Math-Finance seminar

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.

Math-Fi seminar on 14 Apr.

2022.04.14 Thu up
  • Date: 14 Apr. (Thu.)
  • Place: On the Web
  • Time: 17:00-18:30
  • Speaker:  Umut Cetin (London School of Economics)
  • Title:  Speeding up the Euler scheme for killed diffusions
  • Abstract:
Let X be a linear diffusion taking values in  (l,r) and consider the standard Euler discretisation to compute the fair price of a Barrier option written on X that becomes worthless if X hits one of the barriers before the maturity date T. It is well-known since Gobet’s work that the presence of killing introduces a loss of accuracy and reduces the weak convergence rate to N^{-1/2} with N being the number of discretisatons. We introduce a drift-implicit Euler method to bring the convergence rate back to 1/N, i.e. the optimal rate in the absence of killing, using the theory of recurrent transformations. Although the current setup assumes a one-dimensional setting, multidimensional extension is within reach as soon as a systematic treatment of recurrent transformations is available in higher dimensions.
 

Math-Fi seminar on 7 Apr.

2022.04.11 Mon up
  • Date: 7 Apr. (Thu.)
  • Place: On the Web
  • Time: 16:30-18:00
  • Speaker:  Andrey Pilipenko (Ukraine National Academy of Sciences)
  • Title: Limit behavior of perturbed random walks
  • Abstract:
A particle moves randomly over the integer points of the real line. Jumps of the particle outside the membrane (a fixed “locally perturbating set”) are i.i.d., have zero mean and finite variance, whereas jumps of the particle from the membrane have other distributions with finite means which may be different for different points of the membrane; furthermore, these jumps are mutually independent and independent of the jumps outside the membrane. We prove that the weak scaling limit of the particle position is a skew Brownian motion.

Math-Fi seminar on 20 Jan.

2022.01.19 Wed up
  • Date: 20 Jan. (Thu.)
  • Place: On the Web
  • Time: 16:30 – 18:00
  • Speaker: Stefano Pagliarani (University of Bologna)
  • Title: A Yosida’s parametrix approach to Varadhan’s estimates for a degenerate diffusion under the weak Hörmander condition
  • Abstract: 
We adapt and extend Yosida’s parametrix method, originally introduced for the construction of the fundamental solution to a parabolic operator on a Riemannian manifold, to derive Varadhan-type asymptotic estimates for the transition density of a degenerate diffusion under the weak H\”ormander condition. This diffusion process, widely studied by Yor in a series of papers, finds direct application in the study of a class of path-dependent financial derivatives known as Asian options. We obtain a Varadhan-type formula for the asymptotic behavior of the logarithm of the transition density, in terms of the optimal cost function of a deterministic control problem associated to the diffusion. We provide a partial proof of this formula, and present numerical evidence to support the validity of an intermediate inequality that is required to complete the proof. We also derive an asymptotic expansion of the cost function, expressed in terms of elementary functions, which is useful in order to design efficient approximation formulas for the transition density.

Math-Fi seminar on 13 Jan.

2022.01.12 Wed up
Date: 13 Jan. (Thu.)
Place: On the Web
Time: 16:30 – 18:00
Speaker: Benjamin Jourdain (CERMICS)
Title: Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q — L ρDrift Coefficient and Additive Noise
Abstract: 
We are interested in the time discretization of stochastic differential equations with additive d-dimensional Brownian noise and L q — L ρ drift coefficient when the condition d ρ + 2 q < 1, under which Krylov and R{ö}ckner [26] proved existence of a unique strong solution, is met. We show weak convergence with order 1 2 (1 — (d ρ + 2 q)) which corresponds to half the distance to the threshold for the Euler scheme with randomized time variable and cutoffed drift coefficient so that its contribution on each time-step does not dominate the Brownian contribution. More precisely, we prove that both the diffusion and this Euler scheme admit transition densities and that the difference between these densities is bounded from above by the time-step to this order multiplied by some centered Gaussian density.
 

Math-Fi seminar on 6 Jan.

2022.01.06 Thu up
  • Date: 6 Jan. (Thu.)
  • Place: On the Web
  • Time: 18:00 – 19:30
  • Speaker: Xin Chen (Shanghai Jiao Tong University)
  • Title: Some results on backward stochastic differential equation on a Riemannian manifold
  • Abstract: 
In this talk, we will introduce some recent results on backward stochastic differential equation on a Riemannian manifold, including the definition of Riemannian-manifold valued BSDE, the probabilistic  representation for heat flow of harmonic map, the characterization of Navier-Stokes equation on a Riemannian manifold.
The talk is based on a joint work with Wenjie Ye.