# Dr. Miryana Grigorova

Financial and Actuarial Mathematics

## Short CV

Dr. Miryana Grigorova holds a Bachelor's degree in Applied Mathematics from University Paris-Dauphine and a Master's degree in Stochastic Modelling from University Paris-Diderot. She gained a PhD degree in Applied Mathematics with an emphasis on stochastic modelling for financial applications from University Paris-Diderot. Dr. Grigorova worked as a Research Associate at Humboldt University-Berlin, the Center for Excellence in Risk and Insurance in Hannover, and at Bielefeld University. Dr. Grigorova held the position of Lecturer in Financial and Actuarial Mathematics at the University of Leeds. She served as the Programme Director of the MSc Financial Mathematics at the School of Mathematics of the same institution and as the Liaison Manager between the School of Mathematics and Leeds University Business School. She is currently an Assistant Professor at the University of Warwick. Her research interests lie at the interface of probability theory, stochastic analysis, mathematical finance and actuarial science. Her recent research focuses on topics in stochastic control, optimal stopping, game theory, risk measures, and their applications to finance, economics, and insurance.

## Publications

### Stochastic representation under g-expectation and applications: The discrete time case

###

Miryana Grigorova and Hanwu Li

, and

Journal of Mathematical Analysis and Applications

Volume 518 (1), February 2023

In this paper, we address the stochastic representation problem in discrete time under (non-linear) g-expectation. We establish existence and uniqueness of the solution, as well as a characterization of the solution. As an application, we investigate a new approach to the optimal stopping problem under g-expectation and the related pricing of American options under Knightian uncertainty. Our results are also applied to a (non-linear) Skorokhod-type obstacle problem.

### American options in a non-linear incomplete market model with default

###

Miryana Grigorova, Marie-Claire Quenez, and Agnès Sulem

, and

Stochastic Processes and their Applications

Volume 142 (12), Pages 479-512, 2021

We study the superhedging problem for American options with completely irregular payoffs in a non-linear and incomplete market model with default. We give a dual representation of the seller’s (superhedging) price in terms of the value of a non-linear mixed control/stopping problem, involving a suitable set of equivalent probability measures. We characterize the seller’s price process as the minimal supersolution of two types of reflected BSDEs: a constrained one and an optional one. Under some regularity assumptions on the pay-off, we show a duality result for the buyer’s price in terms of the value of a non-linear control/stopping game problem.

European options in a non-linear incomplete market model with default

Miryana Grigorova, Marie-Claire Quenez, and Agnès Sulem

, and

SIAM Journal on Financial Mathematics

Volume 11 (3), Pages 849–880, 2020

We study the superhedging prices and the associated superhedging strategies for European options in a non-linear incomplete market model with default. The underlying market model consists of one risk-free asset and one risky asset, whose price may admit a jump at the default time. The portfolio processes follow non-linear dynamics with a non-linear driver f. By using a dynamic programming approach, we first provide a dual formulation of the seller’s (superhedging) price for the European option as the supremum, over a suitable set of equivalent probability measures Q∈ Q, of the f-evaluation/expectation under Q of the payoff. We also establish a characterization of the seller’s (superhedging) price as the initial value of the minimal supersolution of a constrained backward stochastic differential equation with default. Moreover, we provide some properties of the terminal profit made by the seller, and some results related to replication and no-arbitrage issues. Our results rely on first establishing a non-linear optional and a non-linear predictable decomposition for processes which are Ef-strong supermartingales under Q for all Q∈Q.

Miryana Grigorova, Peter Imkeller, Youssef Ouknine and Marie-Claire Quenez

Electronic Communications in Probability

Volume 25, paper 49, 9 pages, 2020

Miryana Grigorova, Peter Imkeller, Youssef Ouknine and Marie-Claire Quenez

Stochastic Processes and their Applications

Volume 130 (3), Pages 1258-1288, 2020

We consider the optimal stopping problem with non-linear f-expectation (induced by a BSDE) without making any regularity assumptions on the pay-off process $\xi$. We show that the value family can be aggregated by an optional process Y. We characterize the process Y as the $\mathcal{E}^f$-Snell envelope of $\xi$. We also establish an infinitesimal characterization of the value process Y in terms of a Reflected BSDE with $\xi$ as the obstacle. This characterization is established by first showing existence and uniqueness for the Reflected BSDE with irregular obstacle and also a comparison theorem.

Miryana Grigorova, Peter Imkeller, Youssef Ouknine and Marie-Claire Quenez

Electronic Journal of Probability

Volume 23, paper no. 122, 38 pp., 2018

We formulate a notion of doubly reflected BSDE in the case where the barriers ξ and ζ do not satisfy any regularity assumption and with general filtration. Under a technical assumption (a Mokobodzki-type condition), we show existence and uniqueness of the solution. In the case where ξ is right upper-semicontinuous and ζ is right lower-semicontinuous, the solution is characterized in terms of the value of a corresponding Ef-Dynkin game, i.e. a game problem over stopping times with (non-linear) f-expectation, where f is the driver of the doubly reflected BSDE. In the general case where the barriers do not satisfy any regularity assumptions, the solution of the doubly reflected BSDE is related to the value of “an extension” of the previous non-linear game problem over a larger set of “stopping strategies” than the set of stopping times. This characterization is then used to establish a comparison result and a priori estimates with universal constants.

Roxana Dumitrescu, Miryana Grigorova, Marie-Claire Quenez, Agnès Sulem

In: Celledoni E., Di Nunno G., Ebrahimi-Fard K., Munthe-Kaas H. (eds). Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer.

We study (nonlinear) Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale attached to a default jump with intensity process λ. The driver of the BSDEs can be of a generalized form involving a singular optional finite variation process. In particular, we provide a comparison theorem and a strict comparison theorem. In the special case of a generalized λ-linear driver, we show an explicit representation of the solution, involving conditional expectation and an adjoint exponential semimartingale; for this representation, we distinguish the case where the singular component of the driver is predictable and the case where it is only optional. We apply our results to the problem of (nonlinear) pricing of European contingent claims in an imperfect market with default. We also study the case of claims generating intermediate cashflows, in particular at the default time, which are modeled by a singular optional process. We give an illustrating example when the seller of the European option is a large investor whose portfolio strategy can influence the probability of default.

Miryana Grigorova, Peter Imkeller, Elias Offen, Youssef Ouknine, Marie-Claire Quenez

Volume 27(5), 3153-3188, 2017

In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is established by using some results from optimal stopping theory, some tools from the general theory of processes such as Mertens’ decomposition of optional strong supermartingales, as well as an appropriate generalization of Itô’s formula due to Gal’chouk and Lenglart. In the second part of the paper, we provide some links between the RBSDE studied in the first part and an optimal stopping problem in which the risk of a financial position ξ is assessed by an f-conditional expectation Ef(⋅) (where f is a Lipschitz driver). We characterize the “value function” of the problem in terms of the solution to our RBSDE. Under an additional assumption of left upper-semicontinuity along stopping times on ξ, we show the existence of an optimal stopping time. We also provide a generalization of Mertens’ decomposition to the case of strong Ef-supermartingales.

### Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs

Miryana Grigorova and Marie-Claire Quenez

Stochastics: An International Journal of Probability and Stochastic Processes

Volume 89, 2017

We first study an optimal stopping problem in which a player (an agent) uses a discrete stopping time in order to stop optimally a payoff process whose risk is evaluated by a (non-linear) g-expectation. We then consider a non-zero-sum game on discrete stopping times with two agents who aim at minimizing their respective risks. The payoffs of the agents are assessed by g-expectations (with possibly different drivers for the different players). By using the results of the first part, combined with some ideas of S. Hamad{è}ne and J. Zhang, we construct a Nash equilibrium point of this game by a recursive procedure. Our results are obtained in the case of a standard Lipschitz driver g without any additional assumption on the driver besides that ensuring the monotonicity of the corresponding g-expectation.

Miryana Grigorova

Statistics & Risk Modeling, 31(3-4), pp. 259-295, 2014

In our previous work, we have extended the classical notion of increasing convex stochastic dominance relation with respect to a probability to the more general case of a normalized monotone (but not necessarily additive) set function, also called a capacity. In the present paper, we pursue that work by studying the set of monetary risk measures (defined on the space of bounded real-valued measurable functions) satisfying the properties of comonotonic additivity and consistency with respect to the generalized stochastic dominance relation. Under suitable assumptions on the underlying capacity space, we characterize that class of risk measures in terms of Choquet integrals with respect to a distorted capacity whose distortion function is concave. Kusuoka-type characterizations are also established. A generalization to the case of a capacity of the Tail Value at Risk is provided as an example. It is also shown that some well-known results about Choquet integrals with respect to a distorted probability do not necessarily hold true in the more general case of a distorted capacity.

Miryana Grigorova

Statistics & Risk Modeling, 31(2), pp. 183-213, 2014

By analogy with the classical case of a probability measure, we extend the notion of increasing convex (concave) stochastic dominance relation to the case of a normalized monotone (but not necessarily additive) set function also called a capacity. We give different characterizations of this relation establishing a link to the notions of distribution function and quantile function with respect to the given capacity. The Choquet integral is extensively used as a tool. In the second part of the paper, we give an application to a financial optimization problem whose constraints are expressed by means of the increasing convex stochastic dominance relation with respect to a capacity. The problem is solved by using, among other tools, a result established in our previous work, namely a new version of the classical upper (resp. lower) Hardy–Littlewood's inequality generalized to the case of a continuous from below concave (resp. convex) capacity. The value function of the optimization problem is interpreted in terms of risk measures (or premium principles).

Hardy–Littlewoodʼs inequalities, well known in the case of a probability measure, are extended to the case of a monotone (but not necessarily additive) set function, called a capacity. The upper inequality is established in the case of a capacity assumed to be continuous and submodular, the lower — under assumptions of continuity and supermodularity.

### Optimal stopping: Bermudan strategies meet non-linear evaluations

###

Miryana Grigorova, Marie-Claire Quenez, and Peng Yuan

, and

Electronic Journal of Probability

Volume 29: 1-29, 2024

We address an optimal stopping problem over the set of Bermudan-type strategies Θ (which we understand in a more general sense than the stopping strategies for Bermudan options in finance) and with non-linear operators (non-linear evaluations) assessing the rewards, under general assumptions on the non-linear operators ρ. We provide a characterization of the value family V in terms of what we call the (Θ,ρ)-Snell envelope of the pay-off family. We establish a Dynamic Programming Principle. We provide an optimality criterion in terms of a (Θ,ρ)-martingale property of V on a stochastic interval. We investigate the (Θ,ρ)-martingale structure and we show that the “first time” when the value family coincides with the pay-off family is optimal. The reasoning simplifies in the case where there is a finite number n of pre-described stopping times, where n does not depend on the scenario ω. We provide examples of non-linear operators entering our framework..

## Preprints

Miryana Grigorova, Marie-Claire Quenez, Peng Yuan

## Presentations

### 28 June-30June 2023--INFORMS Applied Probability Society Conference, Nancy, speaker at the invited session on Risk Measures in finance and insurance.

### 26 January 2023--Joint Risk & Stochastics and Financial Mathematics Seminar, London School of Economics, London, invited talk.

### 16 January- 21 January 2023--15th Bachelier Colloquium in Mathematical Finance and Stochastic Calculus, Métabief, invited talk.

### October 31-November 11, 2022--Workshop Mathematics of Risk 2022, MATRIX Research Institute for the Mathematical Sciences, Creswick, invited speaker at the mini-symposium Recent Advances in Stochastic Modelling in Finance and Economics.

### June 28, 2022--9th Colloquium BSDEs and Mean Field Systems

Session on BSDEs in Credit and Default Risk, Annecy, invited session organizer and speaker.

### June 8, 2022--International Conference on Computational Finance 2022, Minisymposium on BSDEs: Applications and Numerical Methods, Wuppertal, invited talk.

### March 24, 2022--Workshop on Stochastic Analysis: Theory, Applications and Numerics,

Schloss Rauischholzhausen, University of Giessen, invited talk.

### January 11-12, 2022--London-Oxford-Warwick Workshop on Financial Mathematics, London, invited talk.

### November 10, 2021--Stochastics and Finance Seminar, the University of Sydney, invited talk.

### November 3, 2021 -- Economics Department Seminar Series, University of Verona, invited talk.

### September 30, 2021 -- Joint La Trobe Statistics and Stochastics & Probability Victoria Seminar, invited talk.

### May 21, 2021 -- Warwick Stochastic Finance Seminar, University of Warwick, invited talk.

### April 19-24, 2021 -- Spring Research School in Mathematical Finance, Sochi, lecturer of a mini-course on Non-linear Optimal Stopping and (Reflected) BSDEs for Finance.

### April 16, 2021 -- Seminar Questrom School of Business, Boston University, invited talk.

### April 8, 2021 -- Online Research Seminar Series, AIMS Ghana, invited talk.

### April 7, 2021 -- Seminar in Financial Mathematics, ORFE, Princeton University, invited talk.

### March 27, 2021 -- Global Seminar in Mathematical Finance, Moscow, invited talk.

### February 17, 2021 -- African Mathematics Seminar, virtually hosted by University of Nairobi, invited talk.

### January 13-18, 2020 -- 14th Bachelier Colloquium in Mathematical Finance and Stochastic Calculus, Métabief, invited talk.

### December 20, 2019 -- Seminar Bachelier, Paris, France, invited talk.

### December 19, 2019 -- Seminar Probability and Financial Mathematics, Université d'Evry-Paris Saclay, invited talk.

### October 21-25, 2019 -- Conference Advances in Stochastic Analysis for Handling Risks in Finance and Insurance, CIRM-Luminy, France, invited talk.

### July 15-19, 2019 -- Conference "Equilibria in Markets, Strategic Interactions, and Complex Systems", Bielefeld, invited talk.

### June 3-8, 2019 -- SIAM Conference on Financial Mathematics and Engineering, Toronto, invited talk at a mini-symposium.

### January 7-11 2019 -- 13th Bachelier Colloquium in Mathematical Finance and Stochastic Calculus, Métabief, France, invited talk.

### September 10-12, 2018 --Conference on BSDEs, Information and McKean-Vlasov equations, University of Leeds, Leeds, invited talk.

### September 3-7, 2018 --Conference "Innovative Research in Mathematical Finance" in honour of Yuri Kabanov's 70th birthday, CIRM Luminy, France, invited talk.

### July 16-20, 2018 -- 10th World Congress of the Bachelier Finance Society, Dublin.

### June 18-22, 2018 -- 1st Research School in Financial Mathematics in Nigeria, Ibadan, lecturer of a mini-course on "Optimal Stopping with Financial Applications".

### June 4-8, 2018 -- 3rd International Conference on Stochastic Methods, Divnomorskoye, Russia, plenary talk.

### April 23-27, 2018 -- Fourth Young Researchers Meeting on BSDEs, Nonlinear Expectations and Mathematical Finance, Shanghai.

### March 26 -28, 2018-- Bielefeld-Edinburgh-Swansea Stochastic Spring, held in Bielefeld, invited talk.

### February 27 - March 2, 2018 --13th German Probability and Statistics Days, Freiburg.

### February 20-21, 2018 -- Collaborative Research Centre 1283 Internal Workshop, Bad Salzuflen, invited talk.

### February 8, 2018 -- Seminar on Financial and Mathematics and Numerical Probability of the Universities Paris 6-Paris 7, Paris, invited talk.

### January 15-20, 2018 -- 12th Bachelier Colloquium in Mathematical Finance and Stochastic Calculus, Métabief, invited talk.

### January 10, 2018 -- Bielefeld Stochastic Afternoon, University of Bielefeld, invited talk.

### July 24-28, 2017 -- 39th Conference on Stochastic Processes and their Applications (SPA), Moscow.

### July 3-7, 2017 -- Special Session on "Backward Stochastic Differential Equations and the General Theory of Processes", session organizer and speaker, Workshop on BSDEs, SPDEs and their Applications, Edinburgh.

### January 16-21, 2017 -- Eleventh Bachelier Colloquium on Mathematical Finance and Stochastic Calculus, Métabief.

### October 25, 2016 -- Conference Stochastic analysis of dynamical systems, stochastic control and games, University of Leeds.

### October 18, 2016 -- Oberseminar Stochastik, Leibniz University-Hannover, invited talk.

### September 1, 2016 -- 9th European Summer School in Financial Mathematics, Pushkin, St. Petersburg.

### April 13, 2016 -- Brown Bag Seminar of the Institute for Statistics and Mathematics, Vienna University of Economics and Business, Vienna, invited talk.

### February 1-2, 2016 -- the Actuarial and Financial Mathematics Conference, held at the Royal Flemish Academy of Belgium for Science and the Arts, Brussels.

### October 22-23, 2015 -- Workshop "Junior Female Researchers in Probability", held at the Weierstrass Institute and the Technical University in Berlin.

### June 8, 2015 -- Seminar on mathematical finance of The Hebrew University of Jerusalem, invited talk.

### May 22, 2015 -- Bachelier Seminar, Paris, invited talk.

### April 20-24, 2015 -- Second conference on Stochastics of Environmental and Financial Economics held at The Norwegian Academy of Sciences and Letters, Oslo.

### September 1-5, 2014 -- 7th European Summer School in Mathematical Finance, Oxford.

### June 24, 2014 -- "Arbeitsgruppenseminar" of Prof. P. Imkeller, Humboldt University-Berlin, invited talk.

### April 3, 2014 -- Seminar on probability and statistics of the Université du Maine, invited talk.

### November 25, 2013 -- Seminar on probability of the Mathematics Research Institute of Rennes (IRMAR), University Rennes 1, invited talk.

### November 19, 2013 -- Seminar on probability theory of the University of Vienna, invited talk.

### May 30, 2013 -- 30th International Conference of the French Finance Association, Lyon.

### April 18, 2013 -- Seminar of the working group "Financial mathematics and probability", University Evry-Val d'Essonne, invited talk.

### March 25, 2011 -- Seminar "Stochastic methods and finance", University Paris-Est Marne-La-Vallée, invited talk.

## Teaching

2017

Leibniz University-Hannover

### Introduction to Risk Measures

Master level

2019, 2020, 2021, and 2022

University of Leeds

### Continuous-time finance

Master level

2020, 2021, and 2022

University of Leeds

### Risk Management

Master level

2023

University of Warwick

### Advanced Trading Strategies

Topic 3

Master level

Nominations for the Partnership Award for Inspirational Teaching in 2019, 2020, and 2021 (University of Leeds)

Nomination for the Partnership Award for Supervision in 2019 and 2021 (University of Leeds).

## Students' Testimonials

### CIRM Luminy 2022

Advances in Stochastic Control and Optimal Stopping with Applications in Economics and Finance

## PhD Opportunities

Pricing and hedging in non-linear financial market models

Contemporary mathematical finance, especially in the aftermath of the financial crisis of 2007-2008, faces important challenges which include better taking into account of various market imperfections, thorough understanding of default risk, studying complex strategic interactions between agents, to name a few. Imperfections in the markets might come, for example, from taxes on profits from risky investments, or from the trading impact of a large investor on the market prices and/or on the default probability. Such imperfections typically lead to non-linearities in the pricing rules, and hence to non-linear financial market models. Addressing the above challenges calls for new developments in stochastic analysis and stochastic control.

The aim of the project is to advance our understanding of optimal stopping problems and stopping games

with non-linear expectations, of (reflected) backward stochastic differential equations and of optimal switching problems. The project involves theoretical research leading to applications to pricing and hedging of financial derivatives (American options, game options, Bermudan options, ...) and to risk management in financial markets with imperfections.

Risk measurement and risk management in finance and actuarial science

Risk measurement and risk management have become increasingly important in finance and actuarial science, both from a practical and theoretical point of view, especially since the financial crisis in 2007-2008. The importance of this area is further emphasized by the current Covid-19 induced economic and financial crisis.

This PhD project aims at addressing important challenges in risk measures (design and applications), in systemic risk, and in optimal risk allocation, by combining a variety of mathematical tools from probability theory, capacity theory and non-linear integration, functional analysis and convex analysis, multivariate statistics and copula theory.

Connections to decision theory and welfare economics will also be investigated and the findings of the project will

contribute to the mathematical modelling, treatment, and understanding of problems in these fields.

The successful candidates will have a strong scientific background in probability theory and stochastic processes and/or financial mathematics, as evidenced by very good Bachelor and Master degrees in Mathematics/Applied Mathematics or a closely related area and excellent recommendation letters. Please to first get in touch with the supervisor via e-mail and then submit your application to the Centre for Doctoral Training (CDT) in Probability and Statistics of the University of Warwick.

miryana_grigorova 'at' yahoo.fr

## Contact

Department of Statistics, University of Warwick, Coventry, UK

miryana.grigorova 'at' warwick.ac.uk