## Conference on Stochastic Analysis, Stochastic Dynamical Systems and Stochastic Finance

#### Shanghai Jiao Tong University, School of Mathematical Sciences

Place:
School of Mathematical Sciences
Shanghai Jiao Tong University
Date:
from September 6th to 8th
Room:
706, School of Mathematical Sciences Building, Shanghai Jiao Tong University, Minhang Campus
Registration:
6, 7 and 8th of September starting from 8:00

### Program

Date/Time Fri 6 Sep 2019 Sat 7 Sep 2019 Sun 8 Sep 2019
08:30-09:15 Registration /Opening Michael Roeckner Huaizhong Zhao
09:15-10:00 Mathias Beiglboeck Fengyu Wang Bo Wu
10:00-10:30 Tea Break/Photo Tea Break Tea Break
10:30-11:15 Shanjian Tang Zhenqing Chen Xicheng Zhang
11:15-12:00 Michael Kupper Samuel Drapeau Nizar Touzi
12:00-14:00 Lunch Break Lunch Break Lunch Break
14:00-14:45 Martin Schweizer Kai Du
14:45-15:30 Hao Xing Paolo Guasoni
15:30-16:00 Tea Break Tea Break
16:00-16:45 Guojing Wang Zhenya Liu
16:45-17:30 Zhao Dong Daniel Bartl

### Titles and Abstracts

#### Daniel Bartl

Title
Adapted Wasserstein distance in finance and quantitative estimation
Abstract
What notion of closeness for probabilistic models guarantees closeness of utility maximization/optimal stopping/…? We argue that while Wasserstein distances are not suitable for this task, there is a modified offspring (the adapted Wasserstein distance) that is suitable. We establish Lipschitz continuity w.r.t. adapted Wasserstein distance and show sharpness already in a simple Brownian setting. The picture is completed by providing statistical estimators and studying their convergence rate in adapted Wasserstein distance. Joint with J.Backhoff, M.Beiglboeck, M.Eder, J.Wiesel

#### Mathias Beiglboeck

Title
Abstract
Several researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. We find that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover, we explain how optimal transport theory can be used obtain a compatible metric that is both tractable and highly relevant for a number of questions in mathematical finance.
(J. Backhoff, D. Bartl, D. Lacker, M. Eder)

#### Zhenqing Chen

Title
Stability of heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms
Abstract
In this talk, I will present recent progress in the study of heat kernels and parabolic Harnack inequalities for symmetric Markov processes that have both diffusive and jumping parts on general metric measure spaces. Under general volume doubling condition and some mild assumptions on the scaling functions, we establish stability results for two-sided estimates for heat kernels in terms of the jumping kernels, the generalized capacity inequalities, and Poincare inequalities. Stable characterizations of the associated parabolic Harnack inequalities will also be given. Our results hold on spaces even when the underlying spaces have walk dimensions are larger than 2. Joint work with Takashi Kumagai and Jian Wang.

#### Samuel Drapeau

Title
Computational aspects and sensitivity of robust optimization problems
Abstract
We study generic optimization problems that often appear in finance but in a robust context based on a Wasserstein type of uncertainty. We present how these infinite dimensional optimization problems can be expressed in a finite dimensional optimization problem. To this end we use transport duality. We further extend the study to the characterization of the argmin. The question is more challenging in the sense that unless very special cases are assumed, a full characterization does not seem the be possible. Using perturbation analysis, we provide a sensitivity analysis in terms of first order impact of uncertainty for the optimal value function as well as the corresponding argmin. Based on joint works with Daniel Bartl, Ludovic Tangpi, Jan Obloj and Johannes Wiesel.

#### Kai Du

Title
On stochastic parabolicity conditions for SPDEs
Abstract
Stochastic parabolicity conditions are a class of structural conditions for stochastic PDEs to ensure wellposedness of the equations, which, in contrast to classical parabolicity conditions, involve additionally the coefficients of leading terms in the stochastic integral part. The proper form of stochastic parabolicity condition for weak solutions of SPDEs were found long ago and also valid in constructing L^p theory and Schauder theory of SPDEs of second-order. However, things may dramatically change for other problems, such as complex-valued SPDEs, systems of SPDEs, and higher-order SPDEs. More specifically, those conditions for weak solutions ($L^2$ theory) is not sufficient to ensure $L^p$-integrability of solutions, and certain modified conditions are required in those cases. The talk will present some p-dependent parabolicity conditions imposed on systems of SPDEs and on higher-order SPDEs, which ensure us to construct stochastic Schauder theory for those equations. Examples are discussed for necessity of modified parabolicity conditions and sharpness of our modifications. The talk is based on joint works with Jiakun Liu and Fu Zhang and with Yuxing Wang.

#### Zhao Dong

Title
Large deviation principles for first-order scalar conservation laws with stochastic forcing
Abstract
In this paper, we established the Freidlin-Wentzell type large deviation principles for first-order scalar conservation laws perturbed by small multiplicative noise. Due to the lack of the viscous terms in the stochastic equations, the kinetic solution to the Cauchy problem for Abstract 15 these first-order conservation laws is studied. Then, based on the well-posedness of the kinetic solutions, we show that the large deviations holds by utilising the weak convergence approach. This is joint work with Wu Jiang Lun, Zhang Rang Rang, Zhang Tu sheng.

#### Paolo Guasoni

Title
Asset Prices in Segmented and Integrated Markets
Abstract
This paper evaluates the effect of market integration on prices and welfare, in a model where two Lucas trees grow in separate regions with similar investors. We find equilibrium asset price dynamics and welfare both in segmentation, when each region holds its own asset and consumes its dividend, and in integration, when both regions trade both assets and consume both dividends. Integration always increases welfare. Asset prices may increase or decrease, depending on the time of integration, but decrease on average. Correlation in assets’ returns is zero or negative before integration, but significantly positive afterwards, explaining some effects commonly associated with financialization.

#### Michael Kupper

Title
Nonlinear expectations under Wasserstein uncertainty and convex semigroups
Abstract
We provide general results for convex semigroups, in particular uniqueness in terms of the generator. As an example, we consider scaling limits of Levy processes under distributional uncertainty with respect to (martingale) transport distances. The talk is based on joint works with Daniel Bartl, Robert Denk and Max Nendel.

#### Zhenghu Li

Title
Ergodicities and exponential ergodicities of branching processes with immigration
Abstract
Under natural assumptions, we prove the ergodicities and exponential ergodicities in Wasserstein and total variation distances of Dawson-Watanabe superprocesses without or with immigration. Those processes are infinite-dimensional generalizations of the well-known continuous-state branching processes, which are also known as Cox-Ingersoll-Ross type models and have played important roles in the study of mathematical finance. The strong Feller property of the processes in the total variation distance is derived as a by-product. The key of our approach is a set of estimates for the variations of the transition probabilities. The estimates in Wasserstein distance are derived from an upper bound of the kernels induced by the first moment of the superprocess. Those in total variation distance are based on a comparison of the cumulant semigroup of the superprocess with that of a continuous-state branching process. The results improve and extend considerably those of Stannat (2003a, 2003b) and Friesen (2019+). We also show a connection between the ergodicities of the associated immigration superprocesses and decomposable distributions.

#### Zhenya Liu

Title
The Optimal Selling Price with Endogenous Maximum Drawdown
Abstract
We propose the optimal endogenous drawdown in an investor’s objective function, which is concerned about the profit and the loss. The endogenous drawdown is the solution of an optimal stopping problem, including the maximum running process. The optimal selling price with endogenous maximum drawdown is determined by the weights on profit and loss of investor’s objective function and the stock price process, e.g., Geometric Brownian Motion. With the comparison of the stock markets in China and the United States, investors in China’s stock markets have a significant attitudes to suffer a more massive endogenous drawdown.

#### Michael Roeckner

Title
Strong dissipativity of generalized time-fractional derivatives and quasi-linear (stochastic) partial differential equations
Abstract
In this talk we shall identify generalized time-fractional derivatives as generators of $C_0$-operator semigroups and prove their strong dissipativity on Gelfand triples of properly in time weighted $L^2$-path spaces. In particular, the classical Caputo derivative is included as a special case. As a consequence one obtains the existence and uniqueness of solutions to evolution equations on Gelfand triples with generalized time-fractional derivatives. These equations are of type \begin{equation*} \frac{d}{dt} (k * u)(t) + A(t, u(t)) = f(t), \quad 0 < t < T \end{equation*} with (in general nonlinear) operators $A(t,\cdot)$ satisfying general weak monotonicity conditions. Here $k$ is a non-increasing locally Lebesgue-integrable nonnegative function on $\mathbb{R}_+$ with $\lim_{s \to \infty}k(s)=0$. Analogous results for the case, where $f$ is replaced by a time-fractional additive noise, are obtained as well. Applications include generalized time-fractional quasi-linear (stochastic) partial differential equations. In particular, time-fractional (stochastic) porous medium and fast diffusion equations with ordinary or fractional Laplace operators or the time-fractional (stochastic) $p$-Laplace equation are covered.

#### Shanjian Tang

Title
Stochastic LQ and Associated Riccati equation of PDEs Driven by State- and Control-Dependent White Noise
Abstract
The optimal stochastic control problem with a quadratic cost functional for linear partial differential equations (PDEs) driven by a state- and control-dependent white noise is formulated and studied. Both finite- and infinite-time horizons are considered. The multi- plicative white noise dynamics of the system give rise to a new phenomenon of singularity to the associated Riccati equation and even to the Lyapunov equation. Well-posedness of both Riccati equation and Lyapunov equation are obtained for the first time. The linear feedback coefficient of the optimal control turns out to be singular and expressed in terms of the solution of the associated Riccati equation. The null controllability is shown to be equivalent to the existence of the solution to Riccati equation with the singular terminal value. Finally, the controlled Anderson model is addressed as an illustrating example. This is a joint work with Ying Hu.

#### Nizar Touzi

Title
Continuous-time Principal-Agent problem and optimal planning
Abstract
Motivated by the approach introduced by Sanninkov to solve principal-agent problems, we provide a solution approach which allows to address a wider range of problems. The key argument uses a representation result from the theory of backward stochastic differential equations. This methodology extends to the mean field game version of the problem, and provides a connexion with the P.-L. Lions optimal planning problem.

#### Martin Schweizer

Title
Absence of arbitrage without a numeraire
Abstract
We propose a new approach to defining and characterising absence of arbitrage in a way which does not depend on an a priori chosen numeraire. In spirit, this is similar to Kabanov’s approach to modelling markets with transaction costs. The key idea is that the comparison basis for a profit is no longer wealth, which is numeraire-dependent, but a chosen reference portfolio, which is in units of assets and hence numeraire-independent. We provide dual characterisations in terms of discounting and martingale properties, and we also discuss to which extent the approach and results depend on the reference portfolio. This is based on joint work with Daniel Balint.

#### Fengyu Wang

Title
Asymptotic Formulas for Empirical Measures of (Reflecting) Diffusion Processes on Riemannian Manifolds
Abstract
Consider the (reflecting) diffusion process $(X\_t)\_{t\ge 0}$ generated by $L:= \Delta+\nabla V$ on a compact connected Riemannian manifold $M$ possibly with a boundary $\partial M$, where $V\in C^2(M)$ is such that $\mu(d x) :=e^{V(x)} d x$ is a probability measure. Then the empirical measures $$\mu_{t}:=\frac{1}{t}\int_0^t \delta_{X_s} d s,\ \ t>0$$ satisfy $$\lim_{t\to \infty} \big\{t \mathbb{E}^x \left[W_2(\mu_{t},\mu)^2\right]\big\}= \sum_{i=1}^\infty\frac{2}{\lambda_i^2} \text{ uniformly in } x\in M,$$ where $\mathbb{E}^x$ is the expectation for the diffusion process starting at point $x$, $W_2$ is the $L^2$-Warsserstein distance induced by the Riemannian metric, and the limit is finite if and only if $d\leq 3$. Moreover, when $d\geq 4$ the main order of $\mathbb{E}^x[W_2(\mu_{t},\mu)^2]$ is $t^{- 2/(d-2)}$ as $t\to\infty$.

#### Guojing Wang

Title
Pricing basket CDS with some certain dependent default risk
Abstract
In this talk, we consider the basket CDS spreads under some reduced form credit risk models. We explain how the default intensity of a defaultable firm can be defined as the intensity of a point process. The dependence of default can be described by the dependence among the default intensity processes of defaultable firms with contagion (interacting), the conditional independence, the common shock and the regime switching. We derive some closed form joint distributions for default times and present some analytic pricing formulas for the basket CDS spreads under the proposed credit risk models.

#### Bo Wu

Title
Functional inequalities on general Riemannian loop spaces
Abstract
In this talk, based on a family of cut-off processes, we provide a method for obtaining asymptotic and short time gradient and Hessian estimates for logarithmic heat kernel on a general complete Riemannian manifold. These together with heat kernel estimates and asymptotic comparison estimates yield a reliable way to obtain different gradient and Hessian estimates on general non-compact Riemannian manifolds. Even for the compact manifolds, the expression for Hessian of the heat semigroup with adapted random fields is new. As an application, by using the above estimates we obtain the local Log-Sobolev inequality and the Log-Sobolev inequality with a potential term on general Riemannian loop spaces respectively.

#### Hao Xing

Title
Rational inattention and dynamic discrete choice
Abstract
We adopt the posterior-based approach to study dynamic discrete choice problems with rational inattention. We show that the optimal solution for the Shannon entropy case is characterized by a system of equations that resembles the dynamic logit rule. We propose an efficient algorithm to solve this system and apply our model to explain phenomena such as status quo bias, confirmation bias, and belief polarization. We also study the dynamics of consideration sets. Unlike the choice-based approach, our approach applies to general uniformly posterior-separable information cost functions. A key condition for our approach to work in dynamic models is the convexity of the difference between the discounted (generalized) entropy of the prior beliefs about the future states and the entropy of the current posterior. This is a joint work with Jianjun Miao.

#### Huaizhong Zhao

Title
Periodic measures and stochastic resonance
Abstract
In this talk, I will talk about the periodic measures of Markovian systems and their ergodic theory. Our theory reveals how geometric ergodic periodic measure provides rigorous proof for the transition of the stochastic resonance problem. We derive a parabolic partial differential equation for the expected exit time of the Markovian periodic process and its periodic solution gives the expected transition time of stochastic resonance. We subsequently find the parameters when the expected transition time is equal to the period of the system which is the real meaning of resonance. This is a joint work with Chunrong Feng and Johnny Zhong.

#### Xicheng Zhang

Title
A discretized version of Krylov’s estimate and its applications
Abstract
In this paper we prove a discretized version of Krylov’s estimate for discretized Ito’s processes. As applications, we study the weak and strong convergences for Euler’s approximation of mean-field SDEs with measurable discontinuous and linear growth coefficients. Moreover, we also show the propagation of chaos for Euler’s approximation of mean-field SDEs.