqueness result and a comparison theorem for strong solution of the BSPDVI. Moreover, we study the monotonicity on the strong solution of the BSPDVI by the comparison theorem for BSPDVI and define the free boundaries. Finally, we identify the counterparts for an optimal stopping time problem as a special Dynkin game.Comment: 40 page

Pricing options under rough volatility with backward SPDEs

In this paper, we study the option pricing problems for rough volatility models. As the framework is non-Markovian, the value function for a European option is not deterministic; rather, it is random and satisfies a backward stochastic partial differential equation (BSPDE). The existence and uniqueness of weak solutions is proved for general nonlinear BSPDEs with unbounded random leading coefficients whose connections with certain forward-backward stochastic differential equations are derived as well. These BSPDEs are then used to approximate American option prices. A deep learning-based method is also investigated for the numerical approximations to such BSPDEs and associated non-Markovian pricing problems. Finally, the examples of rough Bergomi type are numerically computed for both European and American options

Random neural networks for rough volatility

We construct a deep learning-based numerical algorithm to solve path-dependent partial differential equations arising in the context of rough volatility. Our approach is based on interpreting the PDE as a solution to an SPDE, building upon recent insights by Bayer, Qiu and Yao, and on constructing a neural network of reservoir type as originally developed by Gonon, Grigoryeva, Ortega. The reservoir approach allows us to formulate the optimisation problem as a simple least-square regression for which we prove theoretical convergence properties.Comment: 33 pages, 3 figure

Nonlinear filtering of high dimensional, chaotic, multiple timescale correlated systems

This dissertation addresses theoretical and numerical questions in nonlinear filtering theory for high dimensional, chaotic, multiple timescale correlated systems. The research is motivated by problems in the geosciences, in particular oceanic or atmospheric estimation and climate prediction. As the capability and need to further resolve the physics models on finer scales continues, greater spatial and temporal scales become present and the dimension of the models becomes increasingly large. In the atmospheric sciences, these models can be of the order $\mathcal{O}(10^9)$ degrees of freedom and require assimilation of the order $\mathcal{O}(10^7)$ observations during a single day. The models are chaotic and the observing sensors may be correlated with the physical processes themselves. The goal of the dissertation is to develop theoretical results that can provide the mathematical justification for new filtering algorithms on a lower dimensional problem, and to develop novel methods for dealing with issues that plague particle filtering when applied to high dimensional, chaotic, multiple timescale correlated systems. The first half of the dissertation is theoretical and addresses the question of approximating the continuous time nonlinear filtering equation for a multiple timescale correlated system by an averaged filtering equation in the limit of large timescale separation. The first result in this direction is within the context of a slow-fast system with correlation between the slow process and the observation process, and when we are only interested in estimating functions of the slow process. The main result is that we can retrieve a rate of convergence and that there is a metric generating the topology of weak convergence, such that the marginal filter converges to the averaged filter at the given rate in the limit of large timescale separation. The proof uses a probabilistic representation (backward doubly stochastic differential equation) of the dual process to the unnormalized filter, and sharp estimates on the transition density and semigroup of the fast process. The second theoretical result of the dissertation addresses the same question for a broader problem, where the slow signal dynamics include an intermediate timescale forcing. We prove that the marginal filter converges in probability to the average filter for a metric that generates the topology of weak convergence. The method of proof is by showing tightness of the measure-valued process, characterizing the weak limits, and proving the limit is unique. The perturbation test function (also known as method of corrector) is used to deal with the intermediate timescale forcing term, where the corrector is the solution of a Poisson equation. The second half of the dissertation develops filtering algorithms that leverage the theoretical results from the first half of the thesis to produce particle filtering methods for the averaged filtering equation. We also develop particle methods that address the issue of particle collapse for filtering on general high dimensional chaotic systems. Using the two timescale Lorenz 1996 atmospheric model, we show that the reduced order particle filtering methods are shown to be at least an order of magnitude faster than standard particle methods. We develop a method for particle filtering when the signal and observation processes are correlated. We also develop extensions to controlled optimal proposal particle filters that improve the diversity of the particle ensemble when tested on the Lorenz 1963 model. In the last chapter of the dissertation, we adopt a dynamical systems viewpoint to address the issue of particle collapse. This time the goal is to exploit the chaotic properties of the system being filtered to perform assimilation in a lower dimensional subspace. A new approach is developed which enables data assimilation in the unstable subspace for particle filtering. We introduce the idea of future right-singular vectors to produce projection operators, enabling assimilation in a lower dimensional subspace. We show that particle filtering algorithms using dynamically generator projection operators, in particular the future right-singular vectors, outperforms standard particle methods in terms of root-mean-square-error, diversity of the particle ensemble, and robustness when applied to the single timescale Lorenz 1996 model