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## Über dieses Buch

This collection of selected, revised and extended contributions resulted from a Workshop on BSDEs, SPDEs and their Applications that took place in Edinburgh, Scotland, July 2017 and included the 8th World Symposium on BSDEs.

The volume addresses recent advances involving backward stochastic differential equations (BSDEs) and stochastic partial differential equations (SPDEs). These equations are of fundamental importance in modelling of biological, physical and economic systems, and underpin many problems in control of random systems, mathematical finance, stochastic filtering and data assimilation. The papers in this volume seek to understand these equations, and to use them to build our understanding in other areas of mathematics.

This volume will be of interest to those working at the forefront of modern probability theory, both established researchers and graduate students.

## Inhaltsverzeichnis

### On the Monotone Stability Approach to BSDEs with Jumps: Extensions, Concrete Criteria and Examples

Abstract
We show a concise extension of the monotone stability approach to backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure of jumps, which could be of infinite activity with a non-deterministic and time-inhomogeneous compensator. The BSDE generator function can be non-convex and needs not satisfy global Lipschitz conditions in the jump integrand. We contribute concrete sufficient criteria, that are easy to verify, for results on existence and uniqueness of bounded solutions to BSDEs with jumps, and on comparison and a-priori $$L^\infty$$-bounds. Several examples and counter examples are discussed to shed light on the scope and applicability of different assumptions, and we provide an overview of major applications in finance and optimal control.
Dirk Becherer, Martin Büttner, Klebert Kentia

### On the Wellposedness of Some McKean Models with Moderated or Singular Diffusion Coefficient

Abstract
We investigate the well-posedness problem related to two models of nonlinear McKean Stochastic Differential Equations with some local interaction in the diffusion term. First, we revisit the case of the McKean-Vlasov dynamics with moderate interaction, previously studied by Méléard and Jourdain in [16], under slightly weaker assumptions, by showing the existence and uniqueness of a weak solution using a Sobolev regularity framework instead of a Hölder one. Second, we study the construction of a Lagrangian Stochastic model endowed with a conditional McKean diffusion term in the velocity dynamics and a nondegenerate diffusion term in the position dynamics.
Mireille Bossy, Jean-François Jabir

### On the Uniqueness of Solutions to Quadratic BSDEs with Non-convex Generators

Abstract
In this paper we prove some uniqueness results for quadratic backward stochastic differential equations without any convexity assumptions on the generator. The bounded case is revisited while some new results are obtained in the unbounded case when the terminal condition and the generator depend on the path of a forward stochastic differential equation. Some of these results are based on strong estimates on Z that are interesting on their own and could be applied in other situations.

### An Example of Martingale Representation in Progressive Enlargement by an Accessible Random Time

Abstract
Given two martingales on the same probability space, both enjoying the predictable representation property with respect to their own filtrations, it can happens that their quadratic covariation process enters in the martingale representation of the filtration obtained as the union of the original ones. This fact on one hand influences the multiplicity of the enlarged filtration and on the other hand it is linked to the behavior of the sharp brackets of the martingales. Here we illustrate these arguments presenting an elementary example of martingale representation in the context of progressive enlargement by an accessible random time.
Antonella Calzolari, Barbara Torti

### European Option Pricing with Stochastic Volatility Models Under Parameter Uncertainty

Abstract
We consider stochastic volatility models under parameter uncertainty and investigate how model derived prices of European options are affected. We let the pricing parameters evolve dynamically in time within a specified region, and formalise the problem as a control problem where the control acts on the parameters to maximise/minimise the option value. Through a dual representation with backward stochastic differential equations, we obtain explicit equations for Heston’s model and investigate several numerical solutions thereof. In an empirical study, we apply our results to market data from the S&P 500 index where the model is estimated to historical asset prices. We find that the conservative model-prices cover 98% of the considered market-prices for a set of European call options.
Samuel N. Cohen, Martin Tegnér

### Construction of an Aggregate Consistent Utility, Without Pareto Optimality. Application to Long-Term Yield Curve Modeling

Abstract
The aim of this paper 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.
Nicole El Karoui, Caroline Hillairet, Mohamed Mrad

### BSDEs and Enlargement of Filtration

Abstract
In this paper we study the solution of a BSDE in a large filtration, and we show that the projection (on a smaller filtration) of its semimartingale part has coefficients that can be explicitely given in terms of the coefficients in the large filtration.
Monique Jeanblanc, Dongli Wu

### An Unbiased Itô Type Stochastic Representation for Transport PDEs: A Toy Example

Abstract
We propose a stochastic representation for a simple class of transport PDEs based on Itô representations. We detail an algorithm using an estimator stemming for the representation that, unlike regularization by noise estimators, is unbiased. We rely on recent developments on branching diffusions, regime switching processes and their representations of PDEs. There is a loose relation between our technique and regularization by noise, but contrary to the latter, we add a perturbation and immediately its correction. The method is only possible through a judicious choice of the diffusion coefficient $$\sigma$$. A key feature is that our approach does not rely on the smallness of $$\sigma$$, in fact, our $$\sigma$$ is strictly bounded from below which is in stark contrast with standard perturbation techniques. This is critical for extending this method to non-toy PDEs which have nonlinear terms in the first derivative where the usual perturbation technique breaks down. The examples presented show the algorithm outperforming alternative approaches. Moreover, the examples point toward a potential algorithm for the fully nonlinear case where the method of characteristics break down.
Gonçalo dos Reis, Greig Smith

### Path-Dependent SDEs in Hilbert Spaces

Abstract
We study path-dependent SDEs in Hilbert spaces. By using methods based on contractions in Banach spaces, we prove the Gâteaux differentiability of generic order n of mild solutions with respect to the starting point and the continuity of the Gâteaux derivatives with respect to all the data.
Mauro Rosestolato
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