# Working Papers Series

The Working Paper Series are pre-publication versions of research reports associated with the Centre's staff, research students and visitors.**Links to abstracts and downloadable pdf versions of the papers can be found below:**

## Research Papers

### 2016

16-01

Sergey Badikov, Antoine Jacquier, Daphne Qing Liu, Patrick Roome

**No-arbitrage bounds for the forward smile given marginals**

We explore the robust replication of forward-start straddles given quoted (Call and Put options) market data. One approach to this problem classically follows semi-infinite linear programming arguments, and we propose a discretisation scheme to reduce its dimensionality and hence its complexity. Alternatively, one can consider the dual problem, consisting in finding optimal martingale measures under which the upper and the lower bounds are attained. Semi-analytical solutions to this dual problem were proposed by Hobson and Klimmek [13] and by Hobson and Neuberger [14]. We recast this dual approach as a finite dimensional linear programme, and reconcile numerically, in the Black-Scholes and in the Heston model, the two approaches.

16-02

Thomas Cass, Nengli Lim**A Stratonovich-Skorohod integral formula for Gaussian rough paths**

Given a Gaussian process X, its canonical geometric rough path lift X, and a solution Y to the rough differential equation (RDE) dYt=V(Yt)∘dXt, we present a closed-form correction formula for ∫Y∘dX−∫YdX, i.e. the difference between the rough and Skorohod integrals of Y with respect to X. When X is standard Brownian motion, we recover the classical Stratonovich-to-It\^o conversion formula, which we generalize to Gaussian rough paths with finite p-variation, 2<p<3, and satisfying an additional natural condition. This encompasses many familiar examples, including fractional Brownian motion with H>13. To prove the formula, we show that ∫YdX is the L2(Ω) limit of its Riemann-sum approximants, and that the approximants can be appended with a suitable compensation term without altering the limit. To show convergence of the Riemann-sum approximants, we utilize a novel characterization of the Cameron-Martin norm using higher-dimensional Young-Stieltjes integrals. For the main theorem, complementary regularity between the Cameron-Martin paths and the covariance function of X is used to show the existence of these integrals. However, it turns out not to be a necessary condition, as in the last section we provide a new set of conditions for their existence.

16-03

Thomas Cass, Martin P. Weidner**Hörmander's theorem for rough differential equations on manifolds**

We introduce a new definition for solutions Y to rough differential equations (RDEs) of the form dYt=V(Yt)dXt,Y0=y0. By using the Grossman-Larson Hopf algebra on labelled rooted trees, we prove equivalence with the classical definition of a solution advanced by Davie when the state space E for Y is a finite-dimensional vector space. The notion of solution we propose, however, works when E is any smooth manifold M and is therefore equally well-suited for use as an intrinsic defintion of an M-valued RDE solution. This enables us to prove existence, uniqueness and coordinate-invariant theorems for RDEs on M bypassing the need to define a rough path on M. Using this framework, we generalise result of Cass, Hairer, Litter and Tindel proving the smoothness of the density of M-valued RDEs driven by non-degenerate Gaussian rough paths under Hörmander's** **bracket condition. In doing so, we reinterpret some of the foundational results of the Malliavin calculus to make them appropriate to the study M-valued Wiener functionals.

16-04

John Armstrong, Damiano Brigo

**Coordinate-free Stochastic Differential Equations as Jets**

We explain how Itô Stochastic Differential Equations on manifolds may be defined as 2-jets of curves. We use jets as a natural language to express geometric properties of SDEs and show how jets can lead to intuitive representations of Itô SDEs, including three different types of drawings. We explain that the mainstream choice of Fisk-Stratonovich-McShane calculus for stochastic differential geometry is not necessary and elaborate on the relationships with the jets approach. We consider the two calculi as being simply different coordinate systems for the same underlying coordinate-free stochastic differential equation. If the extrinsic approach to differential geometry is adopted, then Stratonovich calculus may appear to be necessary when studying SDEs on submanifolds but in fact one can use the Itô/2-jets framework proposed here by recalling that the curvature of the 2-jet follows the curvature of the manifold. We argue that the choice between Itô and Stratonovich is a modelling choice dictated by the type of problem one is facing and the related desiderata. We also discuss the forward Kolmogorov equation and the backward diffusion operator in geometric terms, and consider percentiles of the solutions of the SDE and their properties, leading to fan diagrams and their relationship with jets. In particular, the median of an SDE solution is associated to the drift of the SDE in Stratonovich form for small times. Finally, we prove convergence of the 2-jet scheme to classical Itô SDEs solutions.

16-05

Damiano Brigo, Marco Francischello, Andrea Pallavicini

**Invariance, existence and uniqueness of solutions of nonlinear valuation PDEs and FBSDEs inclusive of credit risk, collateral and funding costs**

We study conditions for existence, uniqueness and invariance of the comprehensive nonlinear valuation equations first introduced in Pallavicini et al (2011) [11]. These equations take the form of semi-linear PDEs and Forward-Backward Stochastic Differential Equations (FBSDEs). After summarizing the cash flows definitions allowing us to extend valuation to credit risk and default closeout, including collateral margining with possible re-hypothecation, and treasury funding costs, we show how such cash flows, when present-valued in an arbitrage free setting, lead to semi-linear PDEs or more generally to FBSDEs. We provide conditions for existence and uniqueness of such solutions in a viscosity and classical sense, discussing the role of the hedging strategy. We show an invariance theorem stating that even though we start from a risk-neutral valuation approach based on a locally risk-free bank account growing at a risk-free rate, our final valuation equations do not depend on the risk free rate. Indeed, our final semilinear PDE or FBSDEs and their classical or viscosity solutions depend only on contractual, market or treasury rates and we do not need to proxy the risk free rate with a real market rate, since it acts as an instrumental variable. The equations derivations, their numerical solutions, the related XVA valuation adjustments with their overlap, and the invariance result had been analyzed numerically and extended to central clearing and multiple discount curves in a number of previous works, including [11], [12], [10], [6] and [4].

16-06

Damiano Brigo, Giovanni Pistone

**Projection based dimensionality reduction for** **measure valued ****evolution equations in statistical manifolds**

We propose a dimensionality reduction method for infinite-dimensional measure-valued evolution equations such as the Fokker-Planck partial differential equation or the Kushner-Stratonovich resp. Duncan-Mortensen-Zakai stochastic partial differential equations of nonlinear filtering, with potential applications to signal processing, quantitative finance, heat flows and quantum theory among many other areas. Our method is based on the projection coming from a duality argument built in the exponential statistical manifold structure developed by G. Pistone and co-authors. The choice of the finite dimensional manifold on which one should project the infinite dimensional equation is crucial, and we propose finite dimensional exponential and mixture families. This same problem had been studied, especially in the context of nonlinear filtering, by D. Brigo and co-authors but the L2 structure on the space of square roots of densities or of densities themselves was used, without taking an infinite dimensional manifold environment space for the equation to be projected. Here we re-examine such works from the exponential statistical manifold point of view, which allows for a deeper geometric understanding of the manifold structures at play. We also show that the projection in the exponential manifold structure is

consistent with the Fisher Rao metric and, in case of finite dimensional exponential families, with the assumed density approximation. Further, we show that if the

sufficient statistics of the finite dimensional exponential family are chosen among the eigenfunctions of the backward diffusion operator then the statistical-manifold

or Fisher-Rao projection provides the maximum likelihood estimator for the Fokker Planck equation solution. We finally try to clarify how the finite dimensional and infinite dimensional terminology for exponential and mixture spaces are related.

16-07

Damiano Brigo, Giovanni Pistone

**Maximum likelihood eigenfunctions of the Fokker Planck equation and Hellinger projection**

We apply the L2 based Fisher-Rao vector-field projection by Brigo, Hanzon and LeGland (1999) to finite dimensional approximations of the Fokker Planck equation on exponential families. We show that if the sufficient statistics are chosen among the diffusion eigenfunctions the finite dimensional projection or the equivalent assumed density approximation provide the exact maximum likelihood density. The same result had been derived earlier by Brigo and Pistone (2016) in the infinite-dimensional Orlicz based geometry of Pistone and co-authors as opposed to the L2 structure used here.

16-08**The Local Fractional Bootstrap**

Mikkel Bennedsen, Ulrich Hounyo, Asger Lunde, Mikko S. Pakkanen

We introduce a bootstrap procedure for high-frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first order validity of the bootstrap method and in simulations we observe that the bootstrap-based hypothesis test provides considerable finite-sample improvements over an existing test that is based on a central limit theorem. This is important when studying the roughness properties of time series data; we illustrate this by applying the bootstrap method to two empirical data sets: we assess the roughness of a time series of high-frequency asset prices and we test the validity of Kolmogorov's scaling law in atmospheric turbulence data.

16-09**Arbitrage without borrowing or short selling?**Jani Lukkarinen, Mikko S. Pakkanen

We show that a trader, who starts with no initial wealth and is not allowed to borrow money or short sell assets, is theoretically able to attain positive wealth by continuous trading, provided that she has perfect foresight of future asset prices, given by a continuous semimartingale. Such an arbitrage strategy can be constructed as a process of finite variation that satisfies a seemingly innocuous self-financing condition, formulated using a pathwise Riemann-Stieltjes integral. Our result exemplifies the potential intricacies of formulating economically meaningful self-financing conditions in continuous time, when one leaves the conventional arbitrage-free framework.

16-10**On the conditional small ball property of multivariate Lévy-driven moving average processes**Mikko S. Pakkanen, Tommi Sottinen, Adil Yazigi

We study whether a multivariate Lévy-driven moving average process can shadow arbitrarily closely any continuous path, starting from the present value of the process, with positive conditional probability, which we call the conditional small ball property. Our main results establish the conditional small ball property for Lévy-driven moving average processes under natural non-degeneracy conditions on the kernel function of the process and on the driving Lévy process. We discuss in depth how to verify these conditions in practice. As concrete examples, to which our results apply, we consider fractional Lévy processes and multivariate Lévy-driven Ornstein-Uhlenbeck processes.

16-11**Joint Asymptotic Distribution of Certain Path Functionals of the Reflected Process**Aleksandar Mijatović, Martijn Pistorius

See link '16-11' above for the abstract and paper.

16-12**On Dynamic Deviation Measures and Continuous-Time Portfolio Optimisation**Martijn Pistorius, Mitja Stadjey

In this paper we propose the notion of dynamic deviation measure, as a dynamic timeconsistent extension of the (static) notion of deviation measure. To achieve time-consistency we require that a dynamic deviation measures satises a generalised conditional variance formula. We show that, under a domination condition, dynamic deviation measures are characterised as the solutions to a certain class of backward SDEs. We establish for any dynamic deviation measure an integral representation, and derive a dual characterisation result in terms of additively m-stable dual sets. Using this notion of dynamic deviation measure we formulate a dynamic mean-deviation portfolio optimisation problem in a jump-diusion setting and identify a subgame-perfect Nash equilibrium strategy that is linear as function of wealth by deriving and solving an associated extended HJB equation.

### 2015

**Working Papers 2015**

15-01

Archil Gulisashvili, Blanka Horvath & Antoine Jacquier**Mass at Zero and Small-Strike Implied Volatility Expansion in the Sabr Model.**

*We study the probability mass at the origin in the SABR stochastic volatility model, and derive several tractable expressions for it, in particular when time becomes small or large. In the uncorrelated case, tedious saddlepoint expansions allow for (semi) closed-form asymp-totic formulae. As an application–the original motivation for this paper–we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by definition arbitrage-free, allow us to quantify the impact of the mass at zero on currently used implied volatility expansions. In particular we discuss how much those expansions become erroneous.*

15-02

Aleksandar Mijatovic, Martijn Pistorius*Buffer-Overflows: Joint Limit Laws of Undershoots and Overshoots of Reflected Processes.*

15-03

Thomas Cass, Bruce K. Drivery, Christian Littererz**Constrained Rough Paths.**

*Abstract: We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way we develop a notion of rough integration and an ecient and intrinsic theory of rough di erential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold valued rough paths. Finally, this framework is used to show there is a one to one correspondence between rough paths on a d { dimensional manifold and rough paths on d { dimensional Euclidean space. This last result is a rough path analogue of Cartan's development map and its stochastic version which was developed by Eells and Elworthy and Malliavin.*

15-04

Thomas Cass, Marcel Ogrodnik*Tail estimates for Markovian rough paths.*

*Abstract: We work in the context of Markovian rough paths associated to a class of uniformly subelliptic Dirichlet forms ([25]) and prove an almost-Gaussian tail- estimate for the accummulated local p-variation functional, which has been intro- duced and studied in [17]. We comment on the signi cance of these estimates to a range of currently-studied problems, including the recent results of Chevyrev and Lyons in [18].*

*To appear in Annals of Probability*

15-05

Antoine Jacquier, Partick Roome*Black-Scholes in a CEV random environment: a new approach to smile modelling.*

*Classical (Itˆo diffusions) stochastic volatility models are not able to capture the steepness of small-maturity implied volatility smiles. Jumps, in particular exponential Levy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see [53] for an overview). A recent breakthrough was made by Gatheral, Jaisson and Rosenbaum [27], who proposed to replace the Brownian driver of the instantaneous volatility by a short-memory fractional Brownian motion, which is able to capture the short-maturity steepness while preserving path continuity. We suggest here a different route, randomising the Black-Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential Levy models and fractional stochastic volatility models. As a by-product, we make a conjecture on the small-maturity forward smile asymptotic s of stochastic volatility models, in exact agreement with the results in [37] for the Heston model.*

15-06** **Mikkel Bennedsen, Asger Lundey, Mikko S. Pakkanenz

Hybrid scheme for Brownian semistationary processes

Hybrid scheme for Brownian semistationary processes

*We introduce a simulation scheme for Brownian semistationary processes, which is based* *on discretizing the stochastic integral representation of the process in the time domain. We* *assume that the kernel function of the process is regularly varying at zero. The novel feature* *of the scheme is to approximate the kernel function by a power function near zero and by a* *step function elsewhere. The resulting approximation of the process is a combination of Wiener* *integrals of the power function and a Riemann sum, which is why we call this method a hybrid* *scheme. Our main theoretical result describes the asymptotics of the mean square error of the* *hybrid scheme and we observe that the scheme leads to a substantial improvement of accuracy* *compared to the ordinary forward Riemann-sum scheme, while having the same computational**complexity. We exemplify the use of the hybrid scheme by two numerical experiments, where we* *examine the nite-sample properties of an estimator of the roughness parameter of a Brownian* *semistationary process and study Monte Carlo option pricing in the rough Bergomi model of* **Bayer et al. (2015)**, respectively.

### 2014

**Working Papers 2014**

14-01

Jean-Francois Chassagneux, Antoine Jacquier, Ivo Mihaylov**An explicit Euler scheme with strong rate of convergence for non-Lipschitz SDEs**

*Abstract: We consider the approximation of stochastic differential equations (SDEs) with non-Lipschitz drift or diffusion coefficients. We present a modified explicit Euler-Maruyama discretisation scheme that allows us to prove strong convergence, with a rate. Under some regularity conditions, we obtain the optimal strong error rate. We consider SDEs popular in the mathematical finance literature, including the Cox-Ingersoll-Ross (CIR), the 3=2 and the Ait-Sahalia models, as well as a family of mean-reverting processes with locally smooth coefficients.*

*Keywords: Stochastic differential equations, non-Lipschitz coefficients, explicit Euler-Maruyama scheme with projection, CIR model, Ait-Sahalia model.*

14-02

Antoine Jacquier and Patrick Roome *Large-maturity regimes of the Heston Forward Smile*

*Abstract: We provide a full characterisation of the large-maturity forward implied volatility smile in the Heston model. Although the leading decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms highly depends on the parameters, and di erent powers of the maturity come into play. As a by-product of the analysis we provide new implied volatility asymptotics, both in the forward case and in the spot case, as well as extended SVI-type formulae. The proofs are based on extensions and re nements of sharp large deviations theory, in particular in cases where standard convexity arguments fail.*

14-03

Jean- François Chassagneux and Adrien Richou *Numerical Stability Analysis of the Euler Scheme for BSDES*

*Abstract: In this paper, we study the qualitative behaviour of approximation schemes for Backward Stochastic Differential Equations (BSDEs) by introducing a new notion of numerical stability. For the Euler scheme, we provide sufficient conditions in the one-dimensional and multidimensional case to guarantee the numerical stability. We then perform a classical Von Neumann stability analysis in the case of a linear driver f and exhibit necessary conditions to get stability in this case. Finally, we illustrate our results with numerical applications.*

14-04

M. Ottobre, G.A. Pavlotis, K. Pravda-Starov *Some remarks on Degenrate Hypoelliptic Ornstein-Uhlenbeck Operators*

*Abstract: We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in L2 spaces with respect to invariant measures. The purpose of this article is to show how recent results on general quadratic operators apply to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators. We rst show that some known results about the spectral and subelliptic properties of Ornstein-Uhlenbeck operators may be directly recovered from the general analysis of quadratic operators with zero singular spaces. We also provide new resolvent estimates for hypoelliptic Ornstein-Uhlenbeck operators. We show in particular that the spectrum of these non-selfadjoint operators may be very unstable under small perturbations and that their resolvents can blow-up in norm far away from their spectra. Furthermore, we establish sharp resolvent estimates in speci c regions of the resolvent set which enable us to prove exponential return to equilibrium.*

14-05

John Armstrong, Damiano Brigo*Stochastic Filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples*

*Abstract: We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite dimensional Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric and on a family of normal mixtures. We compare this method to earlier projection methods based on the Hellinger distance/Fisher metric and exponential families, and we compare the L2 mixture projection filter with a particle method with the same number of parameters, using the Levy metric. We prove that for a simple choice of the mixture manifold the L2 mixture projection filter coincides with a Galerkin method, whereas for more general mixture manifolds the equivalence does not hold and the L2 mixture filter is more general. We study particular systems that may illustrate the advantages of this new filter over other algorithms when comparing outputs with the optimal filter. We finally consider a specific software design that is suited for a numerically efficient implementation of this filter and provide numerical examples.*

*Keywords: Direct L2 metric, Exponential Families, Finite Dimensional, Families of Probability Distributions, Fisher information metric, Hellinger distance, Levy Metric, Mixture Families, Stochastic filtering, Galerkin, AMS Classification codes: 53B25, 53B50, 60G35, 62E17, 62M20, 93E11*

14-06

Damiano Brigo, Jan-Frederik Mai, Matthias Scherer *Consistent iterated simulation of multi-variate default times: a Markovian indicators characterization*

*Abstract: We investigate under which conditions a single simulation of joint default times at a nal time horizon can be decomposed into a set of simulations of joint defaults on subsequent adjacent sub-periods leading to that nal horizon. Besides the theoretical interest, this is also a practical problem as part of the industry has been working under the misleading assumption that the two approaches are equivalent for practical purposes. As a reasonable trade-o between realistic stylized facts, practical demands, and mathematical tractability, we propose models leading to a Markovian multi-variate survival{indicator process, and we investigate two instances of static models for the vector of default times from the statistical literature that fall into this class. On the one hand, the "looping default" case is known to be equipped with this property at least since [Herbertsson, Rootzen (2008), Bielecki et al. (2011b)],and we point out that it coincides with the classical "Freund distribution" in the bivariate case. On the other hand, if all sub-vectors of the survival indicator process are Markovian, this constitutes a new characterization of the Marshall-Olkin distribution, and hence of multi-variate lack-of-memory. A paramount property of the resulting model is stability of the type of multi-variate distribution with respect to elimination or insertion of a new marginal component with marginal distribution from the same family. The practical implications of this "nested margining" property are enormous. To implement this distribution we present an ecient and unbiased simulation algorithm based on the Levy-frailty construction. We highlight dfferent pitfalls in the simulation of dependent default times and examine, within a numerical case study, the effect of inadequate simulation practices.*

*Keywords: Stepwise default simulation, default modeling, credit modeling, default dependence, default correlation, default simulation, arrival times, credit risk, Marshall-Olkin distribution, nested margining, Freund distribution, looping default models.*

14-07

Damiano Brigo, Francesco Rapisarday, Abir Sridiz *The arbitrage-free Multivariate Mixture Dynamics Model: Consistent single-assets and index volatility smiles*

*Abstract: We introduce a multivariate diffusion model that is able to price derivative securities featuring multiple underlying assets. Each asset volatility smile is modeled according to a density-mixture dynamical model while the same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. This allowsto reconcile single name and index/basket volatility smiles in a consistent framework. Our approach could be dubbed a multidimensional local volatility approach with vector-state dependent diffusion matrix. The model is quite tractable, leading to a complete market and not requiring Fourier techniques for calibration and dependence measures, contrary to multivariate stochastic volatility models such as Wishart. We prove existence and uniqueness of solutions for the model stochastic differential equations, provide formulas for a number of basket options, and analyze the dependence structure of the model in detail by deriving a number of results on covariances, its copula function and rank correlation measures and volatilities-assets correlations. A comparison with sampling simply-correlated suitably discretized one-dimensional mixture dynamical paths is made, both in terms of option pricing and of dependence, and first order expansion relationships between the two models’ local covariances are derived. We also show existence of a multivariate uncertain volatility model of which our multivariate local volatilities model is a Markovian projection, highlighting that the projected model is smoother and avoids a number of drawbacks of the uncertain volatility version. We also show a consistency result where the Markovian projection of a geometric basket in the multivariate model is a univariate mixture dynamics model. A few numerical examples on basket and spread options pricing conclude the paper.*

* Key words: Mixture of densities, Volatility smile, Lognormal density, Multivariate local volatility, Complete Market, Option on a weighted Arithmetic average of a basket, Spread option, Option on a weighted geometric average of a basket, Markovian projection, Copula function.*

14-08

Stefano De Marco, Antoine Jacquier, Patrick Roome *Two Examples of Non Strictly Convex Large Deviations*

14-09

Dan Crisan, Yoshiki Otobe, Szymon Peszat*Inverse Problems for Stochastic transport equations*

*Abstract: Inverse problems for stochastic linear transport equations driven by a temporal or spatial white noise are discussed. We analyse stochastic linear transport equations which depend on an unknown potential and have either additive noise or multiplicative noise. We show that one can approximate the potential with arbitrary small error when the solution of the stochastic linear transport equation is observed over time at some fixed point in the state space.*

*Keywords: STOCHASTIC TRANSPORT EQUATIONS, partial observations, inverse problem.*

14-10

Dan Crisan, Christian Litterer, Terry Lyons*Kusuoka-Strook gradient bounds for the solution of the filtering equation*

*Abstract: We obtain sharp gradient bounds for perturbed di¤usion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock [12, 14, 15, 16], and extends their program developed for the heat semi-group to solutions of stochastic partial di¤erential equations. The work is motivated by and applied to nonlinear ...ltering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. It uses a pathwise representation of the per-turbed semigroup following Ocone [21]. The estimates we derive have sharp small time asymptotics.*

*Keywords: Stochastic partial di¤erential equation; Filtering; Zakai equation; Ran- domly perturbed semigroup, gradient bounds, small time asymptotics.*

14-11

Dilip Madan, Martijn Pistorius, Mitja Stadje*Convergence of BSΔEs driven by random walks to BSDES: the caseof (in)finite activity jumps with general driver*

*Abstract: In this paper we present a weak approximation scheme for BSΔEs driven by a Wiener process and an (in) nite activity Poisson random measure with drivers that are general Lipschitz functionals of the solution of the BSDE. The approximating backward stochastic difference equations (BSΔEs) are driven by random walks that weakly approximate the given Wiener process and Poisson random measure. We establish the weak convergence to the solution of the BSDE and the numerical stability of the sequence of solutions of the BSΔEs. By way of illustration we analyse explicitly a scheme with discrete step-size distributions.*

14-12

Zbigniew Michna, Zbigniew Palmowski, and Martijn Pistorius*The distribution of the supremum for spectrally asymmetric L´evy processes*

*Abstract: In this article we derive formulas for the probability IP(suptT X(t) > u) T > 0 and IP(supt<1 X(t) > u) where X is a spectrally positive L´evy process with infinite variation. The formulas are generalizations of the well-known Tak´acs formulas for stochastic processes with non-negative and interchangeable increments. Moreover, we find the joint distribution of inftT Y (t) and Y (T) where Y is a spectrally negative L´evy process.*

*Keywords: L´evy process, distribution of the supremum of a stochastic process, spectrally asymmetric L´evy process*

### 2013

#### Working Papers 2013

13-01

F. Avram, Z. Palmowski, M. Pistorius*On Gerber-Shiu functions and optimal dividend distribution for a Levy risk-process in the presence of a penalty function*

*Abstract: In this paper we consider an optimal dividend problem for an insurance company which risk process evolves as a spectrally negative Levy process (in the absence of dividend payments). We assume that the management of the company controls timing and size of dividend payments. The objective is to maximize the sum of the expected cumulative discounted dividends received until the moment of ruin and a penalty payment at the moment of ruin which is an increasing function of the size of the shortfall at ruin; in addition, there may be a fixed cost for taking out dividends. We explicitly solve the corresponding optimal control problem. The solution rests on the characterization of the value-function as (i) the unique stochastic solution of the associated HJB equation and as (ii) the pointwise smallest stochastic supersolution. We show that the optimal value process admits a dividend-penalty decomposition as sum of a martingale (associated to the penalty payment at ruin) and a potential (associated to the dividend payments). We find also an explicit necessary and sufficient condition for optimality of a single dividend-band strategy, in terms of a particular Gerber-Shiu function. We analyze a number of concrete examples.*

*Keywords: Optimal control, L´evy process, De Finetti model, transaction costs, singular control, variational inequality, barrier policies, band policies, Gerber-Shiu function.*

13-02

D. Madan, M. Pistorius, M. Stadje*On consistent valuations based on distorted expectations: from multinomial random walks to L'{e}vy processes*

*Abstract: A distorted expectation is a Choquet expectation with respect to the capacity induced by a concave probability distortion. Distorted expectations are encountered in various static settings, in risk theory, mathematical finance and mathematical economics. There are a number of different ways to extend a distorted expectation to a multi-period setting, which are not all time-consistent. One time-consistent extension is to define the non-linear expectation by backward recursion, applying the distorted expectation stepwise, over single periods. In a multinomial random walk model we show that this non-linear expectation is stable when the number of intermediate periods increases to infinity: Under a suitable scaling of the probability distortions and provided that the tick-size and time step-size converge to zero in such a way that the multinomial random walks converge to a Levy process, we show that values of random variables under the multi-period distorted expectations converge to the values under a continuous-time non-linear expectation operator, which may be identified with a certain type of Peng's g-expectation. A coupling argument is given to show that this operator reduces to a classical linear expectation when restricted to the set of pathwise increasing claims. Our results also show that a certain class of g-expectations driven by a Brownian motion and a Poisson random measure may be computed numerically by recursively defined distorted expectations.*

*Keywords: g-expectation, non-linear expectation, probability distortion, option pricing, risk measurement, convergence, L?evy process, multinomial tree.*

13-03

A. Mijatovic, M. Urusov*On the loss of the semimartingale property at the hitting time of a level*

*Abstract: This paper studies the loss of the semimartingale property of the process $g(Y)$ at the time a one-dimensional diffusion $Y$ hits a level, where $g$ is a difference of two convex functions. We show that the process $g(Y)$ can fail to be a semimartingale in two ways only, which leads to a natural definition of non-semimartingales of the \textit{first} and \textit{second kind}. We give a deterministic if and only if condition (in terms of $g$ and the coefficients of $Y$) for $g(Y)$ to fall into one of the two classes of processes, which yields a characterisation for the loss of the semimartingale property. A number of applications of the results in the theory of stochastic processes and real analysis are given: e.g. we construct an adapted diffusion $Y$ on $[0,\infty)$ and a \emph{predictable} finite stopping time $\zeta$, such that $Y$ is a semimartingale on the stochastic interval $[0,\zeta)$, continuous at $\zeta$ and constant after $\zeta$, but is \emph{not} a semimartingale on $[0,\infty)$.*

*Keywords: Continuous semimartingale; one-dimensional diffusion; local time; additive functional; Ray-Knight theorem.*

13-04

D. Crisan, S. Ortiz-Latorrey *A Kusuoka-Lyons-Victoir particle filter*

*Abstract: The aim of this paper is to introduce a new numerical algorithm for solving the continuous time non-linear ltering problem. In particular, we present a particle lter that combines the Kusuoka-Lyons-Victoir cubature method on Wiener space (KLV) [13], [18] to approximate the law of the signal with a minimal variance "thining" method, called the tree based branching algorithm (TBBA) to keep the size of the cubature tree constant in time. The novelty of our approach resides in the adaptation of the TBBA algorithm to simultaneously control the computational effort and incorporate the observation data into the system. We provide the rate of convergence of the approximating particle lter in terms of the computational effort (number of particles) and the discretization grid mesh. Finally, we test the performance of the new algorithm on a benchmark problem (the Bene?s filter).*

*Keywords: Cubature on Wiener space; particle filters; TBBA.*

13-05

A. Jacquier, P. Roome *The Small-Maturity Heston Forward Smile*

*Abstract: In this paper we investigate the asymptotics of forward-start options and the forward implied volatility smile in the Heston model as the maturity approaches zero. We prove that the forward smile for out-ofthe- money options explodes and compute a closed-form high-order expansion detailing the rate of the explosion. Furthermore the result shows that the square-root behaviour of the variance process induces a singularity such that for certain parameter con gurations one cannot obtain high-order out-of-the-money forward smile asymptotics. In the at-the-money case a separate model-independent analysis shows that the small-maturity limit is well de ned for any It^o di usion. The proofs rely on the theory of sharp large deviations (and re nements) and incidentally we provide an example of degenerate large deviations behaviour.*

*Keywords: Stochastic volatility model, Heston model, forward implied volatilty, asymptotic expansion.*

13-06

F. Haba, A. Jacquier *Asymptotic arbitrage in the Heston model*

*Abstract: In the context of the Heston model, we establish a precise link between the set of equivalent martingale measures, the ergodicity of the underlying variance process and the concept of asymptotic arbitrage proposed in Kabanov-Kramkov [13] and in Follmer-Schachermayer [8].*

*Keywords: Stochastic volatility model, Heston model, asymptotic arbitrage, large deviations.*

13-07

S. Jacka, A. Mijatovic, D. Siraj *Mirror and Synchronous Couplings of Geometric Brownian Motions*

*Abstract: The paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a characterisation of the optimality of the two couplings over any finite time horizon and show that, unlike in the case of Brownian motion, the optimality fails in general even if the geometric Brownian motions are martingales. On the other hand, we prove that in the cases of the ergodic average and the infinite time horizon criteria, the mirror coupling and the synchronous coupling are always optimal for general (possibly non-martingale) geometric Brownian motions. We show that the two couplings are efficient if and only if they are optimal over a finite time horizon and give a conjectural answer for the efficient couplings when they are suboptimal.*

To appear in Stochastic Processes & Applications.

*Keywords: mirror and synchronous coupling, coupling time, geometric Brownian motion, efficient coupling, optimal coupling, Bellman's principle.*

13-08**A. Jacquier, M. Lorig

*The Smile of certain Lévy-type Models**Abstract: We consider a class of assets whose risk-neutral pricing dynamics are described by an exponential L´evy-type process subject to default. The class of processes we consider features locally-dependent drift, diffusion and default-intensity as well as a locally-dependent L´evy measure. Using techniques from regular perturbation theory and Fourier analysis, we derive a series expansion for the price of a European-style option. We also provide precise conditions under which this series expansion converges to the exact price. *

*Additionally, for a certain subclass of assets in our modeling framework, we derive an expansion for the implied volatility induced by our option pricing formula. The implied volatility expansion is exact within its radius of convergence. As an example of our framework, we propose a class of CEV-like L´evy-type models. Within this class, approximate option prices can be computed by a single Fourier integral and approximate implied volatilities are explicit (i.e., no integration is required). Furthermore, the class of CEV-like L´evy-type models is shown to provide a tight fit to the implied volatility surface of S&P500 index options.*

*Keywords: Regular Perturbation, L´evy-type, Local Volatility, Implied Volatility, Default, CEV.*

13-09

A. Beskos, D. Crisan, A. Jasra, N. Whiteley**Error Bounds and Normalizing Constants for Sequential Monte Carlo Samplers in High Dimensions**

*Abstract: In this article we develop a collection of results associated to the analysis of the Sequential Monte Carlo (SMC) samplers algorithm, in the context of high-dimensional iid target probabilities. The SMC sampler algorithm can be designed to sample from a single probability distribution, using Monte Carlo to approximate expectations with respect to this law. Given a target density in d dimensions our results are concerned with d large while the number of Monte Carlo samples, N, remains fixed. We deduce an explicit bound on the Monte-Carlo error for estimates derived using the SMC sampler and the exact asymptotic relative L2-error of the estimate of the normalizing constant associated to the target. We also establish marginal propagation of chaos properties of the algorithm. These results are deduced when the cost of the algorithm is O(Nd^2).*

*Keywords: Sequential Monte Carlo, High Dimensions, Propagation of Chaos, Normalizing Constants.*

13-10

A. Beskos, D. Crisan, A. Jasra*On the Stability of Sequential Monte Carlo Methods in High Dimensions*

*Abstract: We investigate the stability of a Sequential Monte Carlo (SMC) method applied to the problem of sampling from a target distribution on Rd for large d. It is well known that using a single importance sampling step, one produces an approximation for the target that deteriorates as the dimension d increases, unless the number of Monte Carlo samples N increases at an exponential rate in d. We show that this degeneracy can be avoided by introducing a sequence of artificial targets, starting from a `simple' density and moving to the one of interest, using an SMC method to sample from the sequence. Using this class of SMC methods with a fixed number of samples, one can produce an approximation for which the effective sample size (ESS) converges to a random variable e_N as d increases. with 1 < e_N < N. The convergence is achieved with a computational cost proportional to Nd^2. If e_N<<N, we can raise its value by introducing a number of resampling steps, say m (where m is independent of d). Also, we show that the Monte Carlo error for estimating a fixed dimensional marginal expectation is of order 1/\sqrt{N} uniformly in d. The results imply that, in high dimensions, SMC algorithms can efficiently control the variability of the importance sampling weights and estimate fixed dimensional marginals at a cost which is less than exponential in d and indicate that resampling leads to a reduction in the Monte Carlo error and increase in the ESS. All of our analysis is made under the assumption that the target density is iid.*

*Keywords: Sequential Monte Carlo, High Dimensions, Resampling, Functional CLT.*

13-11

A. Jacquier, M. Lorig*From characteristic functions to implied volatility expansions*

*Abstract: For any strictly positive martingale S = eX for which X has an analytically tractable characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in log(K=S0). We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one nite activity exponential Levy model (Merton), one in nite activity exponential Levy model (Variance Gamma), and one stochastic volatility model (Heston). We show how this technique can be extended to compute approximate forward implied volatilities and we implement this extension in the Heston setting. Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface.*

*Keywords: Implied volatility expansions, exponential Levy, ane class, Heston, additive process.*

13-12

S. De Marco, C. Hillairet, A. Jacquier*Shapes of implied volatility with positive mass at zero*

*Abstract: We study the shapes of the implied volatility when the underlying distribution has an atom at zero. We show that the behaviour at small strikes is uniquely determined by the mass of the atom at least up to the third asymptotic order, regardless of the properties of the remaining (absolutely continuous, or singular) distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one can-a priori-distinguish between mass at the origin and a heavy-left-tailed distribution. An atom at zero is found in stochastic models with absorption at the boundary, such as the CEV process, and can be used to model default events, as in the class of jump-to-default structural models of credit risk. We numerically test our model-free result in such examples. Note that while Lee's moment formula tells that implied variance is \emph{at most} asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as Benaim and Friz (09) or Gulisashvili (10) do not apply in this setting-essentially due to the breakdown of Put-Call symmetry-and we rely here on an alternative treatment of the problem.*

*Keywords: Atomic distribution, heavy-tailed distribution, Implied Volatility, smile asymptotics, absorption at zero, CEV model.*

### 2012

#### Working Papers 2012

12-01

S. Jacka, A. Mijatovic*Coupling and tracking of regime-switching martingales*

*Abstract: This paper describes two explicit couplings of standard Brownian motions $B$ and $V$, which naturally extend the mirror coupling and the synchronous coupling and respectively maximise and minimise (uniformly over all time horizons) the coupling time and the tracking error of two regime-switching martingales.*

*The generalised mirror coupling minimizes the coupling time of the two martingales while simultaneously maximising the tracking error for all time horizons. The generalised synchronous coupling maximises the coupling time and minimises the tracking error over all co-adapted couplings. The proofs are based on the Bellman principle.*

*We give counterexamples to the conjectured optimality of the two couplings amongst a wider classes of stochastic integrals.*

*Keywords: generalised mirror and synchronous coupling of Brownian motion, coupling time and tracking error of regime-switching martingales, Bellman principle, continuous-time Markov chains, stochastic integrals*

12-02

D. Crisan, O. Obanubi*Particle Filters with Random Resampling Times*

*Abstract: Particle filters are numerical methods for approximating the solution of the filtering problem which use systems of weighted particles that (typically) evolve according to the law of the signal process. These methods involve a corrective/resampling procedure which eliminates the particles that become redundant and multiplies the ones that contribute most to the resulting approximation. The correction is applied at instances in time called resampling/correction times. Practitioners normally use certain overall characteristics of the approximating system of particles (such as the effective sample size of the system) to determine when to correct the system. As a result, the resampling times are random. However, in the continuous time framework, all existing convergence results apply only to particle filters with deterministic correction times. In this paper, we analyse (continuous time) particle filters where resampling takes place at times that form a sequence of (predictable) stopping times. We prove that, under very general conditions imposed on the sequence of resampling times, the corresponding particle filters converge.*

*The conditions are verified when the resampling times are chosen in accordance to effective sample size of the system of particles, the coefficient of variation of the particles’ weights and, respectively, the (soft) maximum of the particles’ weights. We also deduce central-limit theorem type results for the approximating particle system with random resampling times.*

*Keywords: Stochastic partial differential equation, Filtering. Zakai equation, Particle filters, Sequential Monte-Carlo, Methods. Resampling, Resampling times, Random times, Effective Sample Size, Coefficient of variation, Soft Maximum, Central Limit Theorem.*

12-03

G. Pavliotis, A. Abdulle*Numerical Methods for Stochastic Partial Differential Equations with Multiple Scales*

*Abstract: A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Commun. Pure Appl. Math., 58(11) (2005) 1544–1585]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blömker, M. Hairer, G.A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20(7) (2007) 1721–1744]. For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method.*

*Keywords: Stochastic partial differential equations, Multiscale methods, Averaging, Homogenization, Heterogeneous multiscale method (HMM)*

12-04

M. Ottobre *Long time asymptotics of a Brownian Particle coupled with a random environment with non-diffusive feedback force*

(Stochastic Processes and their Applications, 122 (2012), 844-884)

*Abstract: We study the long time behavior of a Brownian particle moving in an anomalously diffusing field, the evolution of which depends on the particle position. We prove that the process describing the asymptotic behavior of the Brownian particle has bounded (in time) variance when the particle interacts with a subdiffusive field; when the interaction is with a superdiffusive field the variance of the limiting process grows in time as t2γ−1, 1/2 < γ < 1. Two different kinds of superdiffusing (random) environments are considered: one is described through the use of the fractional Laplacian; the other via the Riemann–Liouville fractional integral. The subdiffusive field is modeled through the Riemann–Liouville fractional derivative.*

*Keywords: Anomalous diffusion; Riemann–Liouville fractional derivative (integral); Fractional Laplacian; Continuous time random walk; Lévy flight; Scaling limit; Interface fluctuations.*

12-05

T. Cass, M. Hairer, C. Litterer, S. Tindel**Smoothness of the density for solutions to Gaussian rough differential equations (arXiv preprint)**

*Abstract: We consider stochastic differential equations of the form dYt = V (Yt) dXt+V0 (Yt) dt driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields V0 and V = (V1, . . . , Vd) satisfy H¨ormander’s bracket condition, we demonstrate that Yt admits a smooth density for any t 2 (0, T], provided the driving noise satisfies certain non-degeneracy assumptions. Our analysis relies on an interplay of rough path theory, Malliavin calculus, and the theory of Gaussian processes. Our result applies to a broad range of examples including fractional Brownian motion with Hurst parameter H > 1/4, the Ornstein-Uhlenbeck process and the Brownian bridge returning after time T.*

12-06

A. Mijatovic, M. Urusov *Convergence of Integral Functionals of One-Dimensional Diffusions*

Electronic Communications in Probability (to appear).

*Abstract: In this paper we describe the pathwise behaviour of the integral functional $\int_0^t f(Y_u)\,\dd u$ for any $t\in[0,\zeta]$, where $\zeta$ is (a possibly infinite) exit time of a one-dimensional diffusion process $Y$ from its state space, $f$ is a nonnegative Borel measurable function and the coefficients of the SDE solved by $Y$ are only required to satisfy weak local integrability conditions.*

*Two proofs of the deterministic characterisation of the convergence of such functionals are given: the problem is reduced in two different ways to certain path properties of Brownian motion where either the Williams theorem and the theory of Bessel processes or the first Ray-Knight theorem can be applied to prove the characterisation. As a simple application of the main results we give a short proof of Feller's test for explosion.*

*Keywords: Integral functional; one-dimensional diffusion; local time; Bessel process; Ray-Knight theorem; Williams theorem.*

12-07

*Markov chain approximations for transition densities of Levy processes*

*Abstract: We consider the convergence of a continuous-time Markov chain approximation $X^h$, $h>0$, to an $\mathbb{R}^d$-valued L'evy process $X$. The state space of $X^h$ is an equidistant lattice and its $Q$-matrix is chosen to approximate the generator of $X$. In dimension one ($d=1$), and then under a general sufficient condition for the existence of transition densities of $X$, we establish sharp convergence rates of the normalised probability mass function of $X^h$ to the probability density function of $X$. In higher dimensions ($d>1$), rates of convergence are obtained under a technical condition, which is satisfied when the diffusion matrix is non-degenerate.*

*Keywords: Levy process, continuous-time Markov chain, spectral representation, convergence rates for semi-groups and transition densities.*

12-08

F. Avram, A. Horvath, M. Pistorius *On matrix exponential approximations of the infimum of a spectrally negative Levy process*

*Abstract: We recall four open problems concerning constructing high-order matrix- exponential approximations for the in?mum of a spectrally negative Levy process (with applications to fi?rst-passage/ruin probabilities, the wait- ing time distribution in the M/G/1 queue, pricing of barrier options, etc).*

*On the way, we provide a new approximation, for the perturbed Cram?er- Lundberg model, and recall a remarkable family of (not minimal order) approximations of Johnson and Taa?e [JT89], which ?fit an arbitrarily high number of moments, greatly generalizing the currently used approximations of Renyi, De Vylder and Whitt-Ramsay. Obtaining such approximations which fit the Laplace transform at in?finity as well would be quite useful.*

**Keywords: **Levy process; ?first passage problem; Pollaczek-Khinchine formula; method of moments; matrix-exponential function; admissible Pad?e approximation; Johnson-Taaff?e approximations; two-point Pad?e approximations.

A. Jacquier, P. Roome*Asymptotics of forward implied volatility*

*Abstract: We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including Heston and time-changed exponential Levy models. This expansion applies to both small and large maturities and is based solely on the knowledge of the forward characteristic function of the underlying process. The method is based on sharp large deviations techniques, and allows us to recover (in particular) many results for the spot implied volatility smile. In passing we show (i) that the small-maturity exploding behaviour of forward smiles depends on whether the quadratic variation of the underlying is bounded or not, and (ii) that the forward-start date also has to be rescaled in order to obtain non-trivial small-maturity asymptotics.*

*Keywords: implied volatility, forward volatility, forward-start, large deviations, saddlepoint methods.*

12-10

G. Guo, A. Jacquier, C. Martini, L. Neufcourt*Generalised arbitrage-free SVI volatility surfaces*

*Abstract: In this article we propose a generalisation of the recent work of Gatheral-Jacquier on explicit arbitrage-free parameterisations of implied volatility surfaces. We also discuss extensively the notion of arbitrage freeness and Roger Lee's moment formula using the recent analysis by Roper. We further exhibit an arbitrage-free volatility surface different from Gatheral's SVI parameterisation.*

*Keywords: implied volatility, no-arbitrage, SVI.*

12-11

J. Gatheral, A. Jacquier*Arbitrage-Free SVI Volatility Surfaces*

*Abstract: In this article, we show how to calibrate the widely-used SVI parameterization of the implied volatility smile in such a way as to guarantee the absence of static arbitrage. In particular, we exhibit a large class of arbitrage-free SVI volatility surfaces with a simple closed-form representation. We demonstrate the high quality of typical SVI fits with a numerical example using recent SPX options data.*

*Keywords: implied volatility, no-arbitrage, SVI.*

12-12

A. Jacquier, A. Mijatovic*Large deviations for the extended Heston model: the large-time case*

*Abstract: We study here the large-time behaviour of all continuous affine stochastic volatility models and deduce a closed-form formula for the large-maturity implied volatility smile. Based on refinements of the Gartner-Ellis theorem on the real line, our proof reveals pathological behaviours of the asymptotic smile. In particular, we show that the condition assumed in [10] under which the Heston implied volatility converges to the SVI parameterisation is necessary and sufficient.*

*Keywords: implied volatility, Heston model, asymptotics, large deviations.*

### 2011

**Working Papers 2011**

11-01

J.D. Deuschel, P.K. Friz, A. Jacquier, S. Violante*Marginal density expansions for diffusions and stochastic volatility*

*Abstract: Density expansions for hypoelliptic diffusions $(X^1,...,X^d)$ are revisited. In particular, we are interested in density expansions of the projection $(X_T^1,...,X_T^l)$, at time $T>0$, with $l \leq d$. Global conditions are found which replace the well-known "not-in-cutlocus" condition known from heat-kernel asymptotics; cf. G. Ben Arous (1988). Our small noise expansion allows for a "second order" exponential factor. Applications include tail and implied volatility asymptotics in some correlated stochastic volatility models; in particular, we solve a problem left open by A. Gulisashvili and E.M. Stein (2009).*

*Keywords: Laplace method onWiener space, generalized density expansions in small noise and small time, sub-Riemannian geometry with drift, focal points, stochastic volatility, implied volatility, large strike and small time asymptotics for implied volatility.*

11-02

M. Forde, A. Jacquier, A. Mijatovic*A note on essential smoothness in the Heston model*

Finance & Stochastics 15 (4): 781-784

*Abstract: This note studies an issue relating to essential smoothness that can arise when the theory of large deviations is applied to a certain option pricing formula in the Heston model. The note identifies a gap, based on this issue, in the proof of Corollary 2.4 in [2] and describes how to circumvent it. This completes the proof of Corollary 2.4 in [2] and hence of the main result in [2], which describes the limiting behaviour of the implied volatility smile in the Heston model far from maturity. *

11-03

A. Beskos, D. Crisan, A. Jasra, N. Whiteley**Error bounds and normalizing constants for sequential Monte Carlo in high dimensions**

*Abstract: In a recent paper [3], the Sequential Monte Carlo (SMC) sampler introduced in [12, 19, 24] has been shown to be asymptotically stable in the dimension of the state space d at a cost that is only polynomial in d, when N the number of Monte Carlo samples, is fixed. More precisely, it has been established that the effective sample size (ESS) of the ensuing (approximate) sample and the Monte Carlo error of fixed dimensional marginals will converge as d grows, with a computational cost of O(Nd2). In the present work, further results on SMC methods in high dimensions are provided as d ! 1 and with N fixed. We deduce an explicit bound on the Monte-Carlo error for estimates derived using the SMC sampler and the exact asymptotic relative L2-error of the estimate of the normalizing constant. We also establish marginal propagation of chaos properties of the algorithm. The accuracy in high-dimensions of some approximate SMC-based filtering schemes is also discussed.*

*Keywords: Sequential Monte Carlo, High Dimensions, Propagation of Chaos, Normalizing Constants, Filtering.*

11-04

G. Pavliotis, G. M. Pradas, D. Tseluiko, S. Kalliadasis, D. Papageorgiou* Noise induced state transitions, intermittency and universality in the noisy Kuramoto-Sivashinksy equation *Phys. Rev. Lett. 106, 060602

*Abstract: Consider the effect of pure additive noise on the long-time dynamics of the noisy Kuramoto- Sivashinsky (KS) equation close to the instability onset. When the noise acts only on the first stable mode (highly degenerate), the KS solution undergoes several state transitions, including critical on-off intermittency and stabilized states, as the noise strength increases. Similar results are obtained with the Burgers equation. Such noise-induced transitions are completely characterized through critical exponents, obtaining the same universality class for both equations, and rigorously explained using multiscale techniques.*

11-05

O. Barndorff-Nielsen, F. Benth, A. Veraart *Modelling Electricity Forward Markets by Ambit Fields*

*Abstract: This paper proposes a new modelling framework for electricity forward markets based on so-called ambit fields. The new model can capture many of the stylised facts observed in energy markets and is highly analytically tractable. We give a detailed account on the probabilistic properties of the new type of model, and we discuss martingale conditions, option pricing and change of measure within the new model class. Also, we derive a model for the typically stationary spot price, which is obtained from the forward model through a limiting argument.*

*Keywords: Electricity Markets, Forward Prices, Random Fields, Ambit Fields, Levy Basis, Samuelson Effect, Stochastic Volatility.*

11-06

O. Barndorff-Nielsen, A. Veraart **Stochastic Volatility of Volatility and Variance Risk Premia**

*Abstract: This paper introduces a new class of stochastic volatility models which allows for stochastic volatility of volatility (SVV): Volatility modulated non-Gaussian Ornstein-Uhlenbeck (VMOU) processes. Various probabilistic properties of (integrated) VMOU processes are presented. Further we study the effect of the SVV on the leverage effect and on the presence of long memory. One of the key results in the paper is that we can quantify the impact of the SVV on the (stochastic) dynamics of the variance risk premium (VRP). Moreover, provided the physical and the risk -- neutral probability measures are related through a structure -- preserving change of measure, we obtain an explicit formula for the VRP.*

*Keywords: Stochastic volatility of volatility, Levy process, Ornstein-Uhlenbeck process, variance risk premium, supOU process.*

11-07

T. Cass, C. Litterer, T. Lyons*Integrability estimates for Gaussian rough differential equations (arXiv preprint)*

*Abstract: We derive explicit tail-estimates for the Jacobian of the solution flow for stochastic differential equations driven by Gaussian rough paths. In particular, we deduce that the Jacobian has finite moments of all order for a wide class of Gaussian process including fractional Brownian motion with Hurst parameter H > 1/4. We remark on the relevance of such estimates to a number of significant open problems.*

### Previous Papers

10-01

Crisan D, Manolarakis K *Second order discretization of Backward SDEs*

*Abstract: In [5] the authors suggested a new algorithm for the numerical approximation of a BSDE by merging the cubature method with the first order discretization developed by [3] and [16]. Though the algorithm presented in [5] compared satisfactorily with other methods it lacked the higher order nature of the cubature method due to the use of the low order discretization. In this paper we introduce a second order discretization of the BSDE in the spirit of higher order implicit-explicit schemes for forward SDEs and predictor corrector methods. <\p>*

*Keywords: Backward SDEs, Second order discretization, Numerical analysis.*

07-01

Pavliotis G, Stuart A.M.**Parameter Estimation for Multiscale Diffusions. J. Stat. Phys. 127(4) 741-781**

*Abstract: We study the problem of parameter estimation for time-series possessing two, widely separated, characteristic time scales. The aim is to understand situations where it is desirable to fit a homogenized single-scale model to suchmultiscale data.We demonstrate, numerically and analytically, that if the data is sampled too finely then the parameter fit will fail, in that the correct parameters in the homogenized model are not identified.We also show, numerically and analytically, that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly. The ideas are studied in the context of thermally activated motion in a two-scale potential. However the ideas may be expected to transfer to other situations where it is desirable to fit an averaged or homogenized equation to multiscale data.*

*Keywords: Parameter estimation, multiscale diffusions, stochastic differentialequations, homogenization, maximum likelihood, subsampling.*

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