# Workshop on Stochastic Control Theory, October 25-26, 2023

The purpose of this workshop is to bring together researchers interested in stochastic control theory and its applications. There will be talks on mathematical theory and methods related to stochastic control and games.

## Speakers

Christian Bayer, Weierstrass Institute, Berlin

Fred Espen Benth, University of Oslo

Amarjit Budhiraja, University of North Carolina, Chapel Hill

Bruno Bouchard, University of Paris Dauphine PSL

Fabio Camilli, University of Rome

Roxana Dumitrescu, King's College, London

Erik Ekström,University of Uppsala

Said Hamadene, University of Le Mans

Sigrid Källblad Nordin, KTH, Stockholm

Huyen Pham, University of Paris Diderot

Raul Tempone, KAUST

### Registration

Please email hult@kth.se for a registration link (registration is free of charge).

### Location

Room K1 , Teknikringen 56, KTH Campus, Valhallavägen, Stockholm.

### Schedule

*Wednesday, October 25*

09.05-09.30 Registration

09.30-10.15 Amarjit Budhiraja

10.15-10.45 Coffee

10.45-11.30 Said Hamadene

11.30-12.15 Bruno Bouchard

12.15-14.00 Lunch

14.00-14.45 Fabio Camilli

14.45-15.30 Roxana Dumitrescu

15:30-16:00 Coffee

16:00-16:45 Sigrid Källblad Nordin

19.30 Conference dinner

*Thursday, October 26*

9.30-10.15 Fred Espen Benth

10.15-10.45 Coffee

10.45-11.30 Huyen Pham

11.30-12.15 Christian Bayer

12.15-14.00 Lunch

14.00-14.45 Erik Ekström

14.45-15.30 Raul Tempone

15.30-16:00 Coffee

### Talks

**Amarjit Budhiraja, University of North Carolina, Chapel Hill****Title:** Large deviations for weakly interacting diffusions and mean field stochastic control problems**Abstract:** Consider a collection of particles whose state evolution is described through a system of interacting diffusions in which each particle is driven by an independent individual source of noise and also by a small amount of noise that is common to all particles. The interaction between the particles is due to the common noise and also through the drift and diffusion coefficients that depend on the state empirical measure. We study large deviation behavior of the empirical measure process which is governed by two types of scaling, one corresponding to mean field asymptotics and the other to the Freidlin–Wentzell small noise asymptotics. The rate functions under the various parameter regimes can be interpreted as the value functions of certain mean field stochastic control problems in which there are two types of controls; one of the controls is random and nonanticipative and arises from the aggregated contributions of the individual Brownian noises, whereas the second control is nonrandom and corresponds to the small common Brownian noise that impacts all particles. This is based on joint work with Conroy.

**Said Hamadene, University of Le Mans****Title:** Stochastic Impulse Control with Delay and Random Coefficients**Abstract:** In this talk we discuss an infinite horizon impulse control problem with execution delay when the dynamics of the system is described by a general stochastic process adapted to the Brownian filtration. The problem is solved by means of probabilistic tools relying on the notion of Snell envelope and infinite horizon reflected backward stochastic differential equations. This allows us to establish the existence of an optimal strategy over all admissible strategies. We consider also the case of exponential utilities. The talk is based on

[1] Djehiche, B., Hamadene, S., Hdhiri, I., & Zaatra, H., Infinite horizon stochastic impulse control with delay and random coefficients, Mathematics of Operations Research, 2022, 47(1), 665-689.

**Bruno Bouchard, University of Paris Dauphine PSL****Title:** On the regularity of solutions of some linear path-dependent parabolic PDEs**Abstract: **We consider a family of linear parabolic partial differential equations defined on the space of continuous path x in C([0,T]), in which coefficients at time t depend on x(t) and the integral of x with respect to A, for some continuous process A with bounded variations. When the equation is uniformly elliptic, we provide conditions on A and the Hölder regularity of the coefficients under which existence of a smooth solution holds, when appealing to the notion of Dupire's derivatives. It provides a generalization to the existing literature considering situations in which A admits a density with respect to the Lebesgue's measure, and complements the recent work of Bouchard, Loeper and Tan (2022) on the regularity of approximate viscosity solutions for path-dependent parabolic partial differential equations. We shall also review some recent results on the Dupire-Itô’s formula for path-dependent functionals that are only C^{0,1}, in the sense of Dupire.

**Fabio Camilli, University of Rome****Title:** Continuous dependence estimates for viscous Hamilton-Jacobi equations on networks with applications**Abstract:** After a brief review of the study of partial differential equations on networks, we present a result concerning a continuous dependence estimates for viscous Hamilton-Jacobi equations defined on such geometric structures. Given two Hamilton-Jacobi equations, we prove an estimate on the C2-norm of the difference between the corresponding solutions in terms of the distance among the coefficients. Then, we

provide two applications of the previous estimate: the first one is an existence and uniqueness result for a quasi-stationary Mean Field Games defined on a network; the second one is an estimate of the rate of convergence for homogenization of viscous Hamilton-Jacobi equations defined on a periodic lattice, when the size of the cells vanishes and the limit problem is defined in the whole Euclidean space.

**Roxana Dumitrescu, King's College****Title:** A new Mertens decomposition of Y g,ξ-submartingale systems and applications**Abstract:** We introduce the concept of Y g,ξ-submartingale systems, where the nonlinear operator Y g,ξ corresponds to the first component of the solution of a reflected BSDE with generator g and lower obstacle ξ. We first show that, in the case of a left-limited right-continuous obstacle, any Y g,ξ-submartingale system can be aggregated by a process which is right-lower semicontinuous. We then prove a Mertens decomposition, by using an original approach which does not make use of the standard penalization technique. These results are in particular useful for the treatment of control/stopping game problems and, to the best of our knowledge, they are completely new in the literature. We finally present two applications in Finance.

**Sigrid Källblad Nordin, KTH Stockholm****Title:** Adapted Wasserstein distance between the laws of SDEs**Abstract:** We consider an adapted optimal transport problem between the laws of Markovian stochastic differential equations (SDEs) and establish optimality of the so-called synchronous coupling between the given laws. The proof of this result is based on time-discretisation methods and reveals an interesting connection between the synchronous coupling and the celebrated discrete-time Knothe–Rosenblatt rearrangement. We also provide a related result on equality of various topologies when restricted to certain laws of continuous-time stochastic processes. The result is of relevance for the study of stability with respect to model specification in mathematical finance. The talk is based on joint work with Julio Backhoff-Veraguas and Ben Robinson.

**Raul Tempone, RWTH Aachen****Title: **Stochastic Optimal Control with Exercise Rate Optimization and Markovian Projections: Pricing American Options and Importance Sampling Applications**Abstract:** This talk begins with the problem of pricing American basket options in a multivariate setting. In high dimensions, nonlinear PDE methods for solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation become expensive due to the curse of dimensionality. In the first part, we present a novel method [1] for the numerical pricing of American options based on Monte Carlo Simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal *exercise regions*, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the *exercise rates* of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate, even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Natural parameters for discretization are the number of time-discretization steps and the required degrees of freedom in the parametrization of the exercise rates. Numerical experiments on vanilla put options in the multivariate Black--Scholes model, and a preliminary theoretical analysis underline the efficiency of our method. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black--Scholes model and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model.

In the second part of the presentation, we propose to use a stopping rule that depends on the dynamics of a low-dimensional Markovian projection of the given basket of assets [2]. It is shown that the ability to approximate the original value function by a lower-dimensional approximation is a feature of the system's dynamics and is unaffected by the path-dependent nature of the American basket option. Assuming that we know the density of the forward process and using the Laplace approximation, we first efficiently evaluate the diffusion coefficient corresponding to the low-dimensional Markovian projection of the basket. Then, we approximate the optimal early-exercise boundary of the option by solving an HJB partial differential equation in the projected, low-dimensional space. The resulting near-optimal early-exercise boundary is used to produce an exercise strategy for the high-dimensional option, thereby providing a lower bound for the price of the American basket option. A corresponding upper bound is also provided. These bounds allow us to assess the accuracy of the proposed pricing method. Indeed, our approximate early-exercise strategy provides a straightforward lower bound for the American basket option price. Following a duality argument due to Rogers [3], we derive a corresponding upper bound solving only the low-dimensional optimal control problem. Numerically, we show the method's feasibility using baskets with dimensions up to fifty. In these examples, the resulting option price relative errors are only a few percent. In the last part of the presentation, we will discuss briefly Importance Sampling based on Stochastic Optimal Control techniques. We will discuss two cases, namely i) Stochastic reaction networks (SRNs) [4] and ii) McKean Vlasov Equations [5] Our numerical experiments verify that the proposed Importance Sampling substantially reduces the Monte Carlo estimator's variance, resulting in a lower computational cost in the rare event regime than standard Monte Carlo estimators.

References

[1] C. Bayer, R. Tempone and S. Wolfers (2020). Pricing American Options by Exercise Rate Optimization, arXiv:1809.07300 (2018). Quantitative Finance, 2020. https://doi.org/10.1080/14697688.2020.1750678

[2] C. Bayer, J. Häppölä and R. Tempone, (2019). Implied Stopping Rules for American Basket Options from Markovian Projection, Quantitative Finance, 19(3) , 371--390.

[3] Rogers, L.C., (2002). Monte Carlo valuation of American options. Mathematical Finance, 12, 271–286.

[4] Ben Hammouda, C., Ben Rached, N., Tempone, R., and Wiechert, S. (2023). Automated Importance Sampling via Optimal Control for Stochastic Reaction Networks: A Markovian Projection-based Approach. *arXiv preprint arXiv:2306.02660*.

[5] Ben Hammouda, C., Ben Rached, N., Tempone, R., and Wiechert, S. (2023). Learning-based importance sampling via stochastic optimal control for stochastic reaction networks. *Statistics and Computing*, *33*(3), 58.

[6] Ben Rached, N., Haji-Ali, A. L., Pillai, S. M. S., and Tempone, R. (2022). Single Level Importance Sampling for McKean-Vlasov Stochastic Differential Equation. *arXiv preprint arXiv:2207.06926*.

[7] Ben Rached, N. , Haji-Ali, A. L., Pillai, S. M. S., and Tempone, R. (2022). Multilevel Importance Sampling for McKean-Vlasov Stochastic Differential Equation. *arXiv preprint arXiv:2208.03225*.

[8] Ben Rached, N. , Haji-Ali, A. L., Pillai, S. M. S., and Tempone, R. (2023). Multi-index Importance Sampling for McKean-Vlasov Stochastic Differential Equation. *arXiv preprint arXiv:2307.05149*.

**Huyen Pham, University of Paris Diderot****Title:** Nonparametric generative modeling for time series via Schrödinger bridge**Abstract:** We propose a novel generative model for time series based on Schrödinger bridge (SB) approach. This consists in the entropic interpolation via optimal transport between a reference probability measure on path space and a target measure consistent with the joint data distribution of the time series. The solution is characterized by a stochastic differential equation on finite horizon with a path-dependent drift function, hence respecting the temporal dynamics of the time series distribution. We estimate the drift function from data samples by nonparametric, e.g. kernel regression methods, and the simulation of the SB diffusion yields new synthetic data samples of the time series.

The performance of our generative model is evaluated through a series of numerical experiments. First, we test with autoregressive models, a GARCH Model, and the example of fractional Brownian motion, and measure the accuracy of our algorithm with marginal, temporal dependencies metrics, and predictive scores. Next, we use our SB generated synthetic samples for the application to deep hedging on real-data sets.

**Christian Bayer, Weierstrass Institute, Berlin****Title:** Optimal stopping with signatures**Abstract:** We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process X. We consider classic and randomized stopping times represented by linear and non-linear functionals of the rough path signature X^\infty associated to X, and prove that maximizing over these classes of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature. By applying a deep neural network approach to approximate the non-linear signature functionals, we can efficiently solve the optimal stopping problem numerically. The only assumption on the process X is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion, which fail to be either semi-martingales or Markov processes, and can be used, in particular, for American-type option pricing in fractional models, e.g. of financial or electricity markets. (Based on joint work with Paul Hager, Sebastian Riedel, and John Schoenmakers)

**Erik Ekström,University of Uppsala****Title**: Irreversible investment under incomplete information and learning-by-doing**Abstract:** We introduce a notion of learning-by-doing for irreversible investment decisions, where an investor faces a project with unknown profitability. By investing in the project, the learning rate of the unknown profitability is gradually increased; however, due to irreversibility, there is an intrinsic cost of investment, which needs to be balanced against the potential profits of investment. We formulate this problem as a singular control problem with incomplete information. In our main result, we provide conditions under which the no-action region and the action region

are separated by a boundary, which can be characterized in terms of an ODE. The optimal investment policy is then to gradually invest so that the sufficient statistic reflects along this boundary.

**Fred Espen Benth, University of Oslo****Title:** Pricing options on flow forwards by neural networks in Hilbert space**Abstract: **Flow forwards are forward contracts delivering the underlying commodity over a period. In electricity markets, say, call and put options are traded on such forwards. Using the HJM-approach, one can view the dynamics of flow forwards as a stochastic process with values in a Hilbert space of functions on the positive real line. Option prices depend on the whole forward curve rather than a single price. We approach pricing of options on flow forwards by a neural net, which is defined on the infinite dimensional Hilbert space. Using knowledge of the basis on this space, which consists of financially interpretable functions (term structures), we can design an efficient neural network which improves the performance of a classical network for this pricing problem. This talk is based on joint work with Nils Detering (Duesseldorf) and Luca Galimberti (King's College London).