I will present a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. It turns out that these transformations are very useful in Euler schemes for killed diffusions, simplifying the solutions of optimal stopping problems with discounting, and characterising the stochastic solutions of Cauchy problems defined by the generators of strict local martingales, which are well-known for not having unique solutions. I will give a description of these transformations and discuss their connections with h-transforms and Schroedinger semigroups, and how one can use them to solve the above problems.

## Recurrent and transient transformations for one-dimensional di

## Approximate Local Volatility Model for Vanilla Rates Options

In this presentation we analyse a model for the pricing of vanilla interest rate options (e.g caps/floors and European swaptions). Within that model we specify a parametric form of the terminal distribution of the underlying rate. The driver of the distribution is a Brownian motion and the parametric form is closely linked to local volatility models. We choose the local volatility function such that the model allows analytic pricing of vanilla and CMS options.

The parametrisation in terms of a local volatility function provides transparent intuition of the model parameters as well as high flexibility for smile calibration. Moreover, the linkage to an underlying Brownian motion may be used as a normalising basis for interpolation between expiries and swap terms.

## Lifting the Heston model

How to reconcile the classical Heston model with its rough counterpart? We introduce a lifted version of the Heston model with n multifactors sharing the same Brownian motion but mean reverting at different speeds. Our model nests as extreme cases the classical Heston model (when n=1) and the rough Heston model (when n goes to infinity). We show that the lifted model enjoys the best of both worlds: Markovianity and satisfactory fits of implied volatility smiles for short maturities. Further, our approach speeds up the calibration time and opens the door to time-efficient simulation schemes.

## The Deutsche Bank Risk Center

**Bernadette Walter:**

The Quant Institute at Deutsche Bank Risk Center: A center for development and validation of models for Risk Management across major risk types

**TBD:**

Model risk in the model landscape for the trading book

We give an overview of the challenges banks face in identifying and measuring model risk in the complex model landscape required to run a trading operation. We explain the changes in the assessment and regulation of model risk since the Global Financial Crisis, and the consequences this has had on modelling and model governance in banks. We will use the example of pricing models to illustrate this and to articulate the problems that remain to be solved.

**Michael Kalkbrener:**

Credit rating migration processes based on economic-state-dependent transition matrices

We develop a model for rating migration. The objective is to study rating migration processes and corresponding default rates. It is generally accepted that rating migrations depend on economic factors. We use the theory of time-homogeneous Markov chains to jointly model the rating process and the state of the economy. Although the rating process itself is neither Markovian nor time-homogeneous in general, we show that sequence of the rating process’ transition matrices converges to a limit. We further analyse the properties of different rating methods, namely point-in-time (PIT) ratings and through-the-cycle (TTC) ratings. Although these rating philosophies have become important from a regulatory perspective, to the best of our knowledge, no formal definition exists yet. We further discuss if and how a rating philosophy can be detected from given rating transition time series.

## Dynamics of a Well-Diversiﬁed Equity Index and Martingale Inference

The paper derives an endogenous model for the long-term dynamics of a well-diversiﬁed equity index with rough volatility, the S&P500. It assumes that the index is a proxy of the respective growth optimal portfolio, the variance of its increments evolves in some market time proportionally to the index value and the derivative of market time is a linear function of the squared derivative of a smoothed proxy of the single driving Brownian motion. The resulting model is highly tractable, allows almost exact simulation and leads beyond classical ﬁnance theory. Its parameters are estimated via a novel martingale inference method, which employs higher-strong order, implicit approximations of the increments of the system of stochastic diﬀerential equations.

## CANCELLED

## Space-time localisation for the dynamic |Phi^4_3 model

In this talk I will present a new method to derive a priori estimates for singular SPDE concentrating on the dynamic Φ^{4}_{3 }equation.

Recently several methods to show non-explosion for this equation in the framework of paracontrolled distributions were put forward by several groups. Here I will show how to prove bounds in the framework of regularity structures. The main result is a space-time version of the coming down from inﬁnity property, i.e. a bound on solutions of the equation on a compact set in space-time which only depends on the stochastic data on a slightly larger set, but is uniform over all possible choices of space-time boundary data.

Our method signiﬁcantly simpliﬁes previous proofs and extensions to (much) more singular equations seem within reach.

This is joint work with Augustin Moinat.

## Large urn model with local mean-field interactions

We study a large urn model connected by an underlying symmetrical graph. After some exponentially distributed amount of time all the balls of one of the urns are redistributed among the connected urns. The allocation of balls is done at random according to a set of weights which depend on the state of the system. The degree of the graph, which is the range of interaction, is assumed large, but is not necessarily linear with respect to the number of urns. Moreover, the number of simultaneous jumps of the process is not bounded due to the redistribution of all balls of an urn at the same time. We describe the dynamic by using the local empirical distributions associated to the state of urns in the neighborhood of a given urn. Under some conditions, we are able to establish a mean-field convergence result for this measure-valued system. Convergence results of the corresponding invariant distributions are obtained for several classes of allocation policies. For the class of power of choices policies, we show that the invariant measure has an asymptotic finite support property when the average load per urn gets large. This result differs somewhat from the classical double exponential decay property usually encountered in the literature for power of choices policies. This finite support property has interesting consequences in practice. (This is a joint work with Philippe Robert.)

## Computation of optimal transport and related hedging problems via penalization and neural networks

We present a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it to a nite dimensional one which corresponds to optimizing a neural network with smooth objective function. We present numerical examples from optimal transport, and bounds on the distribution of a sum of dependent random variables. As an application we focus on the problem of risk aggregation under model uncertainty. The talk is based on joint work with Stephan Eckstein and Mathias Pohl.

## Optimal order placement and limit order book modelling

Optimal order placement is a key aspect of market making, and more generally, of liquidity providing strategies in electronic markets. With this motivation in mind, we study the optimal placement of limit orders from theoretical and numerical points of view, in the context of Markovian limit order book models. The theoretically optimal strategies are then backtested using real data, providing results that advocate for the design of better order book models. Some extensions are made, based either on Hawkes processes, or on processes with finite memory (joint works with C. Huré, X. Lu, H. Pham).