We consider two models of observable process X = (X t ):
Model A: X t = μt + B t ,
Model B: X t = μt + ν(t − θ)^+ + B t ,
where B = (B t ) is a standard Brownian motion, μ and ν are unknown parameters,
and θ is a disorder time.
For Model A, we consider some sequential statistical problems with different
risk functions.
For Model B, we deal with sequential problems of the following type:
H 1 =...
We discuss adaptive and anticipating couplings on Wiener space. In particular we take a fresh look at the connection between the Wasserstein metric and the relative entropy with respect to Wiener measure provided by Talagrand’s inequality and its extension to Wiener space by Feyel and Ustunel. Using results of Nina Gantert for large deviations in the quadratic variation of Brownian motion, we...
We study a Edgeworth-type refinement of the central limit theorem for the discretizacion error of Itô integrals. Towards this end, we introduce a new approach, based on the anticipating I Itô formula. This alternative technique allows us to compute explicitly the terms of the corresponding expansion formula. A joint work with E. Alos.
Moving boundary problems allow to model macroscopic systems with phase transition at an inner boundary. Motivated by problems in economics and finance, more explicitely price-time continuous modelling of the limit order book, we consider a stochastic and non-linear extension of the classical Stefan-problem in one space dimension. More precisely, the dynamics on buy and sell side in an...
Many problems of statistical inference can be solved, using spectral decomposition
of stochastic processes. The principal difficulty with this approach is that eigenproblems
are notoriously hard to solve in a reasonably explicit form. In this talk I will survey some
recent results on the exact asymptotics in eigenproblems for fractional processes and
discuss their applications to parameter...
Working in the Merton's optimal consumption framework with continuous time we consider an optimization problem for a portfolio with an illiquid, a risky and a risk-free asset. Our goal in this paper is to carry out a complete Lie group analysis of PDEs describing value function and investment and consumption strategies for a portfolio with an illiquid asset that is sold in an exogenous random...
We provide a general model of default time, extending the models of Jiao and
Li (modelling sovereign risks) and Gehmlich and Schmidt (dynamic defaultable
term structure modelling beyond intensity paradigm).
We show that any random time τ can be decomposed in two parts as τ = τ 1 ∧τ 2
under the condition that the first random time τ 1 avoids stopping times in the ref-
erence filtration F, and...
The additional information carried by enlarged filtration and its measurement was studied by several authors. Already Meyer (Sur un theoreme de J. Jacod, 1978) and Yor (Entropie d'une partition, et grossissement initial d'une filtration, 1985), investigated stability of martingale spaces with respect to initial enlargement with atomic sigma-field. We extend these considerations to the case...
This article is devoted to the maximization of HARA utilities of Lévy
switching process on finite time interval via dual method. We give the description
of all f-divergence minimal martingale measures in progressively enlarged
filtration, the expression of their Radon-Nikodym densities involving Hellinger
and Kulback-Leibler processes, the expressions of the optimal strategies for
the...
Let $\overline N_t = \sup_{s\leq t} N_s$ be a running maximum of a local martingale $N$. We assume that $N$ is max-continuous, i.e. $\overline N$ is continuous. The Skorokhod embedding problem corresponds to a special case where $N$ is a Brownian motion stopped at a finite stopping time $\tau$. Consider the change of time generated by the running maximum:
$
\sigma_t:=\inf\,{s\colon...
We discuss relations between sets of laws of stochastic integrals with respect to a Wiener process
and general continuous martingales having quadratic characteristics whose RN-derivatives evolve in the
same convex set of positive semidefinite symmetric matrices.
In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models and, while (i) has been explained by using...
We investigate the Esscher pricing rule and the Esscher prices, when the ``public" flow information denoted by $\mathbb F$ is progressively enlarged by a random time $\tau$, for both discrete-time and continuous-time settings. $\tau$ can represent the death time of an agent, default time of a firm, or more generally the occurrence time of an even that might impact the market somehow. Thus, by...
The aim of this talk is to describe globally the behavior and preferences of heterogeneous agents. Our starting point is the aggregate wealth of a given economy, with a given repartition of the wealth among investors, which is not necessarily Pareto optimal.
We propose a construction of an aggregate forward utility, market consistent,
that aggregates the marginal utility of the...
In the last decades solving multisided optimal stopping problems was of interest.
We will show how to approach the above problems for Levy processes if the payoff function is an exponential polynomial (possibly multidimensional), and present several examples. The method we use is based on Appell integral transform.
Motivated from high and infinite dimensional problems in mathematical finance, we consider infinite dimensional polynomial processes taking values in certain space of measures or functions. We have two concrete applications in mind: first, modeling high or even potentially infinite dimensional financial markets in a tractable and robust way, and second analyzing stochastic Volterra processes,...
How to calculate the essential uncertainty of probability distributions hidden behind a real data sequence is a theoretically and practically important challenging problem.
Recently some fundamentally important progresses have been achieved in the domain of law of large numbers (LLN) and central limit theorem (CLT) with a much weaker assumption of independence and identical distribution...
We consider the mixed optimal control of a linear stochastic system with a quadratic cost functional, with two controllers---one can choose only deterministic time functions, called the deterministic controller, while the other can choose adapted random processes, called the random controller. The optimal control is shown to exist under suitable assumptions. The optimal control is...
This talk is concerned with Schur-constant survival models for discrete random variables. Our main purpose is to prove that the associated partial sum process is a non- homogeneous Markov chain. This is shown in different cases as the random variables take values in the set of nonnegative integers or in the set of integers smaller than $m\geq 1$. The property of Schur-constancy is also...
ABSTRACT: We consider the problem faced by a Central Bank of optimally
controlling the exchange rate over a finite time horizon, whereby it can use
two non-excluding tools: controlling directly the exchange rate in the
form of an impulse control; controlling it indirectly via the domestic
exchange rate in the form of a continuously acting control. In line
with existing literature we consider...
This talk is concerned with stochastic optimal control problems with a certain homogeneity. For such problems, a novel dual problem is formulated. The results are applied to a stochastic volatility variant of the classical Merton problem. Another application of this duality is to the study the right-most particle of a branching Levy process.
We consider a class of linear-quadratic-Gaussian mean-field games with a major agent and considerable heterogeneous minor agents with mean-field interactions. The individual admissible controls are constrained in closed convex subsets of the full space. The decentralized strategies for individual agents and the consistency condition system are represented in an unified manner via a class of...
In our previous paper, we have introduced a general class of systemic risk measures that allow random allocations to individual banks before aggregation of their risks. In the present paper, we address the question of fairness of these allocations and we propose a fair allocation of the total risk to individual banks. We show that the dual problem of the minimization problem which identify the...
We investigate geometric properties of Weierstrass curves with two
components, representing series based on trigonometric functions. They
are seen to be 12 − Hölder continuous, and are not (para-)controlled
with respect to each other in the sense of the recently established
Fourier analytic approach of rough path analysis. Their graph is rep-
resented as an attractor of a smooth random...
In this talk, i will introduce a type of mass-conserving stochastic partial differential equations which can be connected with a type of mass-conserving backward doubly stochastic differential equations. The Poincare’s inequality is used in the estimates to relax the monotonic condition of backward doubly stochastic differential equations.
Among other factors, the difficult market environment with its sustained low interest rates triggers certain adjustments of investment and product strategies of life insurance companies and pension funds. In this context, the role of life insurance companies and pension funds as long-term investors has increasingly been discussed among the industry and financial market supervisory authorities....
In this talk I will present some "folk" results in insider trading literature. In particular, I will discuss conditions on pricing functional that are necessary for existence of equilibrium, as well as the ones that are necessary for existence of inconspicuous equilibrium. I will prove that one can restrict insider trading strategies to absolutely continuous ones.
Given discrete time observations over a fixed time interval, we study a nonparametric Bayesian approach to estimation of the volatility coefficient of a stochastic differential equation. We postulate a histogram-type prior on the volatility with piecewise constant realisations on bins forming a partition of the time interval. The values on the bins are assigned an inverse Gamma Markov chain...
We examine the interactions between financing (capital structure) and investment decisions of a firm under asymmetric information about collateral (liquidation) value between well-informed managers and less-informed investors. We show that asymmetric information reduces the amount of debt issuance to finance the cost of investment, that leads to delay corporate investment. In particular, an...
We consider stochastic (partial) differential equations appearing as Markovian lifts of affine Volterra processes with jumps from the point of view of the generalized Feller property which was introduced in, e.g., Dörsek-Teichmann (2010). In particular we provide new existence, uniqueness and approximation results for Markovian lifts of affine rough volatility models of general jump diffusion...
In this talk we
analyze perpetual American call and put options in an exponential L\'evy model.
We consider a negative effective discount rate which arises in a number of financial applications
including stock loans and real options, where the strike price can potentially grow at a higher rate than
the original discount factor. We show that in this case a double continuation region arises and...
We consider a version of the stochastic control problem, in which control opportunities arrive only at the jump times of an independent Poisson process. We consider perpetual American options, optimal dividend problems, and inventory control problems, driven by a spectrally one-sided Levy process. In particular, we show that barrier-type strategies are optimal under suitable conditions. The...
We consider a geometric Lévy market with asset price S t = S 0 exp(X t ), where X is a
general Lévy process on (Ω, F, P), and interest rate equal to zero. As it is well known,
except for the cases that X is a Brownian motion or a Poisson process, the market is
incomplete. Therefore, if the market is arbitrage-free, there are many equivalent mar-
tingale measures and the problem arises to...
In this talk, we study the ruin problem with investment in a general framework where the business part X is a Lévy process and the return on investment R is a semimartingale. We obtain upper bounds on the finite and infinite time ruin probabilities that decrease as a power function when the initial capital increases. When R is a Lévy process, we retrieve the well-known results. Then, we show...
In this talk, we study a class of combined regular and singular stochastic control problems that can be expressed as constrained BSDEs. In the Markovian case, this reduces to a characterization through a PDE with gradient constraint. But the BSDE formulation makes it possible to move beyond Markovian models and consider path-dependent problems. We also provide an approximation of the original...
In this talk, we present some recent results on obliquely reflected BSDEs. In particular we are able to deal with assumptions on the generator weaker than in currently known results. An existence and uniqueness result is obtained in a non Markovian framework by assuming some regularity on the terminal condition. Moreover, a general existence result is obtained in the Markovian framework. We...
In the talk we consider a variant of the basic problem of the calculus of variations, where the Lagrangian is convex and subject to randomness adapted to a Brownian filtration. We solve the problem by reducing it, via a limiting argument, to an unconstrained control problem that consists in finding an absolutely continuous process minimizing the expected sum of the Lagrangian and the deviation...
The starting point of the talk is a recent work with Juan Li (Shandong University,
Weihai, P.R.China) and Jin Ma (University of South California, Los Angeles, U.S.A.), “A mean-
field stochastic control problem with partial observations” (Annals of Appl.Probability, 2017: [1]) in
which we studied Pontryagin’s optimality principle for a stochastic control problem whose dynamics
are given by the...
A classical problem in ergodic control theory consists in the study of the limit behaviour of
λV λ (·) as λ ↘ 0, when V λ is the value function of a deterministic or stochastic control problem
with discounted cost functional with infinite time horizon and discount factor λ. We study this
problem for the lower value function V λ of a stochastic differential game with recursive cost, i.e.,
the...
This talk will focus on a new type of quantile optimization problems arising
from insurance contract design models. This type of optimization problems is
characterized by a constraint that the derivatives of the decision quantile
functions are bounded. Such a constraint essentially comes from the
“incentive compatibility” constraint for any optimal insurance contract to
avoid the potential...
We prove the existence and uniqueness of weak solution of a Neumann boundary problem for an elliptic partial differential equation (PDE for short) with a singular divergence term which can only be understood in a weak sense. A probabilistic approach is applied by studying the backward stochastic differential equation (BSDE for short) corresponding to the PDE.
