## Math-Finance seminar

### Math-Fi seminar on 6 Spe.

2022.09.05 Mon up
• Date: 6 Sep. (Tue.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Dan Crisan (Imperial College London)
• Title: Classical and modern results in the theory and applications of stochastic filtering
• Abstract:
Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this talk is to give a historical account of the subject concentrating on the continuous-time framework. I will also present a recent application of filtering to the estimation of partially observed high dimensional fluid dynamics models. In particular, I will introduce a so-called particle filter that incorporates a nudging mechanism. The nudging procedure is used in the prediction step. In the absence of nudging, the particles have trajectories that are independent solutions of the model equations. The nudging presented here consists in adding a drift to the trajectories of the particles with the aim of maximising the likelihood of their positions given the observation data. This introduces a bias in the system that is corrected during the resampling step.  The methodology is tested on a two-layer quasi-geostrophic model for a beta-plane channel flow with O(10^6) degrees of freedom out of  which only a minute fraction are noisily observed.

The talk is based on the papers:

[1] D Crisan, The stochastic filtering problem: a brief historical account, Journal of Applied Probability 51 (A), 13-22
[2] C Cotter, D Crisan, D Holm, W Pan, I Shevchenko, Data assimilation for a quasi-geostrophic model with circulation-preserving stochastic transport noise,  Journal of Statistical Physics, 1-36, 2020.
[3] D Crisan, I Shevchenko, Particle filters with nudging, work in progress.

### Math-Fi seminar on 26 Jul.

2022.07.25 Mon up
• Date: 26 Jul. (Tue.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Kei Noba (The Institute of Statistical Mathematics)
• Title: Optimality of classical or periodic barrier strategies for Lévy processes
• Abstract:
We revisit the stochastic control problem in two cases with Lévy processes that minimize running and controlling costs. Existing studies have shown the optimality of classical or periodic barrier strategies when driven by Brownian motion or Lévy processes with one-sided jumps. Under the assumption that we can be controlled at any time or only at Poissonian dividend-decision times, we show the optimality of classical or periodic barrier strategies for a general class of Lévy processes.

### Math-Fi seminar on 14 Jul.

2022.07.13 Wed up
• Date: 14 Jul. (Thu.)
• Place: On the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Takuji Arai (Keio University)
• Title: Constrained optimal stopping under a regime-switching model
• Abstract:
We investigate an optimal stopping problem for the expected value of a discounted payoff on a regime-switching geometric Brownian motion under two constraints on the possible stopping times: only at exogenous random times and only during a specific regime. The main objectives are to show that an optimal stopping time exists as a threshold type under some boundary conditions and to derive expressions of the value functions and the optimal threshold. To this end, we solve the corresponding variational inequality and show that its solution coincides with the value functions. Some numerical results are also introduced. Furthermore, we investigate some asymptotic behaviors. This talk is based on joint work with Masahiko Takenaka.

### Math-Fi seminar on 7 Jul.

2022.07.06 Wed up
• Date: 7 Jul. (Thu.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Pierre Bras (Sorbonne Université)
• Title: Asymptotics for the total variation distance between an SDE and its Euler-Maruyama scheme in small time
• Abstract:
We give bounds for the total variation distance between the law of an SDE and the law of its one-step Euler-Maruyama scheme as $t \to 0$. The case of the total variation is more complex to deal with than the classic case of Wasserstein ($L^p$) distances. We show that this distance is of order $t^{1/3}$, and more generally of order $t^{r/(2r+1)}$ for any $r \in \mathbb{N}$. Improving the bounds from $1/3$ to $r/(2r+1)$ relies on a weighted multi-level Richardson-Romberg extrapolation which consists in linear combination annealing the terms of a Taylor expansion, up to some order. This method was introduced for bias reduction in practical problems, but is used here for theoretical purposes.

### Math-Fi seminar on 23 Jun.

2022.06.23 Thu up
• Date: 23 Jun. (Thu.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Kosuke Yamato (Kyoto University)
• Title: A unifying approach to non-minimal quasi-stationary distributions for one-dimensional diffusions
• Abstract:
In the present talk, we consider convergence to non-minimal quasi-stationary distributions for one-dimensional diffusions. I will explain a method of reducing the convergence to the tail behavior of the lifetime via a property which we call the first hitting uniqueness. We apply the results to Kummer diffusions with negative drifts and give a class of initial distributions converging to each non-minimal quasi-stationary distribution.

### Math-Fi seminar on 16 Jun.

2022.06.14 Tue up
• Date: 16 Jun. (Thu.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 16:30-18:30

• Speaker 1: Tomoyuki Ichiba (University of California, Santa Barbara)
• Title: Stochastic Differential Games on Random Directed Trees
• Abstract:
We consider stochastic differential games on a random directed tree with mean-field interactions, where the network of countably many players is formulated randomly in the beginning and each player in the network attempts to minimize the expected cost over a finite time horizon. Here, the cost function is determined by the random directed tree. Under the setup of the linear quadratic stochastic game with directed chain graph, we solve explicitly for an open-loop Nash equilibrium for the system, and we find that the dynamics under the equilibrium is an infinite-dimensional Gaussian process associated with a Catalan Markov chain. We extend it to the random directed tree structures and discuss convergence results.

• Speaker 2: Noriyoshi Sakuma (Nagoya City University)
• Title: Selfsimilar free additive processes and freely selfdecomposable distributions
• Abstract:
In the paper by Fan(2006), he introduced the marginal selfsimilarity of non-commutative stochastic processes and proved the marginal distributions of selfsimilar processes with freely independent increments are freely selfdecomposable. In this talk, we, first, give a short introduction of free probability. Then we introduce a new definition of selfsimilarity via linear combinations of non-commutative stochastic processes and prove the converse of Fan’s result, to complete the relationship between selfsimilar free additive processes and freely selfdecomposable distributions. Furthermore, we construct stochastic integrals with respect to free additive processes for constructing the background driving free L{\’e}vy processes of freely selfdecomposable distributions. A relation in terms of their free cumulant transforms is also given and several examples are also discussed. This talk is based on a joint work arXiv:2202.11848 with Makoto Maejima.

### Math-Fi seminar on 9 Jun.

2022.06.08 Wed up
• Date: 9 Jun. (Thu.)
• Place: On the Web
• Time: 16:30-18:00
• Speaker: Toshiyuki Nakayama (MUFG, Bank, Ltd.)
• Title: Distance between closed sets and the solutions to SPDEs
• Abstract:
The goal of this talk is to clarify when the solutions to stochastic partial differential equations stay close to a given subset of the state space for starting points which are close as well. This includes results for deterministic partial differential equations. As an example, we will consider the situation where the subset is a finite dimensional submanifold with boundary. We also discuss applications to mathematical finance, namely the modeling of the evolution of interest rate curves. This talk is based on a co-authored paper with Stefan Tappe “Distance between closed sets and the solutions to stochastic partial differential equations”, arXiv:2205.00279v1, 30 Apr 2022 (https://arxiv.org/abs/2205.00279).

### Math-Fi seminar on 2 Jun.

2022.06.01 Wed up
• Date: 2 Jun. (Thu.)
• Place: W.W. 6th-floor, Colloquium Room & On the Web
• Time16:30-18:00
• Speaker: Kiyoiki Hoshino (Osaka Metropolitan University)
• Title: Extraction of random functions from the stochastic Fourier coefficients by the process with quadratic variation
• Abstract:
Let V_t be a real stochastic process with quadratic variation. Our concern is whether and how a noncausal type stochastic differential dX_t:=a(t) dV_t+b(t) dt is determined from its stochastic Fourier coefficients (SFCs for short) with respect to a CONS B of L^2[0,L]. In this talk, we use the notion of stochastic derivative to show the following: (i) when B is the Haar system, any stochastic differential dX is determined from its SFCs, (ii) when B is composed of functions of bounded variation, dX is determined from its SFCs under a certain continuity, where dX is defined by an arbitrary stochastic integral which is the inverse of the stochastic derivative.

### Math-Fi seminar on 28 Apr.

2022.04.27 Wed up
• Date: 28 Apr. (Thu.)
• Place: On the Web
• Time: 17:00-18:30
• Speaker:  Arturo Kohatsu-Higa (Ritsumeikan University)
• Title:   Simulation of Reflected Brownian motion on two dimensional wedges
• Abstract:
We study Brownian motion in two dimensions, which is reflected, stopped or killed in a wedge represented as the intersection of two half-spaces. First, we provide explicit density formulas, hinted by the method of images. These explicit expressions rely on infinite oscillating sums of Bessel functions and may demand computationally costly procedures. We propose suitable recursive algorithms for the simulation of the laws of reflected and stopped Brownian motion which are based on generalizations of the reflection principle in two dimensions. We study and give bounds for the complexity of the proposed algorithms. (Joint with P. Bras.)

### Math-Fi seminar on 21 Apr.

2022.04.20 Wed up
• Date: 21 Apr. (Thu.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 17:00-18:00
• Speaker:  Jiro Akahori (Ritsumeikan University)
• Title:  Variational approach to optimal stopping problems revisited
• Abstract:
After reviewing besoussan-lions’s variational approach, I will discuss its applications to numerical problems; discretization error, deep solver, and so on.  The talk will be in English.