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The following seminars have been jointly organised by the Risk and Stochastics Group and the Department of Mathematics. The series aims to promote communication and discussion of research in the mathematics of insurance and finance and their interface, to encourage interaction between practice and theory in these areas, and to support academic students in related programmes at postgraduate level. All are welcome to attend. Please contact Emily Jackson, the Research Manager, on E.Jackson2@lse.ac.uk for further information about any of these seminars.

The Seminar takes place on **alternate Thursdays from 12.00 - 13:00**, unless stated below. N.B. room information is not included below for campus security reasons. If you would like to attend a seminar please contact Emily Jackson, Research Manager, on E.Jackson2@lse.ac.uk, in good time so this can be accommodated.

**Thursday 16 May - Sergio Pulido (École Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise)**

**Polynomial Volterra processes**

Recent studies have extended the theory of affine processes to the stochastic Volterra equations framework. In this talk, I will describe how the theory of polynomial processes extends to the Volterra setting. In particular, I will explain the moment formula and an interesting stochastic invariance result in this context. Potential applications to fractional volatility models will be discussed. This is joint work with Eduardo Abi Jaber, Christa Cuchiero, Luca Pelizzari and Sara Svaluto-Ferro.

**Previous seminars in the series:**

**Thursday 9 May - Johanne Wiesel (Carnegie Mellon University)**

**Empirical martingale projections via the adapted Wasserstein distance**

Given a collection of multidimensional pairs {(Xi,Yi):1≤i≤n}, we study the problem of projecting the associated suitably smoothed empirical measure onto the space of martingale couplings (i.e. distributions satisfying [Y|X]=X) using the adapted Wasserstein distance. We call the resulting distance the smoothed empirical martingale projection distance (SE-MPD), for which we obtain an explicit characterization. We also show that the space of martingale couplings remains invariant under the smoothing operation. We study the asymptotic limit of the SE-MPD, which converges at a parametric rate as the sample size increases if the pairs are either i.i.d. or satisfy appropriate mixing assumptions. Additional finite-sample results are also investigated. Using these results, we introduce a novel consistent martingale coupling hypothesis test, which we apply to test the existence of arbitrage opportunities in recently introduced neural network-based generative models for asset pricing calibration.

This talk is based on joint work with Jose Blanchet, Erica Zhang and Zhenyuan Zhang.

**Thursday 2 May 2024 - Carlos Oliveira (Norwegian University of Science and Technology)**

**Managing the Occurrence of Adverse Events by Investing or Exiting**

We consider a firm that may face sudden decreases in its revenue. Its revenue is modeled by a geometric Brownian motion, and the occurrence of adverse events is modeled by a Poisson process. The firm has two options: either to leave the market immediately or to invest in risk mitigation measures to reduce the negative impact on the revenue. After the investment occurs, the firm can still leave the market. The firm’s option value is modeled as an optimal stopping problem. We prove that the optimal strategy is characterized by a disconnected stopping region.

**Thursday 21 March 2024 - Gilles Pagès (Sorbonne Université) **

**Functional convex ordering and convexity propagation for stochastic processes: a constructive (and simulable) approach**

After a few reminders on the convex ordering between two random vectors U and V defined by

U <=_cv V if E f(U) <= E f(V) for every convex function f: IR^d --> IR

(and some variants like monotonic convex order) and their first applications in finance (from option pricing to expected shortfall), we will explain how to extend this order to path-dependent functional of stochastic processes. In particular to diffusions (Brownian but also with jumps, McKean Vlasov type). We will also consider non-Markovian processes like the solutions of Volterra equations with singular kernels involved in rough volatility modeling in Finance. We systematically establish our comparison results by an simulable time discretization procedure of Euler scheme type. Thus, this approach makes it possible in Finance to ensure that the prices of derivative products computed by simulation cannot give rise to convexity arbitrages. On our way, we will also establish as the convexity of functions x —> E F(X^x) of such stochastic processes X^x when F is convex and x is the starting value of X^x with obvious applications. (Includes joint works with B. Jourdain and Y. Liu).

**Thursday 7 March - Tiziano de Angelis (University of Turin) *17.00-18.00***

**A quickest detection problem with false negatives**

We formulate and solve a quickest detection problem with false negatives. A standard Brownian motion acquires a drift at an independent exponential random time which is not directly observable. Based on the observation in continuous time of the sample path of the process, an optimiser must detect the drift as quickly as possible after it has appeared. The optimiser can inspect the system multiple times upon payment of a fixed cost per inspection. If a test is performed on the system {\em before} the drift has appeared then, naturally, the test will return a negative outcome. However, if a test is performed {\em after} the drift has appeared, then the test may fail to detect it and return a false negative with probability $\epsilon\in(0,1)$. The optimisation ends when the drift is eventually detected. The problem is formulated mathematically as an optimal multiple stopping problem and it is shown to be equivalent to a recursive optimal stopping problem. Exploiting such connection and free boundary methods we find explicit formulae for the expected cost and the optimal strategy. We also show that when $\epsilon = 0$ our expected cost coincides with the one in Shiryaev's classical optimal detection problem. The talk is based on joint work [1] with Quan Zhou and Jhanvi Garg (Texas A&M University).

**Thursday 22 February 2024 - Christoph Kühn (Goethe University Frankfurt)**

**The fundamental theorem of asset pricing with and without transaction costs**

We present a version of the fundamental theorem of asset pricing (FTAP) in continuous time that is based on the strict no-arbitrage condition and that is applicable to both frictionless markets and markets with proportional transaction costs. We consider a market with a single risky asset whose ask price process is higher than or equal to its bid price process. Neither the concatenation property of the set of wealth processes, that is used in the proof of the frictionless FTAP, nor some boundedness property of the trading volume of admissible strategies usually argued with in models with a nonvanishing bid-ask spread need to be satisfied in our model.

**Wednesday 7 February 2024 - Eduardo Abi Jaber (École Polytechnique, Paris)**

**Signature volatility models: fast pricing and hedging with Fourier**

We consider a stochastic volatility model where the dynamics of the volatility are given by a possibly infinite linear combination of the elements of the time extended signature of a Brownian motion. First, we show that the model is remarkably universal, as it includes, but is not limited to, the celebrated Stein-Stein, Bergomi, GARCH and Heston models, together with certain path-dependent variants. Second, we derive the joint characteristic functional of the log-price and integrated variance provided that some infinite-dimensional tensor-valued Riccati equation admits a solution. This allows us to price and (quadratically) hedge certain European and path-dependent options using Fourier inversion techniques. We highlight the efficienty and accuracy of these Fourier techniques in a comprehensive numerical study. This is joint work with Louis-Amand Gérard.

**Friday 2 February - Alexandru Hening (Texas A&M University), 10.45 - 11.45am**

**Stochastic optimal control and harvesting of stochastic populations**

We consider the harvesting of a population in a stochastic environment whose dynamics in the absence of harvesting is described by a diffusion. Using ergodic optimal control, we find the optimal harvesting strategy which maximizes the asymptotic yield of harvested individuals. When the yield function is the identity, we show that the optimal strategy has a bang-bang property: there exists a threshold L>0 such that whenever the population is under the threshold the harvesting rate must be zero, whereas when the population is above the threshold the harvesting rate must be at the upper limit. We show that, if the yield function is smooth enough and strictly concave, then the optimal harvesting strategy is continuous, whereas when the yield function is convex the optimal strategy is of bang-bang type. This shows that one cannot always expect bang-bang type optimal controls. In addition, singular and multi-dimensional harvesting problems will be explored.

**Thursday 1 February 2024 - Ruixun Zhang (Peking University)**

**On Consistency of Signatures Using Lasso**

Signature transforms are iterated path integrals of continuous and discrete-time time series data, and their universal nonlinearity linearizes the problem of feature selection. This paper revisits some statistical properties of signature transform under stochastic integrals with a Lasso regression framework, both theoretically and numerically. Our study shows that, for processes and time series that are closer to Brownian motion or random walk with weaker inter-dimensional correlations, the Lasso regression is more consistent for their signatures defined by It\^o integrals; for mean reverting processes and time series, their signatures defined by Stratonovich integrals have more consistency in the Lasso regression. Our findings highlight the importance of choosing appropriate definitions of signatures and stochastic models in statistical inference and machine learning. This is joint work with Xin Guo and Chaoyi Zhao.

**Thursday 7 December 2023 - Andrew Allan (Durham University)**

**Efficient Itô rough paths: From stochastic portfolio theory to the Euler scheme for SDEs**

It's well-known that rough path theory provides a pathwise counterpart to stochastic calculus, which yields a robust solution theory for SDEs, and comes with strong pathwise continuity results. On the other hand, the financial interpretation of a rough integral is typically unclear. In this talk we'll consider a class of Itô rough paths for which the theories of Itô, Föllmer and rough integration all marry up efficiently, with all the power of rough path theory, and with the financial interpretation of the Itô integral intact. Based on this framework, we'll then look at applications to model-free stochastic portfolio theory, and to establishing pathwise convergence of the Euler scheme for rough and stochastic differential equations. This talk is based on joint works with Christa Cuchiero, Anna Kwossek, Chong Liu and David Prömel.

**Thursday 23 November - Xiling Zhang (University of Leicester)**

**Limit points of external DLA models in the plane **

We will explore particle systems closely related to recent work on systemic risk, wherein a default boundary grows as firms reach the boundary. More generally, Diffusion-Limited Aggregations (DLA) are random growth models on the lattice, where a set grows at the boundary sites when triggered by the absorption of a sequence of lattice-valued, diffusion-limited processes (particles); a typical example is the self-avoiding random walk. We are interested in the limiting behaviour of such sets as the grid size decreases, with the empirical law of the underlying particles converging to the Wiener measure. The main discovery of this work is that almost surely the planar Wiener process makes a loop around itself infinitely many times, and based on this geometric observation we show that the limit points of a certain class of external DLA models coincide with a Wiener process stopped upon hitting the limiting set. Furthermore I will talk about the connection between this problem and the super-cooled Stefan problem. In particular we will see that the latter in two dimensions cannot be approximated by external DLA models. This is a joint work with Sergey Nadtochiy and Mykhaylo Shkolnikov.

**Thursday 9 November - Luca Galimberti (King's College London)**

**Pricing Options on Flow Forwards by Neural Networks in Hilbert Spaces**

In commodity markets, options are typically written on forward and futures contracts. In some markets, like electricity and gas, as well as freight and weather markets on temperature and wind, the forwards deliver the underlying commodity or service over a contracted delivery period, and not at a specified delivery time in the future. Such forwards are sometimes referred to as flow forwards. There is a large literature on neural networks and financial derivatives, mostly focusing on approximating option prices, which, as it is well-known, can be recovered as solutions of PDEs: typically, these equations are high-dimensional, and the main argument for intro- ducing deep neural networks is to overcome the curse of dimensionality. In this talk, we bring this perspective to the “ultimate” high-dimensional case, considering deep neural networks approximating option prices on infinite-dimensional underlyings. Indeed, option prices on flow forwards are in general functions of functions, as the underlying will be a curve (i.e., the term structure) rather than a vector of points (i.e., prices of the underlying assets). We propose to approximate this non-linear option price functional by a neural network in Hilbert space, appealing to the general neural nets in Frechet spaces and their universal approximation properties studied in an earlier work. We test our methodology in some numerical case studies, and find that it works very well with high-dimensional noise which is in line with the general perception that numerical methods based on neural networks can often overcome the curse of dimensionality. This is a joint work with Fred Espen Benth (UiO) and Nils Detering (HHU).

**Thursday 26 October - Aleksey Kolokolov (University of Manchester)**

**Cryptocrashes**

This paper proposes a new nonparametric test for detecting short-lived locally explosive trends (drift bursts) in pure-jump processes. The new test is designed specifically to detect intraday flash crashes and gradual jumps in cryptocurrency prices recorded at a high frequency. Empirical analysis shows that drift bursts in bitcoin price occur on average at every second day. Their economic importance is highlighted by showing that hedge funds holding cryptocurrency in their portfolios are exposed to a risk factor associated with the intensity of bitcoin crashes. On average, hedge funds do not profit from intraday bitcoin crashes and do not hedge against the associated risk.

**Thursday 12 October 2023 - Rishabh Gvalani (Max Planck Institute)**

**An SPDE for Stochastic Gradient Descent**

We study an SPDE with conservative noise structure which arises naturally as a continuum description of the continuous-time analogue of the stochastic gradient descent (SGD) algorithm for a 2-layer neural network. We study well-posedness for this equation under a range of assumptions on the coefficients and initial data. We also provide sharp quantitative versions of a law of large numbers and central limit theorem for this SPDE in the joint limit of overparameterisation and vanishing learning rate. Combining our results with previous results in the literature, we show that the SPDE provides a higher-order approximation of the original discrete-time SGD algorithm than just the law of large numbers. This is joint work with Benjamin Gess (Bielefeld/MPI-MiS) and Vitalii Konarovskyi (Hamburg).

**2022/23, 2021/22, 2019/20, 2018/19, 2017/18, 2016/17, 2015/16 **

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