International Seminar on SDEs and Related Topics

    This online seminar takes place every four weeks on Friday at
   12:30 UTC         11:30 UTC during European daylight saving times (until Oct 29)          

  12:30 noon       1:30 pm       2:30 pm       8:30 pm (7:30 pm until Oct 29)    
    London        Berlin, Paris         Helsinki         Beijing    

                    Zoom link       Meeting ID: 618 9100 7917

No registration required. To get an e-mail reminder before each event write to sde-seminar[at]


  • Stefan Ankirchner   (FSU Jena, Germany)
  • Christian Bender   (Saarland University, Germany)
  • Rainer Buckdahn   (Universite de Bretagne Occidentale, France)
  • Dan Crisan   (Imperial College London, UK)
  • Christel Geiss   (University of Jyväskylä, Finland)
  • Stefan Geiss   (University of Jyväskylä, Finland)
  • Céline Labart   (Université Savoie Mont-Blanc, France)
  • Juan Li   (Shandong University, China)
  • Andreas Neuenkirch   (University of Mannheim, Germany)
  • Shige Peng   (Shandong University, China)
  • Adrien Richou   (University of Bordeaux, France)

Schedule 2023 Autumn

Oct 20, 2023

Theodor Sturm                        (University of Bonn, Germany)

Nov 17, 2023

Dec 15, 2023


Schedule 2024 Spring

Jan 12, 2024

Xiaolu Tan (The Chinese University of Hong Kong)

Jan 12, 2024

Feb 09, 2024

Mar 08, 2024

Apr 05, 2024

May 03, 2024

May 31, 2024

Contact and links

    Stefan Geiss: stefan.geiss[at] or stefanfriedrich.geiss[at]
    The online seminar is hosted by the Department of Mathematics and Statistics (University of Jyväskylä) and supported by the journal Probability, Uncertainty and Quantitative Risk.

Schedule 2023 Spring

Jan 13, 2023

Terry Lyons                        Poster:    
(University of Oxford, United Kingdom)

Rough paths, controlled differential equations, and more scalable data science

Abstract: We will survey the growing contribution of rough path based ideas, such as the signature of a path, to the understanding real world streamed data.
A more comprehensive account can be found at

Jan 27, 2023

Jean-François Chassagneux                        Poster:     Slides
(Université Paris Cité, France)

A dual approach to partial hedging

Abstract: We introduce a class of 'weak hedging problems’ which contains as special examples the quantile hedging problem (Föllmer & Leukert 1999) and PnL (Profit and Loss) matching problem (introduced in Bouchard & Vu 2012). We show that they can generally be rewritten as a kind of Monge transport problem. Using this observation, we introduce a Kantorovich version of the problem and, in some cases, we are able to prove a dual formulation. This allows us to design numerical methods based on SGD (stochastic gradient descent} algorithms to compute the weak hedging price.

Feb 10, 2023

Martin Hairer                        Poster:     Slides
(EPFL, Switzerland and Imperial College London, United Kingdom)

Stochastic quantisation of Yang-Mills

Abstract: We report on recent progress on the problem of building a stochastic process that admits the hypothetical Yang-Mills measure as its invariant measure. One interesting feature of our construction is that it preserves gauge-covariance in the limit even though it is broken by our UV regularisation. This is based on joint work with Ajay Chandra, Ilya Chevyrev, and Hao Shen.

Feb 24, 2023

Rainer Buckdahn                       Poster:     Slides
(Universite de Bretagne Occidentale, France)

Mean field stochastic control under sublinear expectation

Abstract: The talk is devoted to the study of Pontryagin's stochastic maximum principle for a mean-field optimal control problem under Peng's sublinear G-expectation. The dynamics of the controlled state process is given by a SDE driven by a G-Brownian motion, whose coefficients depend on the control, the controlled state process but also on its law under the G-expectation. Also the associated cost functional is of mean-field type. Under the assumption of a convex control state space we study Pontryagin's stochastic maximum principle, which gives a necessary optimality condition for the control process. Under additional convexity assumptions on the Hamiltonian it is shown that this necessary condition is also a sufficient one. The main difficulty which we had to overcome in our work consists in the differentiation of the G-expectation of parametrised random variables. As particularly delicate turns out to handle the G-expectation of a function of the controlled state process inside the running cost of the cost functional. For this we had to study a measurable selection theorem for set-valued functions whose values are subsets of the representing set of probability measures for the G-expectation. The talk is based on a recent joint work with Juan Li and Bowen He (Shandong University, Weihai, China).

Mar 10, 2023

Christa Cuchiero                        Poster:     Slides Video   (expires September 2023)
(University of Vienna, Austria)

From Lévy's stochastic area formula to universality of affine and polynomial processes

Abstract: A plethora of stochastic models used in diverse areas, like mathematical finance, population genetics or physics, stems from the class of affine and polynomial processes. The history of these processes is on the one hand closely connected with the important concept of tractability, that is a substantial reduction of computational efforts due to special structural features, and on the other hand with a unifying framework for a large number of probabilistic models. One early instance in the literature where this unifying affine and polynomial point of view can be applied is Lévy's stochastic area formula. Starting from this example, we present a guided tour through the main properties and results as well as classical and recent applications, which culminates in the surprising insight that infinite dimensional affine and polynomial processes are actually close to generic stochastic processes.

Mar 24, 2023

Antoine Lejay                        Poster:     SlidesVideo   (expires September 2023)
(Université de Lorraine, France)

Estimation of the parameter of the Skew Brownian motion

Abstract: The skew Brownian motion (SBM) is a useful process to deal with diffusion in media presenting some interface. Actually, it behaves like a Brownian motion away from 0. When reaching 0, its behavior is ruled by a single parameter θ ∈ [-1,1] which affects the tendency of the particle to go upward or downward. When θ =±1, the SBM is a reflected Brownian motion while for θ = 0, the SBM is a Brownian motion.
In this talk, we discuss the main properties of the Maximum Likelihood Estimator (MLE), which is consistent and asymptotically mixed normal with a non standard rate of 1/4. In particular, we study its behavior with respect to the value of the true parameter θ and we give an infinite series expansion of the MLE thanks to a recent asymptotic inverse function theorem.

From joint works with E. Mordecki, S. Torres, and S. Mazzonetto.

Mar 31, 2023 11:30 UTC European daylight saving times have started

Andreas Neuenkirch                        Poster:     Slides Video   (expires September 2023)

(University on Mannheim, Germany)

Strong approximation of the CIR process: A never ending story?

Abstract: The CIR process is the prototype stochastic differential equation (SDE) for the class of square root diffusions. These equations have widespread applications, in particular in finance, biology and chemistry. Moreover, since the diffusion coefficient contains a square root and is not Lipschitz continuous, the CIR process is also the prototype example for an SDE whose coefficients do not satisfy the so-called standard assumptions for numerical analysis. Due to these reasons, the approximation of the CIR process has attracted a lot of attention in the last 20 years. In this talk, I will give a state-of-the-art summary and will present some of the latest developments for the strong approximation of the CIR process.

Apr 14, 2023

Konstantinos Dareiotis                        Poster:     Slides Video   (expires September 2023)

(University of Leeds, United Kingdom)

Regularisation of differential equations by multiplicative fractional noises

Abstract: In this talk, we consider differential equations perturbed by multiplicative fractional Brownian noise. Depending on the value of the Hurst parameter H, the resulting equation is pathwise viewed as an ordinary ( H>1), Young ( H ∈ (1/2, 1)) or rough (H ∈ (1/3, 1/2)) differential equation. In all three regimes we show regularisation by noise phenomena by proving the strongest kind of well-posedness for equations with irregular drifts: strong existence and path-by-path uniqueness. In the Young and smooth regime H>1/2 the condition on the drift coefficient is optimal in the sense that it agrees with the one known for the additive case. In the rough regime H ∈ (1/3,1/2) we assume positive but arbitrarily small drift regularity for strong well-posedness, while for distributional drift we obtain weak existence.

This is a joint work with Máté Gerencsér.

Apr 21, 2023

René Schilling                        Poster:     Slides

(Technische Universität Dresden, Germany)

On Liouville's Theorem for Nonlocal Operators

Abstract: We discuss a proof of Liouville's theorem (all bounded harmonic functions are constant) for a class of Fourier multiplier operators; this class includes Lévy-operators. We also show extensions where the harmonic functions are allowed to grow.

This is joint work with D. Berger (TU Dresden) und E. Shargorodsky (King's College, London).

May 05, 2023

Máté Gerencsér                        Poster:     Slides Video   (expires October 2023)

(TU Wien, Austria)

Integration along stochastic processes

Abstract: We consider integrals of expressions of the form f(X), where X is a stochastic process, and f is only a distribution. We overview some recent results on defining such integrals and discuss a variety of their applications in regularisation by noise for stochastic differential equations such as existence, uniqueness, stability, regularity, and approximation properties.

May 19, 2023

Christoph Reisinger                       Poster:     Slides Video   (expires October 2023)

(University of Oxford, United Kingdom)

A posteriori error estimates for fully coupled McKean–Vlasov forward-backward SDEs

Abstract: Fully coupled McKean–Vlasov forward-backward stochastic differential equations (MV-FBSDEs) arise naturally from large population optimization problems. Judging the quality of given numerical solutions for MV-FBSDEs, which usually require Picard iterations and approximations of nested conditional expectations, is typically difficult. In this talk, we propose an a posteriori error estimator to quantify the L2-approximation error of an arbitrarily generated approximation on a time grid. We establish that the error estimator is equivalent to the global approximation error between the given numerical solution and the solution of a forward Euler discretized MV-FBSDE.
We further demonstrate that, for sufficiently fine time grids, the accuracy of numerical solutions for solving the continuous MV-FBSDE can also be measured by the error estimator. The error estimates justify the use of residual-based algorithms for solving MV-FBSDEs. Numerical experiments for MV-FBSDEs arising from mean field control and games confirm the effectiveness and practical applicability of the error estimator.

Schedule 2022 Autumn

Oct 28, 2022 @ 12:30 UTC (3:30pm Helsinki), one hour later than usual!

David Nualart                        Poster:     Slides
(Kansas University):

Limit theorems for additive functionals of the fractional Brownian motion

Abstract: In this talk we will present some recent results on first and second order fluctuations of a class of additive functionals of a fractional Brownian motion. Two different behaviors arise depending of the value of the Hurst parameter H. When the Hurst parameter is larger or equal than 1/3, the limit in distribution turns out to be an independent Brownian motion subordinated to the local time. When H is less than 1/3, the limit is a constant multiple of the derivative of the local time.

Nov 11, 2022 @ 12:30 UTC (3:30pm Helsinki), one hour later than usual!

Yaozhong Hu                        Poster:     Slides
(University of Alberta, Canada)

Nonlinear Stochastic Wave Equation Driven by Rough Noise

Abstract: This talk wil be concerned with the existence and uniqueness of a strong solution to the one-dimensional stochastic wave equation

assuming , where is a mean zero Gaussian noise which is white in time and fractional in space with Hurst parameter . The idea is to decompose the simple one-dimensional Green kernel into some more complicated ones which make the thing work
This talk is based on joint work with Shuhui Liu and Xiong Wang.

Nov 25, 2022

Ying Jiao                        Poster:     Slides
(Université Claude Bernard Lyon 1, France)

Socioeconomic Pathways of Carbon Emission and Credit Risk

Abstract: As the world is facing global climate changes, the Intergovernmental Panel on Climate Change (IPCC) has set the idealized carbon-neutral scenario around 2050. In the meantime, many carbon reduction scenarios, known as Shared Socioeconomic Pathways (SSPs) have been proposed in the literature. We aim to investigates the impact of transition risk on a firm’s low-carbon production. On the one hand, we consider a firm that searches to optimize its emission level under the double objectives of maximizing its production profit and respecting the emission mitigation scenarios. Solving the penalized optimization problem provides the optimal emission according to a given SSPs benchmark. On the other hand, such transitions affect the firm’s credit risk. We model the default time by using the structural default approach and are particularly concerned with how the adopted strategies following different SSPs scenarios may influence the firm’s default probability. Finally we discuss possible extensions of the model to a large portfolio of climate concerned firms and propose numerical methods to compute the cumulative losses.
This is joint work with Florian Bourgey and Emmanuel Gobet.

Dec 09, 2022

Yuri Kabanov                        Poster:     Slides

(University of Franche-Comté, Besançon, France)

Recent results in the ruin theory with investments

Abstract: In the classical collective risk theory it is usually assumed that the capital reserve of a company is placed in a bank account paying zero interest. In the recent three decades the theory was extended to cover a more realistic situation where the reserve is invested, fully or partially, in a risky asset (e.g., in a portfolio evolving as a market index). This natural generalization generates a huge variety of new ruin problems which can be considered as the exit problem for a semimartingale Ornstein-Uhlenbeck process. Roughly speaking, each “classical” ruin problem, e.g., a version of the Cramer-Lundberg model (for the non-life insurance, for the annuity payments etc.) can be combined with a model of price of the risky security (geometric Brownian motion, geometric Lévy process, various models with stochastic volatilities, etc.). In the talk we present new asymptotic results for the ruin probabilities, in particular, for the Sparre Andersen type models with risky investments having the geometric Lévy dynamics and for Cramér-Lundberg type models with investments in a risky asset with a regime switching price.

Schedule 2022 Spring

Jan 14, 2022

Arturo Kohatsu-Higa                        Poster:     Slides
(Ritsumeikan University, Kusatsu):

Acceleration of convergence and regularity of the law for jump processes

Abstract: Recently, I have been interested in how acceleration of approximation methods can be used in order to derive theoretical properties about the laws of their limits. In most situations, numerical approximations do not have good properties to assure that their properties transfer to their limit laws. In this talk, I will give two examples with different acceleration techniques. In the first, we will consider a jump driven sde with purely atomic Lévy measure with infinite activity. We will apply the Asmussen-Rosinki approximation for small jumps of infinite activity Lévy processes and argue that the use of this approximation in order to derive properties of the limits is not possible. Instead, we will combine this method with the moment method in order to improve the approximation and finally obtain results on the regularity of the laws of stochastic differential equations with jumps. This approach is interesting as it only uses Malliavin calculus with respect to the Brownian motion associated to the Asmussen-Rosinki approach. This is joint work with V. Bally and L. Caramellino.
In a second part, we will discuss how to obtain optimal upper bounds for the joint density of a stable process and its supremum using Multi-level Monte Carlo method techniques. We build an ad-hoc Malliavin Calculus method based on the Chamber-Mallows-Struck/Kanter simulation approach for stable increments.
This is joint work with Jorge Gonzales-Cazares and Alex Mijatovic.

Jan 28, 2022

Denis Talay                        Poster:     Slides
(INRIA & École Polytechnique):

First hitting time Laplace transforms of solutions of SDEs are Lipschitz continuous in the Hurst parameter of the driving fractional Brownian noise

Abstract: Sensitivity analysis w.r.t. the long-range/memory noise parameter for probability distributions of functionals of solutions to stochastic differential equations is an important stochastic modeling issue in many applications. In this talk we consider solutions {X​tH} t ∈ R+ to stochastic differential equations driven by fractional Brownian motions. We examine Laplace transforms of functionals which are irregular with regard to Malliavin calculus, namely, first passage times of X H at a given threshold. We will present parts of the machinery necessary to prove the Lipschitz continuity w.r.t. H around the value ½. This result implies that, for applications where first hitting times are a crucial information, the Markov Brownian model is a good proxy model as long as the Hurst parameter remains close to ½.
This is joint work with Alexandre Richard (Ecole Centrale-Supelec, France).

Feb 11, 2022

Mireille Bossy                             Poster:     Slides
(INRIA, Université Côte d'Azur, France):

Weak convergence rate approximation for some SDEs with superlinear growth coefficients

Abstract: In this talk, I would like to discuss around the approximation of the solution of the following class of 1D-SDEs
                                     dXt = b(Xt )dt + σ Xt α dWt ,     X0 =x>0,
when α >1. I will first present some motivating examples where such SDEs arise in modelling approaches, as long-time limit approximation of McKean SDEs of CIR-type for the amplitude of local wind fluctuations. Of course, a priori knowledge on such SDEs about the conditions on the coefficients ensuring well-posedness and some control on the moments is required, not only for their use as models but also to study the convergence of time-integration schemes.
Here, we focus on the weak convergence rate. First, a set of conditions for the C1,4 regularity of the Feynman-Kac formula is proposed using stochastic tools. While direct derivation of the Feynman-Kac formula requires some smoothness for the drift b, we show how to avoid the costly control of the moments of the successive derivatives of the flow process by using a change of measure technique, allowing b to be taken superlinear too. We then introduce an exponential scheme for the time integration of the SDE, which reproduces well the control of the moments of the exact process and for which we prove a convergence rate of order one.
This talk is based on two recent papers with Kerlyns Martínez Rodríguez (University of Vaparaíso) and Jean Francois Jabir (HSE University Moscow).

Feb 25, 2022 @ 12:30 UTC, one hour later than usual!

Xin Guo                             Poster:     Slides
(University of California at Berkeley):

Generative adversarial models: an analytical perspective

Abstract: Recently, the popularity and successes of Generative Adversarial Networks (GANs) in computer vision and image generation have attracted intense attention from the mathematical finance community. GANs have since then been applied to financial data generation and lately shown capable of computing solutions for high dimensional mean-field games.
In this talk, we will discuss the connection of GANs with optimal transport, game structure of GANs in an SDE framework, and present our latest work on the stochastic control approach for the stability of GANs training.
Based on joint work with Haoyang Cao of Ecole Polytechnique, and Othmane Mounjid of UC Berkeley.

Mar 11, 2022

Lukasz Szpruch                             Poster:     Slides
(University of Edinburgh):

Gradient Flows for Regularized Stochastic Control Problems

Abstract: We study stochastic control problems regularized by the relative entropy, where the action space is the space of measures. By exploiting the Pontryagin optimality principle, we identify a suitable metric space on which we construct the gradient flow for the measure-valued control process along which the cost functional is guaranteed to decrease.
It is shown that under appropriate conditions, this gradient flow has an invariant measure which is the optimal control for the regularized stochastic control problem. If the problem we work with is sufficiently convex, the gradient flow converges exponentially fast. Furthermore, the optimal measure valued control admits Bayesian interpretation which means that one can incorporate prior knowledge when solving the stochastic control problem.
This work is motivated by a desire to extend the theoretical underpinning for the convergence of stochastic gradient type algorithms widely used in the reinforcement learning community to solve control problems.

Mar 25, 2022 @ 12:30 UTC, one hour later than usual!

Jianfeng Zhang                             Poster:     Slides
(University of Southern California):

Propagation of Monotonicity for Mean Field Game Master Equations

Abstract: It is well known in the mean field game literature that a certain monotonicity condition is crucial for the uniqueness of mean field equilibria and for the wellposedness of the associated master equation. One interesting observation is that the propagation of the monotonicity (either in Lasry-Lions sense or in displacement sense) of the value function plays the key role here. We shall introduce a method to find conditions on the coefficients which ensure that any solution of the master equation will maintain the monotonicity property. This method also allows us to consider anti-monotonicity and obtain the desired wellposedness provided the coefficients are sufficient anti-monotone in appropriate sense. We finally extend our results to mean field games of controls. The talk is based on a joint work with Gangbo-Meszaros-Mou and two joint works with Mou.

Apr 08, 2022

Krzysztof Bogdan                             Poster:     Slides
(Wrocław University of Science and Technology):

Self-similar solution for Hardy operator

Abstract: We will discuss the large-time asymptotics of solutions to the heat equation for the fractional Laplacian with added subcritical or even critical Hardy-type potential. The asymptotics is governed by a self-similar solution of the equation, obtained as a normalized limit at the origin of the kernel of the corresponding Feynman-Kac semigroup. This will be our focus. Interestingly, an Ornstein-Uhlenbeck semigroup turns out to be an important tool for the analysis. The paper is on arXiv and it is joint work with P. Kim (Seoul), T. Jakubowski, and D. Pilarczyk (Wrocław).

Apr 22, 2022

Étienne Pardoux                             Poster:     Slides

(Aix-Marseille Université):

Epidemic models with varying infectivity and varying susceptibility

Abstract: Almost a century ago, Kermack and McKendrick suggested to take into account the fact that the infectivity of infectious individuals vary with the time elapsed since their infection, the duration of the infectious period can have a very general distribution, and in case of loss of immunity, it is progressive.
Unfortunately, almost all the literature on mathematical models of epidemics concentrate on simpler ODE models, which are the law of large numbers limits, as the size of the population tends to infinity, of finite population stochastic Markovian models.
We consider the models suggested by Kermack and McKendrick, and obtain those models (or in one case a generalization of their model), which are convolution type equations, as law of large numbers limits of general non Markov stochastic finite population models. We also derive some associated Central Limit Theorems. Our approach involves the study of a class of mean field Poisson driven SDEs.
This is joint work with Guodong Pang, Raphaël Forien and Arsene Brice Zotsa-Ngoufack.

May 06, 2022

François Delarue                             Poster:     Slides
(Université Côte d’Azur, Nice):

Weak solutions to the master equation of potential mean field games

Abstract: The talk is motivated by the theory of mean field games, initiated by Lasry and Lions and by Caines, Huang and Malhamé. The very purpose of it is to address the so-called master equation, which describes the value of the game, when equilibria may not be unique. In order to do so, we restrict ourselves to a class of mean field games that coincide with the first order condition of an optimal control problem set over McKean-Vlasov dynamics. Such games are called potential. We then introduce a notion of weak solution to the master equation and prove that existence and uniqueness hold under quite general assumptions. The key point is to interpret the master equation in a conservative sense and then to adapt to the infinite dimensional setting earlier arguments for hyperbolic systems deriving from a Hamilton-Jacobi-Bellman equation. This a joint work with Alekos Cecchin (Padova, Italy), see arXiv 2204.04315

May 20, 2022

Ying Hu                             Poster:     SlidesVideo   (expires November 2022)

(Université de Rennes 1, CNRS):

Scalar valued BSDEs with sublinear, superlinear and subquadratic growth

Abstract: The aim of this talk is to give some well-posedness results for scalar valued Backward Stochastic Differential Equations (BSDEs) when the generator has a sublinear growth, superlinear growth, subquadratic growth in the second variable. In each of these cases, we give some precise conditions for terminal random variables to guarantee the existence and uniqueness of the solution. Joint works with Shengjun Fan and Shanjian Tang.

Schedule 2021

Oct 29, 2021

Peter Friz (TU Berlin and WIAS Berlin):           Poster:          

On rough SDEs

Abstract: A hybrid theory of rough stochastic analysis is built that seamlessly combines the advantages of both Itô's stochastic - and Lyons' rough differential equations. A major role is played by a new stochastic variant of controlled rough paths spaces, with norms that reflect some generalized stochastic sewing lemma, and which may prove useful whenever rough paths and Itô integration meet. We will mentioned several applications. Joint work with Antoine Hocquet and Khoa Lê (both TU Berlin).

Nov 12, 2021

Xunyu Zhou (Columbia University, New York):           Poster:          

Policy Evaluation and Temporal-Difference Learning in Continuous Time and Space:
A Martingale Lens

Abstract: We propose a unified framework to study policy evaluation (PE) and the associated temporal difference (TD) methods for reinforcement learning in continuous time and space. Mathematically, PE is to devise a data-driven Feynman--Kac formula without knowing any coefficients of a PDE. We show that this problem is equivalent to maintaining the martingale condition of a process. From this perspective, we present two methods for designing PE algorithms. The first one, using a "martingale loss function", interprets the classical gradient Monte-Carlo algorithm. The second method is based on a system of equations called the "martingale orthogonality conditions". Solving these equations in different ways recovers various classical TD algorithms, such as TD, LSTD, and GTD. We apply these results to option pricing and portfolio selection. This is joint work with Yanwei Jia.

Nov 26, 2021

Michael Scheutzow             Poster:     Slides
(Technische Universität Berlin):

Generalized couplings and stochastic functional differential equations

Abstract: We provide an introduction to generalized couplings and present a recent result [contained in S.: Couplings, generalized couplings and uniqueness of invariant measures. ECP, 2020] which says that the existence of a generalized coupling for a Markov chain implies uniqueness of an invariant probability measure even if the state space is just a metric space without requiring separability or completeness as in previous works. The proof turns out to be rather elementary. We show how this result can be applied to show uniqueness of an invariant measure for the solution process of a stochastic functional differential equation (SFDE) and we show how generalized couplings can be employed to show weak uniqueness of solutions of an SFDE with Hölder continuous coefficients.
Parts of the talk are based on joint work with Alex Kulik (Wroclaw).

Dec 10, 2021

Nizar Touzi                 Poster:     Slides
(CMAP & Polytechnique Paris):

Entropic mean field optimal planning

Abstract: The problem of optimal planning was introduced by P.-L. Lions in the context of a mean field game, by fixing a target distribution in the Focker-Planck equation and relaxing the boundary condition in the HJB equation. We analyze an extension of this problem to the path-dependent setting which has remarkable connections with optimal transport and optimal incentive theory in economics. We provide a general characterization of mean field optimal planning solutions, and we solve explicitly the minimum entropy optimal planning problem.