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The international conference on "Advanced Methods in Mathematical Finance" will take place in Angers, France, from the 28th to the 31th of August 2018.
This conference is dedicated to innovations in the mathematical analysis of financial data, new numerical methods for finance and applications to risk modeling. The selected topics include actuarial theory, risk measures, ruin theory, credit default models, stochastic control and its applications to portfolio choice and liquidation, models of liquidity and with transaction costs, pricing and hedging of financial instruments. During this conference we plan to discuss new models, new methods and new results in mathematical finance, to answer questions related to actuarial theory, to analyse new financial instruments and to consider several applications of mathematical finance. More precisely, we will consider submissions in the following topics:
Actuarial Theory and Mathematical Finance
Ruin Theory
Models for Interest Rates
Default Models
Financial Markets with Transation Costs
SDEs and BSDEs in Mathematical Finance
Stochastic Analysis, Optimal Control and their Applications
Statistical Methods in Mathematical Finance
The conference starts in the morning of August 28th and ends on the evening of August 31st. The departure day is in the morning of September 1st. If you plan to participate only for a part of the week, please enter the corresponding dates when you register. For the convenience of participants we precise that the arrival day is August 27th. The accomodations will be reserved by the organizers in the hotels situated near the Angers train station, namely in the hotels Progrès, Iena and Bon Pasteur. To cover a part of the conference expenses we ask a registration fee of 100 euros which can be paid by "bon de commande", bank tranfer or credit card (see Registration fee).
To register, it is necessary to create a Indico login and password. This login and password will also allow you to view or edit your abstract submission.
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 = sup EX τ
or H 2 = sup EE(X τ ),
τ ≤1
τ ≤1
where τ is a stopping time. We show that for such functionals H 1 and H 2 optimal
stopping times have the following form:
τ ∗ = inf{t ≤ 1: ψ(t) ≥ a ∗ (t)},
where ψ(t) is some statistic of observations and a ∗ (t) is a curvilinear boundary
satisfying the Fredholm integral equation of second order. These problems will
be applied to the real asset price models (Apple, Nasdaq).
The talk will gives a survey of the joint papers of authors with Četin,
Novikov, Zhitlukhin, and Muravlev.
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 extend this inequality beyond the absolutely continuous case, using the notion of specific relative entropy.
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 electronic financial markets are modeled by respective second order stochastic partial differential equations which are separated by an inner interface: the mid-price. We discuss new results beyond existence theory, such as approximations of the solution.
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 estimation and filtering.
In the real world, decision making under uncertainty is often viewed as an opti-
mization problem under choice criterium, and most of theory focuses on the deriva-
tion of the "optimal decision" and its out-comes. But, poor information is available
on the criterium yielding to these observed data. The interesting problem to infer
the unknown criterium from the known results is an example of inverse problem.
Here we are concerned with a very simple version of the problem: what does ob-
servation of the "optimal" out-put tell us about the preference, expressed in terms
of expected utility; in Economics, this question was pioneered by the american
economist Samuelson in1938.
Typically we try to reproduce the properties of the stochastic value function of
a portfolio optimization problem in finance, which satisfies the first order condi-
tion U (t, z)). In particular, the utility process U is a strictly concave stochastic
family, parametrized by a number z ∈ R + (z 7→ U (t, z)), and the characteris-
tic process X c = (X t c (x)) is a non negative monotonic process with respect to
its initial condition x, satisfying the martingale condition U (t, X t c (x)) is a martin-
gale, with initial condition U (0, x) = u(x). We first introduce the adjoint process
Y t (u x (x)) = U x (t, X t c (x)) which is a characteristic process for the Fenchel transform
of U if and only if X t c (x)Y t (u x (x)) is a martingale. The minimal property is the
martingale property of Y t (u x (x)) with the x-derivative of X t c (x), which is sufficient
to reconstruct U from U x (t, x) = Y t (u x ((X t c ) −1 (x))). Obviously, in general, with-
out additional constraints, the characterization is not unique. Various example are
given, in general motivated by finance or economics: contraints on a characteristic
portfolio in a economy at the equilibrium, thoptimal portfolio for a in complete
financial market, under strongly orthogonality between X and Y , the mixture of
different economies....In any case, the results hold for general but monotonic pro-
cesses, without semimartingale assumptions.
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 moment of time $T$ with a prescribed liquidation time distribution. The problem of such type leads to three dimensional nonlinear Hamilton-Jacobi-Bellman (HJB) equations. Such equations are not only tedious for analytical methods but are also quite challenging form a numeric point of view. To reduce the three-dimensional problem to a two-dimensional one or even to an ODE one usually uses some substitutions, yet the methods used to find such substitutions are rarely discussed by the authors.
We use two types of utility functions: general HARA type utility and logarithmic utility. We carry out the Lie group analysis of the both three dimensional PDEs and are able to obtain the admitted symmetry algebras. Then we prove that the algebraic structure of the PDE with logarithmic utility can be seen as a limit of the algebraic structure of the PDE with HARA-utility as $\gamma \to 0$. Moreover, this relation does not depend on the form of the survival function $\overline{\Phi} (t)$ of the random liquidation time $T$.
We find the admitted Lie algebra for a broad class of liquidation time distributions in cases of HARA and log utility functions and formulate corresponding theorems for all these cases.
We use found Lie algebras to obtain reductions of the studied equations. Several of similar substitutions were used in other papers before whereas others are new to our knowledge. This method gives us the possibility to provide a complete set of non-equivalent substitutions and reduced equations.
We also show that if and only if the liquidation time defined by a survival function $\overline{\Phi} (t)$ is distributed exponentially, then for both types of the utility functions we get an additional symmetry. We prove that both Lie algebras admit this extension, i.e. we obtain the four dimensional $L^{HARA}_4$ and $L^{LOG}_4$ correspondingly for the case of exponentially distributed liquidation time.
We list reduced equations and corresponding optimal policies that tend to the classical Merton policies as illiquidity becomes small.
This research was supported by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE - Novel Methods in Computational Finance)
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 second time τ 2 is thin, i.e., its graph is included in a
countable union of graphs of stopping times in the reference filtration F. Under the
condition τ 1 ∨ τ 2 = ∞, the decomposition is unique. This decomposition is based
on a study of the dual optional projection of τ , as the decomposition of a stopping
time into accessible and totally inaccessible is based on the dual predictable pro-
jection. We show that for a thin time τ 2 , any F-martingale is a semimartingale in
its progressive enlargement with τ 2 and we give its semimartingale decomposition.
We prove that any martingale in the reference filtration is a semimartingale in
the progressive enlargement with τ if and only if the same property holds for the
progressive enlargement with τ 1 and we give its semimartingale representation.
We establish in that the immersion property holds for τ if and only if it holds
for τ 1 .This is a joint work with Anna Aksamit and Tahir Choulli.
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 where information is disclosed progressively in time. We define the entropy of such information and we prove that its finiteness is enough for stability of some martingale spaces in progressive setting. Finally we calculate additional logarithmic utility of a discrete information disclosed progressively in time.
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 maximization of HARA utilities as well as the values of the corresponding
maximal expected utilities. The example of Brownian switching models is presented.
This is common work with Lioudmila Vostrikova.
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 \overline N_s>t\}.
$
Then the time-changed process $M:=N\circ\sigma$ has a simple structure:
$
M_t=N_{\sigma_t}= t\wedge W - V1_{\{t\geq W\}},
$
where $W:=\overline N_\infty$ and $V:=\overline N_\infty-N_\infty$ ($V$ is correctly defined on the set $\{\overline N_\infty < \infty\}$). Besides, $M_\infty=N_\infty$ and $\overline M_\infty=\overline N_\infty$. This simple observation explains how we can use single jump martingales $M$ of the above form to describe properties of $N$. For example, $N$ is a closed supermartingale if and only $M$ is a martingale and the negative part of $W-V$ is integrable. Another example shows how to connect the Dubins-Gilat construction of a martingale whose supremum is given by the Hardy-Littlewood maximal function and the Azéma-Yor construction in the Skorokhod embedding problem.
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 a fractional volatility model with Hurst index $H>1/2$, (ii) is proved to be satisfied by a {\it rough} volatility model with $H<1/2$ under a risk-neutral measure. This paper provides a solution to this fractional puzzle in the implied volatility. Namely, we construct a two-factor fractional volatility model and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle, when the correlations between the underlying asset process and the factors of rough volatility and persistence belong to a certain range. More specifically, depending on the three correlation values, the implied volatility surface is classified into four types: (1) the roughness exists, but the persistence does not; (2) the persistence exists, but the roughness does not; (3) both the roughness and the persistence exist; and (4) neither
the roughness nor the persistence exist. (Joint work with H. Funahashi)
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 considering the new flow of information $\mathbb G$ resulting from the expansion of the flow $\mathbb F$ with $\tau$, we address the stopped model $(S^{\tau}$,$\mathbb{G})$ in different directions and various frameworks. In discrete time, for instance, we describe the Esscher martingale measure for the general case in different manners, and we illustrate the results on particular cases of models for the pair $(S,\tau)$. To well illustrate the impact of $\tau$ on the Esscher pricing rules and/or prices, we consider the Black-Scholes model for $S$ and a class of models for $\tau$. For these models, we describe the Esscher martingale measures, the Esscher prices for some death-linked contracts, the Greeks of these obtained Esscher prices, and we compare the Esscher prices with the Black-Scholes pricing formula. This talk is based on joint work with Haya Alsemary (University of Alberta).
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 heterogeneous agents. This construction
is based on the aggregation of the pricing kernels of each investor. As an application
we analyze the impact of the heterogeneity and of the wealth market on the yield curve.
This is a joint work with Nicole El Karoui and Mohamed Mrad.
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, which recently gained popularity through rough volatility models and ambit processes. The first question leads to probability measure valued polynomial diffusions and the second one to Markovian lifts of polynomial Volterra processes. For both cases we provide existence results and a moment formula.
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 (i.i.d.) under a sublinear expectation.
These new LLN and CTL can be applied to a significantly wide classes of data sequence to construct the corresponding optimal estimators. In particular, many distribution uncertainties hidden behind data sequences are able to be quantitatively calculated by introducing a new algorithm of phi-max-mean type.
In this talk, I take some typical examples to provide a more concrete explanation of the above mentioned LLN and CLT, the key idea of their proofs, as well as the new phi-max-mean estimators.
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 characterized via a system of fully coupled forward-backward stochastic differential equations (FBSDEs) of mean-field type. We solve the FBSDEs via solutions of two (but decoupled) Riccati equations, and give the respective optimal feedback law for both deterministic and random controllers, using solutions of both Riccati equations. The optimal state satisfies a linear stochastic differential equation (SDE) of mean-field type. Both the singular and infinite time-horizonal cases are also addressed.
This is a joint work with Ying HU, Universite de Rennes 1.
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 compared for thesecases. We also present a few additional results on Schur-constant vectors. This is based on joint works with Castaner, Claramunt, Lefèvre and Utev.
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 as a mixed
classical-impulse control problem for which, on the basis of a
quasi-variational inequality, we search for an analytic solution within a
specific class of value functions and controls. Besides the finite
horizon, the main novelty here is the assumption that the drift in the
exchange rate dynamics is not directly observable and has thus to be
filter-estimated from observable data. The problem becomes thus time
inhomogeneous and the Markovian state variables have to include also
the filter of the drift. This is a joint work with
Kazuhiro Yasuda.
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.
This work studies a class of continuous-time scalar-state stochastic Linear-Quadratic (LQ) optimal control problem with the linear control constraints. Using the state separation theorem induced from its special structure, we derive the analytical solution for this class of problem. The revealed optimal control policy is a piece-wise affine function of system state. This control policy can be computed efficiently by solving two Riccati equations off-line. Under some mild conditions, the stationary optimal control policy can be also achieved for this class of problem with infinite horizon. This result can be applied to solve the constrained dynamic mean-variance portfolio selection problem. Examples shed light on the solution procedure of implementing our method.
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 mean-field forward-backward stochastic differential equation involving projection operators. The well-posedness of consistency condition system is established and the related ε−Nash equilibrium property is also verified.
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 systemic risk measure, provides a valuation of the random allocations which is fair both from the point of view of the society/regulator and from the individual financial institutions. The case with exponential utilities which allows for explicit computation is treated in details.
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 dynamical system. For
one-dimensional versions we show existence of a local time and smooth-
ness of the Sinai-Bowen-Ruelle (SBR) measure. Our argument that its
graph has Hausdorff dimension 2 is in the spirit of Ledrappier-Young’s
approach of the Hausdorff dimension of attractors. This is joint work
with G. dos Reis (U Edinburgh) and A. Réveillac (U Toulouse).
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. These discussions are often focused on the idea of trying to benefit from the possibility of long-term hold to maturity strategies partially based on assets providing a certain illiquidity premium. This idea is compared to alternative ideas regarding investment or resolution plans for life insurance portfolios, some of which are based on recent developments in quantitative finance. Furthermore, the link between investment plans and product design will also be briefly discussed.
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 (IGMC) prior. Posterior inference is straightforward to implement via Gibbs sampling, as the full conditional distributions are available explicitly and turn out to be inverse Gamma. We also discuss in detail the hyperparameter selection for our method. Our nonparametric Bayesian approach leads to good practical results in representative simulation examples. Finally, we apply it on a classical data set in change-point analysis: weekly closings of the Dow-Jones industrial averages. [Joint work with Shota Gugushvili, Moritz Schauer and Frank van der Meulen.]
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 increase in the degree of asymmetric information forces the firm to be a risk-free debt-equity financing (ultimately be the all-equity financing) by reducing the amount of debt issuance. In addition, an increase in the cash flow volatility decreases the amount of debt issuance, credit spread, and leverage under asymmetric information. Our results fit well with empirical studies. This is a joint work with Michi Nishihara.
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 type. We demonstrate that in this Markovian light the theory of stochastic Volterra processes becomes almost classical.
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 identify the two critical prices.
We also generalize this result to multiple stopping problems of swing type, that is, when
successive exercise opportunities are separated by i.i.d. random
refraction times. We conduct numerical analysis for the Black-Scholes model and
the jump-diffusion model with exponentially distributed jumps.
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 optimal strategies and value functions are concisely written in terms of the scale functions. This talk is based on the joint work with A. Bensoussan and J.L. Perez.
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 choose an appropriate martingale measure
for pricing contingent claimes.
One way is to choose the equivalent martingale measure Q ∗ which minimizes the
relative entropie to P, if it exists. Another choice is the famous Esscher martingale
measure Q E , if it exists.
The main objective of the present talk is to discuss a simple and rigorous approach
for proving the fact that the entropie minimal martingale measure Q ∗ and the Esscher
martingale measure Q E actually coincide: Q ∗ = Q E . Our method consists of a suit-
able approximation of the physical probability measure P by Lévy preserving probaility
measures P n.
The problem was treated in several earlier papers but more heuristally or in a so-
phisticated way.
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 that these bounds are asymptotically optimal in the finite time case, under some simple conditions on the characteristics of X. Finally, we obtain a condition for ruin with probability one when X is a Brownian motion with negative drift and express it explicitly using the characteristics of R. (The results were obtained as a joint work with L. Vostrikova.)
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 control problem with standard BSDEs that yield a characterization of approximately optimal values and controls.
This is a joint work with Bruno Bouchard and Patrick Cheridito.
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 also present an application to some new optimal switching problems called randomised switching problems.
This is a joint work with Jean-François Chassagneux (University of Paris 7)
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 of the terminal state from a given target position. Using the Pontryagin maximum principle one can characterize a solution of the unconstrained control problem in terms of a fully coupled forward-backward stochastic differential equation (FBSDE). We use the method of decoupling fields for proving that the FBSDE has a unique solution.
The talk is based on joint work with Alexander Fromm, Thomas Kruse and Alexandre Popier.
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 stochastic differential equation (SDE)
Z t
Z t
X|Y
X t = x +
b(s, X .∧s , μ s , u s (Y ))ds +
σ(s, X .∧s , μ s X|Y , u s (Y ))dB s 1 ,
0
0
R t
Y t = 0 h(s, X s )ds + B t 2 , t ∈ [0, T ], P -a.s.,
where (B 1 , B 2 ) is a P -Brownian motion, the controlled state process X is only observable through
the observation process Y and so the control process u = u(Y ) is a non anticipating functional of
the observation process Y . Moreover, unlike classical controlled dynamics, the coefficients σ and b
do not only depend on the paths of the controlled state process X and the control u(Y ) but also
X|Y
on the law μ s = P ◦ [E[X s |Y r , r ≤ s]] −1 , s ∈ [0, T ]. Motivations for such a type of dynamics are
given in [1]. However, in [1] the dependence of the law is linear; the talk will study the case where
the coefficients are non linear functions of the law. Moreover, unlike in [1] the coefficients b and σ
are only supposed to be continuous in the law w.r.t. the 1-Wasserstein metric and on h we only
impose boundedness and Lipschitz continuity in the state variable. The main objective of the talk
is to prove the weak existence and the uniqueness in law for the above dynamics.
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 cost functional is defined through a backward stochastic differential equation with infinite
time horizon. But unlike the ergodic control approach, we are interested in the case where the
limit can be a function depending on the initial condition. For this we extend the so-called
non-expansivity assumption from the case of control problems to that of stochastic differential
games.
Based on a joint work with Rainer Buckdahn (Brest, France), Nana Zhao (Weihai, China).
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 severe problem of moral hazard in insurance contract
design models. By a further development of the relaxation method, we
provide a systemic approach to solving this new type of quantile optimization
problems. The optimal quantile is expressed via the solution of a free
boundary problem for a second-order nonlinear ordinary differential equation
(ODE), which is similar to the Black-Scholes ODE for perpetual American
options.
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.