Pricing contingent convertibles. An Intensity approach.
José Manuel Corcuera
University of Barcelona
Abstract
Contingent Convertible Bonds, or CoCos, are contingent capital instruments which are converted into shares, or may suffer a principal write-down, if certain trigger event occurs. They are a kind of credit risk contracts, and consequently we can use the different approaches in Credit Risk theory to price them. After an introduction on the Reduce Form Approach we shall concentrate our discussion in the Intensity Approach to model the trigger event. As we shall see, in this context, by using the abstract Bayes rule with a non-equivalent change of probability we can price contingent convertibles in an easier way. In particular when the underlying stock jumps down at the conversion time, giving an alternative way of treating the dilution problem.
Progressive Enlargement of Filtrations and Control Problems for Step Processes
Paolo Di Tella
TU Dresden
Abstract
In the present paper we address stochastic optimal control problems for a step process \( (X,\mathbb{F}) \) under a progressive enlargement of the filtration. The global information is obtained adding to the reference filtration \( \mathbb{F} \) the point process \( H=1_{[\tau,+\infty)} \). Here \( \tau \) is a random time that can be regarded as the occurrence time of an external shock event. We study two classes of control problems, over \( [0,T] \) and over the random horizon \( [0,T \wedge \tau] \). We solve these control problems following a dynamical approach based on a class of BSDEs driven by the jump measure \( \mu^ Z \) of the semimartingale \( Z=(X,H) \), which is a step process with respect to the enlarged filtration \( \mathbb G \). The BSDEs that we consider can be solved in \( \mathbb{G} \) thanks to a martingale representation theorem which we also establish here. To solve the BSDEs and the control problems we need to ensure that \( Z \) is quasi-left continuous in the enlarged filtration \( \mathbb{G}\). Therefore, in addition to the \( \mathbb{F} \)-quasi left continuity of \( X \), we assume some further conditions on \( \tau \).
This is a joint work with Elena Bandini (Uni Bologna) and Fulvia Confortola (PoliMi).
Interference of time-change Lévy noises on characterisation of reflected backward stochastic differential equations
Jasmina Đorđević
University of Oslo
Abstract
We analyze reflected backward stochastic differential equations with a left/lower barrier (RBSDEs with lower barrier) and time-changed Lévy noise, when time-change and Lévy process are independent. Existence and uniqueness of the solution of RBSDE with left barrier under natural filtration \(\mathbb{F}\) and under enlarged filtration \(\mathbb{G}\) (which contains additional informations with respect to time-change process) are proved. Further, comparison principle is obtained. Possible applications in stochastic control as well as in risk theory are marked.
This is joint work with Prof. Giulia Di Nunno.
Control and stopping mean-field games: the linear programming approach
Roxana Dumitrescu
King's College London
Abstract
In this talk, we present recent results on the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via other approaches used in the previous literature. We then present a fictitious play algorithm to approximate the mean-field game population dynamics in the context of the linear programming approach. Finally, we give an application of the theoretical and numerical contributions introduced in the first part of the talk to an entry-exit game in electricity markets.
The talk is based on several works, joint with R. Aïd, G. Bouveret, M. Leutscher and P. Tankov.
Hypothetical Treatment Accelerations
Haris Fawad
University of Oslo
Abstract
What would have happened if we managed to reduce the waiting times for kidney transplants? We use stochastic time changes to model hypothetical scenarios in which patients receive treatment faster than their observed treatment trajectory. This time change leads to a different probability measure, which we can use to formulate interesting parameters such as the marginalised survival function. We demonstrate this model using observational data from a cohort of patients with end-stage kidney disease, who were on a waiting list for kidney transplants.
Higher-order adaptive methods for exit times of Itô diffusions
Håkon Andreas Hoel
University of Oslo
We present an higher-order method for strong approximations of exit times of diffusion processes in the form of Itô stochastic differential equations (SDE). The method employs and Itô-Taylor scheme for simulating SDE paths and carefully decreases the step-size in the numerical integration as the solution approaches the boundary of the domain. These techniques turn out to complement each other nicely: adaptive time-stepping improves the accuracy of the exit time by reducing the magnitude of the overshoot of the numerical solution when it exits the domain, and higher-order scheme improve the approximation of the state of the diffusion process. We will describe theoretical results on convergence rates and computational cost, and discuss how the method can be combined with multilevel Monte Carlo to produce an efficient method for estimating mean exit times.
Non-linear Young equations in the plane and pathwise regularization by noise
Fabian Harang
BI Norwegian Business School
Abstract
Regularization by noise for stochastic differential equations has been a long studied topic in the field of stochastic analysis. After the work of Gubinelli and Catellier in 2016 on a pathwise analogue to the probabilistic analysis of regularization by noise based on what they called averaged fields in combination with the concept of non-linear Young equations, this area of study has recently received much attention. In this talk we will discuss an extension of this approach to pathwise differential equations in the plane. These are essentially hyperbolic non-linear PDEs with additive noise. We extend the so-called local time formulation of the regularization by noise approach to these equations, and extend the concept of non-linear Young equations to rectangular domains in order to prove wellposedness of these equations, even when the nonlinear coefficient is a generalized function (e.g. distribution). We illustrate the application of this construction by proving wellposedness of a wave equation with a noisy boundary constructed from two independent fractional Brownian motions. In the end we will discuss further potential applications and extensions of this framework and present several open challenges. This talk is based on a joint collaboration with Florian Bechtold (Bielefeld University) and Nimit Rana (Bielefeld University), see arXiv:2206.05360.
Multi-currency reserving for coherent risk measures
Saul Jacka
University of Warwick
Abstract
We examine the problem of dynamic reserving for risk in multiple currencies under a general coherent risk measure. The reserver requires to hedge risk in a time-consistent manner by trading in baskets of currencies. We show that reserving portfolios in multiple currencies \(\mathbf{V}\) are time-consistent when (and only when) a generalisation of Delbaen's m-stability condition, which we term optional \( \mathbf{V}\)-m-stability, holds. We prove a version of the Fundamental Theorem of Asset Pricing in this context.
Dynamical analysis of a stochastic delayed epidemic model with Lévy jumps and regime switching
Bojana Jovanović
University of Niš
Abstract
A delayed stochastic SLVIQR epidemic model is derived. Model is constructed by assuming that transmission rate satisfies the mean-reverting Ornstein-Uhlenbeck process. Besides a standard Brownian motion, another two driving processes are considered: a stationary Poisson point process and a continuous finite-state Markov chain. For the constructed model, the existence and uniqueness of a positive global solution is proven. Also, sufficient conditions under which the disease would lead to extinction or be persistent in the mean are established and it is shown that constructed model has a richer dynamic analysis compared to existing models. In addition, numerical simulations are given to illustrate the theoretical results.
Controlled measure-valued martingales: a viscosity solution approach
Sigrid Källblad Nordin
KTH Royal Institute of Technology
Abstract
We consider a class of stochastic control problems where the state process is a probability measure-valued process satisfying an additional martingale condition on its dynamics, called measure-valued martingales (MVMs). We establish the ‘classical’ results of stochastic control for these problems: specifically, we show that the value function for the problem can be characterised as the unique solution to a Hamilton-Jacobi-Bellman equation in the sense of viscosity solutions. In order to obtain this, we exploit structural properties of the MVM processes; in particular, our results include existence of controlled MVMs and an appropriate version of Itô’s formula for such processes. We also illustrate how problems of this type arise in a number of applications including model-independent derivatives pricing.
The talk is based on joint work with Alex Cox, Martin Larsson and Sara Svaluto-Ferro.
Regularity and efficient simulation of Gaussian processes defined through SPDEs
Kristin Kirchner
TU Delft
Abstract
Many models in spatial statistics are based on Gaussian random fields (GRFs). Motivated by the SPDE approach which exploits a well-known relation between the Matérn class of GRFs and stochastic partial differential equations (SPDEs), in this talk I will consider GRFs on a bounded Euclidean domain which are solutions of fractional-order elliptic SPDEs driven by additive spatial white noise. I will discuss the regularity of these GRFs in Sobolev and Hölder spaces and propose a numerical approximation, which provably converges in these spaces at explicit and sharp rates. Furthermore, I will address the computational benefits of the proposed method compared to kernel-based approaches, illustrated by several numerical experiments. Finally, I will give an outlook on spatiotemporal models which are based on SPDEs involving fractional powers of parabolic space-time differential operators.
This talk is based on joint works with David Bolin, Sonja Cox, Mihály Kovács, Christoph Schwab and Joshua Willems.
High-Frequency Trading with Fractional Brownian Motion
Yuliya Mishura
Taras Shevchenko National University of Kyiv
Abstract
This talk is the result of common research with Paolo Guasoni and Miklos Rasonyi.
We find an explicit formula for locally mean-variance optimal strategies and their performance for an asset price that follows fractional Brownian motion. Without trading costs, risk-adjusted profits are linear in the trading horizon and rise asymmetrically as the Hurst exponent departs from Brownian motion, remaining finite as the exponent reaches zero while diverging as it approaches one. Trading costs penalize numerous portfolio updates from short-lived signals, leading to a finite trading frequency, which can be chosen so that the effect of trading costs is arbitrarily small, depending on the required speed of convergence to the high-frequency limit.
The fractional heat equation driven by time-space white noise
Bernt Øksendal
University of Oslo
Abstract
We study the stochastic time-fractional stochastic heat equation driven by time-space white noise \(W(t,x)\) , where the space dimension of \(x\) is denoted by \(d\).
The time-fractional derivative is the Caputo derivative of order \(\alpha \in (0,2)\).
- In the classical case, when \(\alpha=1\), this equation models the normal diffusion of heat in a random or noisy medium, the noise being represented by the time-space white noise \(W(t,x)\).
- When \(\alpha > 1\) the equation models superdiffusion or enhanced diffusion, where the particles spread faster than in regular diffusion. This occurs for example in some biological systems.
- When \(\alpha < 1 \) the equation models subdiffusion, in which travel times of the particles are longer than in the standard case. Such situation may occur in transport systems.
We consider the equation in the sense of distribution, and we find an explicit expression for the \(S'\)-valued solution \(Y(t,x)\), where \(S'\) is the space of tempered distributions.
Following the terminology of Yaozhong Hu, we say that the solution is mild if \(Y(t,x) \in L^2(P)\) for all \(t\in [0,\infty, x \in R^d\). It is well-known that in the classical case with \(\alpha = 1\), the solution is mild if and only if the space dimension \(d=1\). We prove that if \(\alpha > 1\) the solution is mild both for \(d=1\) and for \(d=2\). We also give results in the case \(\alpha < 1\).
The presentation is based on joint work in progress with Yasmine Moulay Hachemi, University of Saida, Algeria.
Epidemic models with varying infectivity/susceptibility
Étienne Pardoux
Aix-Marseille University
Abstract
In three papers between 1927 and 1933, Kermack and McKendrick proposed models with infection age dependent infectivity and recovery rate, as well as recovery age dependent susceptibility. We prove that the large population limits of adequate stochastic non Markov epidemic models converge to the models of Kermack and McKendrick, or extensions of those. The limits of those non Markov models are either integral equations with memory or else ODE/PDE models.
This is joint work with Guodong Pang (Rice Univ., Houston, USA), Raphaël Forien (INRAE, Avignon) and in part with Arsene Brice Zotsa-Ngoufack (Phd student, AMU).
Interpolation of noise in finite element approximation of stochastic PDEs
Andreas Petersson
University of Oslo
Abstract
We consider a fully discrete finite element approximation of a semilinear stochastic reaction-advection-diffusion equation on a convex polygon driven by multiplicative Gaussian noise that is white in time and correlated by a continuous kernel in space. Traditionally, in the numerical analysis of such an approximation one assumes that the noise is sampled by projecting it, in an L^2 sense, onto the finite element space. Without additional assumptions on the kernel, this necessitates the computation of a large number of integrals when calculating the covariance matrix for the noise, which can be very costly. In this talk we consider the questions of 1) what the effect of interpolating the noise instead of projecting it has (meaning that we generate samples directly from the specified covariance kernel) and 2) what the effect of sampling the noise on a larger more regular domain and then interpolating it onto the polygon has. The latter approach allows for the application of circulant embedding methods for the noise sampling. We provide a comprehensive answer to these questions. Specifically, we show that 1) the interpolation of noise does not cause an additional error if the noise is of Matérn type, 2) there exist commonly considered kernels where the interpolation of noise causes an additional error and 3) sampling the noise on a coarser grid than the one used for the finite element space can be computationally beneficial. Simulations illustrate the results.
This is based on ongoing joint work with Gabriel Lord, Radboud University.
Numerical Analysis of the Stochastic Navier-Stokes equation
Andreas Prohl
University of Thübingen
Abstract
By now, numerical tools to obtain convergent discretizations for deterministic incompressible Navier-Stokes equation are well-known - as opposed to its stochastic version: relevant new key difficulties which prevent their immediate use in the stochastic case are: the non-trivial interplay of the (multiplicative) noise with the Lagrange multiplier ('pressure'), the nonlinearity, and Dirichlet boundary data.
In this talk, I detail construction of schemes, and new tools needed to show convergence with rates for them, addressing the problem in 2D and for periodic BC's. Then, I sketch further tools which (1) show improved rates for only additive noise, (2) are able to address Dirichlet BC's, and (3) motivate why the problem in 3D again requires different tools for its numerical analysis.
Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths
Frank Proske
University of Oslo
Abstract
In this talk we want to discuss a path-by-path uniqueness result in the sense of A. M. Davie [1] for SDEs driven by a fractional Brownian motion with a Hurst parameter \(H \in (0,1/2)\), where the drift vector field is allowed to be merely bounded and measurable.
Using this result, we construct unique weak regular solutions to the classical transport transport equation with singular velcocity fields perturbed along fractional Brownian paths.
The approach used for proving our results is based on supremum concentation inequalities and certain variational techniques. The results presented in this talk are based on a joint work with O. Amine (University of Oslo) and A.-R. Mansouri (Queenís University).
Reference: [1] Davie, A. M.: Uniqueness of solutions of stochastic differential equations. International Mathematics Research Notices (2007).
SPDEs with linear multiplicative fractional noise: Continuity in law with respect to the Hurst index
Lluis Quer-Sardanyons
The Autonomous University of Barcelona
Abstract
We consider the one-dimensional stochastic wave and heat equations driven by a linear multiplicative Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index \( H \in (\frac14,1)\). We prove that the solution of each of the above equations is continuous in terms of the index \( H\), with respect to the convergence in law in the space of continuous functions. The proof is based on a tightness criterion on the plane and Malliavin calculus techniques in order to identify the limit law.
The talk is based on joint work with Luca M. Giordano (Università degli Studi di Milano) and Maria Jolis (Universitat Autònoma de Barcelona).
Quasi-logconvex risk measures and applications
Emanuela Rosazza Gianin
University of Milano-Bicocca
Abstract
This talk introduces and fully characterizes the novel class of quasi-logconvex measures of risk, to stand on equal footing with the rich class of quasi-convex measures of risk. Quasi-logconvex risk measures naturally generalize logconvex return risk measures, just like quasi-convex risk measures generalize convex monetary risk measures. We establish their dual representation and analyze their taxonomy in a few (sub)classification results. Furthermore, we characterize quasi-logconvex risk measures in terms of properties of families of acceptance sets and provide their law-invariant representation. Examples and applications to portfolio choice and capital allocation are also discussed.
Based on a joint work with Roger Laeven.
Guaranteed Minimum Maturity Benefit under self-exciting mortality
Åsmund Hausken Sande
University of Oslo
Abstract
The Guaranteed Minimum Maturity Benefit is quite a popular feature embedded in several unit-linked policies offered by insurance companies. The value of this benefit depends on several processes assumed to describe both the mortality and the financial dynamics, typically represented by interest rates and by the fund associated to the unit-linked policy. A large literature is devoted to the valuation of GMMB for different mortality models, in particular when the mortality dynamics is described by affine models of diffusion type. In the present paper we assume for the mortality dynamics a self-exciting behaviour, described by a Hawkes type process with exponential kernel, which allows to keep both the Markov and the affine features, but introduces jumps with a stochastic intensity. This type of dynamics, exhibiting a jump clustering property, is quite convenient in order to describe mortality in some critical situations, like epidemics, when contagions phenomena make the probability of jumps arrivals higher when a jump is already occurred. By assuming a diffusion type dynamics for both the fund and the interest rates and introducing all the possible correlations among the diffusion processes necessary in order to get a realistic dynamics, we take advantage of the affine features of the model proposed and compute in a semi-explicit form the GMMB.
An Application of Markov Regime-Switching Models: Bancassurance
Emel Savku
University of Oslo
Abstract
We develop an approach for two player constraint nonzero.sum stochastic differential game, which is modelled by Markov regime-switching jump-diffusion processes. We provide the relations between a usual stochastic optimal control setting and a Lagrangian method. In this context, we prove corresponding theorems for two different type of constraints, which lead us to find real valued and stochastic Lagrange multipliers, respectively. Then, we illustrate our results for an example of cooperation between a bank and an insurance company with a popular, well-known business agreement type, called Bancassurance. It is well known that the timing and the amount of dividend payments are strategic decisions for companies. The announcement of a dividend payment may reduce or increase the stock prices of a company. From the side of the bank, it is clear that creating a cash flow with high returns would be the main goal. By using stochastic maximum principle, we investigate optimal dividend strategy for the company as a best response according to the optimal mean rate of return choice of a bank for its own cash flow and vice versa. We find out a Nash equilibrium for this game and solve the adjoint equations explicitly for each state. Hence, in our formulation, we provide an insight to both of the bank and the insurance company about their best moves in a bancassurance commitment under specified technical conditions.
Reference: E. Savku, A stochastic control approach for constrained stochastic differential games with jump and regimes. Submitted 2022.
Optimal reinsurance via BSDEs in a partially observable contagion model with jump clusters
Carlo Sgarra
Polytechnic University of Milan
Abstract
We investigate the optimal reinsurance problem when the loss process exhibits jump clustering features and the insurance company has restricted information about the loss process. We maximize expected exponential utility of terminal wealth and show that an optimal solution exists. By exploiting both the Kushner-Stratonovich and Zakai approaches, we provide the equation governing the dynamics of the (infinite-dimensional) filter and characterize the solution of the stochastic optimization problem in terms of a BSDE, for which we prove existence and uniqueness of solution. After discussing the optimal strategy for a general reinsurance premium, we provide more explicit results for proportional reinsurance under the expected value premium principle.
Time-changed SIRV model for epidemic of SARS-CoV-2 virus
Nenad Šuvak
Josip Juraj Strossmayer University of Osijek
Abstract
The stochastic version of the SIRV (susceptible-infected-recovered-vaccinated) model for the epidemic of the SARS-CoV-2 virus in the population of non-constant size and finite period of immunity is considered. Among many parameters influencing the dynamics of this model, the most important is the contact rate, i.e. the average number of adequate contacts of an infective person, where an adequate contact is one which is sufficient for the transmission of an infection if it is between a susceptible and an infected individual. It is expected that this parameter exhibits time-space clusters which reflect interchanging of periods of low and steady transmission and periods of high and volatile transmission of the disease.
The stochastics in the SIRV model considered here comes from the transmission coefficient, which is modeled as the mean reverting stochastic process governed by the time-changed Lévy noises, where the time-change is independent of the driving Lévy process. More precisely, this noise can be represented as the sum of the conditional Brownian motion and Poisson random field, closely related to the corresponding time-changed Brownian motion and the time-changed Poisson random measure, see [1].
The existence and uniqueness of positive global solution of the stochastic SIRV process is proven by classical techniques, see [2] for a rough idea. Furthermore, persistence and extinction of infection in population in long-run scenario are analyzed. In particular, conditions depending on parameters of the model and the underlying measure, under which the persistence and the extinction of the disease appear, are derived.
The theoretical results are illustrated via simulations. Transmission coefficient is simulated as the mean-reverting CIR diffusion with jumps with different propositions for the absolutely continuous time-change process: integrated deterministic periodic function, integrated subordinator, integrated compound Poisson process and integrated Ornstein-Uhlenbeck diffusion. The recovery problem of the transmission coefficient and the corresponding time-change process from numbers of susceptible, vaccinated, infected, and recovered individuals is briefly discussed.
Joint work with Giulia Di Nunno and Jasmina Đorđević.
References
[1] Di Nunno, G. & Sjursen, S. (2014) BSDEs driven by time-changed Lévy noises and optimal control. Stochastic Processes and their Applications 124, 1679-1709
[2] Đordjević, J., Papić, I. & Šuvak, N. (2021) A two diffusions stochastic model for the spread of the new corona virus SARS-CoV-2. Chaos, Solitons and Fractals 148: 110991
Fluid Models with Pseudo-Differential Noise
Hao Tang
University of Oslo
Abstract
We propose a general framework of proper regularization to solve nonlinear SPDEs with singularities included in both drift and noise coefficients. As applications, the (local and global) existence is presented for a broad class of fluid models driven by pseudo-differential noise, which include the stochastic magnetohydrodynamics (hence Navier-Stokes/Euler) equations, stochastic Camassa-Holm type equations, stochastic aggregation-diffusion equation and stochastic surface quasi-geostrophic equation. Thus, some recent results derived in the literature are considerably extended in a unified way.
Volterra sandwiched volatility model: Markovian approximation and hedging
Anton Yurchenko-Tytarenko
University of Oslo
Abstract
We propose a new market model with a stochastic volatility driven by a general Hölder continuous Gaussian Volterra process, i.e. the resulting price is not a Markov process. On the one hand, it is consistent with empirically observed phenomenon of market memory, but, on the other hand, brings a vast amount of issues of a technical nature, especially in optimization problems. In the talk, we describe a way to obtain a Markovian approximation to the model as well as exploit it for the numerical computation of the optimal hedge. Two numerical methods are considered: Nested Monte Carlo and Least Squares Monte Carlo. The results are illustrated by simulations.
Change of measure in a Heston-Hawkes stochastic volatility model
Oriol Zamora Font
University of Oslo
Abstract
We consider the stochastic volatility model obtained by adding a compound Hawkes process to the volatility of the well-known Heston model. A Hawkes process is a self-exciting counting process with many applications in mathematical finance, insurance, epidemiology, seismology and other fields. We prove a general result on the existence of a family of equivalent (local) martingale measures. We apply this result to a particular example where the size of the jumps are exponentially distributed. Finally, we also give the dynamics of the forward variance which can be used to add a tradable asset in this model. This is a work in progress.