2006 stochastics of environmental and financial economics

It has the same simple properties as the trawl process.Trawl processes can be used to directly model an object of interest, for example,the exponential of the trawl process has been used

Springer Proceedings in Mathematics & Statistics

Fred Espen Benth

Giulia Di Nunno Editors

Stochastics of Environmental and Financial Economics

Centre of Advanced Study,

Oslo, Norway, 2014–2015

Springer Proceedings in Mathematics & Statistics Volume 138

This book series features volumes composed of selected contributions fromworkshops and conferences in all areas of current research in mathematics andstatistics, including operation research and optimization In addition to an overallevaluation of the interest, scientific quality, and timeliness of each proposal at thehands of the publisher, individual contributions are all refereed to the high qualitystandards of leading journals in the field Thus, this series provides the researchcommunity with well-edited, authoritative reports on developments in the mostexciting areas of mathematical and statistical research today.

More information about this series at http://www.springer.com/series/10533

Fred Espen Benth • Giulia Di Nunno

Editors

Stochastics of Environmental and Financial Economics

Centre of Advanced Study, Oslo, Norway,

Fred Espen Benth

Norway

ISSN 2194-1009 ISSN 2194-1017 (electronic)

Springer Proceedings in Mathematics & Statistics

ISBN 978-3-319-23424-3 ISBN 978-3-319-23425-0 (eBook)

DOI 10.1007/978-3-319-23425-0

Library of Congress Control Number: 2015950032

Mathematics Subject Classification: 93E20, 91G80, 91G10, 91G20, 60H30, 60G07, 35R60, 49L25, 91B76

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Norway is a country rich on natural resources Wind, rain and snow provide us with

a huge resource for clean energy production, while oil and gas have contributedsignificantly, since the early 1970s, to the country’s economic wealth Nowadaysthe income from oil and gas exploitation is invested in the world’s financial markets

to ensure the welfare of future generations With the rising global concerns aboutclimate, using renewable resources for power generation has become more andmore important Bad management of these resources will be a waste that is anegligence to avoid given the right tools

This formed the background and motivation for the research group Stochasticsfor Environmental and Financial Economics (SEFE) at the Centre of AdvancedStudies (CAS) in Oslo, Norway During the academic year 2014–2015, SEFEhosted a number of distinguished professors from universities in Belgium, France,Germany, Italy, Spain, UK and Norway The scientific purpose of the SEFE centrewas to focus on the analysis and management of risk in the environmental andfinancial economics New mathematical models for describing the uncertaindynamics in time and space of weather factors like wind and temperature werestudied, along with sophisticated theories for risk management in energy, com-modity and more conventionalfinancial markets

In September 2014 the research group organized a major international ence on the topics of SEFE, with more than 60 participants and a programmerunning over five days The present volume reflects some of the scientific devel-opments achieved by CAS fellows and invited speakers at this conference All the

confer-14 chapters are stand-alone, peer-reviewed research papers The volume is dividedinto two parts; thefirst part consists of papers devoted to fundamental aspects ofstochastic analysis, whereas in the second part the focus is on particular applications

to environmental andfinancial economics

v

We thank CAS for its generous support and hospitality during the academic year

we organized our SEFE research group We enjoyed the excellent infrastructureCAS offered for doing research

Part I Foundations

Some Recent Developments in Ambit Stochastics 3Ole E Barndorff-Nielsen, Emil Hedevang, Jürgen Schmiegel

and Benedykt Szozda

Functional and Banach Space Stochastic Calculi:

Path-Dependent Kolmogorov Equations Associated

with the Frame of a Brownian Motion 27Andrea Cosso and Francesco Russo

Nonlinear Young Integrals via Fractional Calculus 81Yaozhong Hu and Khoa N Lê

A Weak Limit Theorem for Numerical Approximation

of Brownian Semi-stationary Processes 101Mark Podolskij and Nopporn Thamrongrat

Non-elliptic SPDEs and Ambit Fields: Existence of Densities 121Marta Sanz-Solé and André Süß

Part II Applications

Dynamic Risk Measures and Path-Dependent Second

Order PDEs 147Jocelyne Bion-Nadal

PricingCoCos with a Market Trigger 179José Manuel Corcuera and Arturo Valdivia

vii

Quantification of Model Risk in Quadratic Hedging in Finance 211Catherine Daveloose, Asma Khedher and Michèle Vanmaele

Risk-Sensitive Mean-Field Type Control Under Partial

Observation 243Boualem Djehiche and Hamidou Tembine

Risk Aversion in Modeling of Cap-and-Trade Mechanism

and Optimal Design of Emission Markets 265Paolo Falbo and Juri Hinz

Exponential Ergodicity of the Jump-Diffusion CIR Process 285Peng Jin, Barbara Rüdiger and Chiraz Trabelsi

Optimal Control of Predictive Mean-Field Equations

and Applications to Finance 301BerntØksendal and Agnès Sulem

Modelling the Impact of Wind Power Production

on Electricity Prices by Regime-Switching Lévy

Semistationary Processes 321Almut E.D Veraart

Pricing Options on EU ETS Certificates with a Time-Varying

Market Price of Risk Model 341

Ya Wen and Rüdiger Kiesel

Part I Foundations

Ole E Barndorff-Nielsen, Emil Hedevang, Jürgen Schmiegel

and Benedykt Szozda

Abstract Some of the recent developments in the rapidly expanding field of Ambit

Stochastics are here reviewed After a brief recall of the framework of Ambit chastics, two topics are considered: (i) Methods of modelling and inference forvolatility/intermittency processes and fields; (ii) Universal laws in turbulence andfinance in relation to temporal processes This review complements two other recentexpositions

1 Introduction

Ambit Stochastics is a general framework for the modelling and study of dynamicprocesses in space-time The present paper outlines some of the recent developments

in the area, with particular reference to finance and the statistical theory of

Stochastics

O.E Barndorff-Nielsen (B) · E Hedevang · B Szozda

Department of Mathematics, Aarhus University, Ny Munkegade 118,

F.E Benth and G Di Nunno (eds.), Stochastics of Environmental

and Financial Economics, Springer Proceedings in Mathematics and Statistics 138,

DOI 10.1007/978-3-319-23425-0_1

3

4 O.E Barndorff-Nielsen et al.

A key characteristic of the Ambit Stochastics framework, which distinguishes thisfrom other approaches, is that beyond the most basic kind of random input it alsospecifically incorporates additional, often drastically changing, inputs referred to asvolatility or intermittency

Such “additional” random fluctuations generally vary, in time and/or in space, in

regard to intensity (activity rate and duration) and amplitude Typically the

volatil-ity/intermittency may be further classified into continuous and discrete (i.e jumps)elements, and long and short term effects In turbulence the key concept of energydissipation is subsumed under that of volatility/intermittency

The concept of (stochastic) volatility/intermittency is of major importance in many fields of science Thus volatility/intermittency has a central role in mathematical finance and financial econometrics, in turbulence, in rain and cloud studies and other aspects of environmental science, in relation to nanoscale emitters, magne- tohydrodynamics, and to liquid mixtures of chemicals, and last but not least in the physics of fusion plasmas.

As described here, volatility/intermittency is a relative concept, and its

mean-ing depends on the particular settmean-ing under investigation Once that meanmean-ing isclarified the question is how to assess the volatility/intermittency empirically andthen to describe it in stochastic terms, for incorporation in a suitable probabilisticmodel Important issues concern the modelling of propagating stochastic volatil-ity/intermittency fields and the question of predictability of volatility/intermittency

the concepts involved by two examples The modelling of volatility/intermittencyand energy dissipation is a main theme in Ambit Stochastics and several approaches

Stochastics has been to take the cue from recognised stylised features—or ity traits—in various scientific areas, particularly turbulence, as the basis for modelbuilding; and in turn to seek new such traits using the models as tools We discusscertain universal features observed in finance and turbulence and indicate ways to

time

A(x, t)

X (x, t)

Fig 1 A spatio-temporal ambit field The value Y (x, t) of the field at the point marked by the black

region The circles of varying sizes indicate the stochastic volatility/intermittency By considering the field along the dotted path in space-time an ambit process is obtained

where

Q (x, t) =



Here t denotes time while x gives the position in d-dimensional Euclidean space.

are deterministic weight functions, and L denotes a Lévy basis (i.e an independently

fields representing aspects of the volatility/intermittency In Ambit Stochastics the

shows a sketch of the concepts

The development of Y along a curve in space-time is termed an ambit process.

As will be exemplified below, ambit processes are not in general semimartingales,

even in the purely temporal case, i.e where there is no spatial component x.

6 O.E Barndorff-Nielsen et al.

In a recent extension the structure (1) is generalised to

Y (x, t) = μ +



a further volatility/intermittency trait The relatively new concept of metatime isinstrumental in generalising subordination of stochastic processes by time change

measures We return to this concept and its applications in the next section and referalso to the discussion given in [8]

Note however that in addition to modelling volatility/intermittency through the

It might be thought that ambit sets have no role in purely temporal modelling

In many cases it is possible to choose specifications of the volatility/intermittency

making the models especially tractable analytically We recall that the importance

of the concept of selfdecomposability rests primarily on the possibility to representselfdecomposable variates as stochastic integrals with respect to Lévy processes,see [32]

So far, the main applications of ambit stochastics has been to turbulence and, to

a lesser degree, to financial econometrics and to bioimaging An important potentialarea of applications is to particle transport in fluids

2.2 Existence of Ambit Fields

The paper [25] develops a general theory for integrals

X (x, t) =



Rd×Rh (x, y, t, s) M(dx dx) where h is a predictable stochastic function and M is a dispersive signed random

measure Central to this is that the authors establish a notion of characteristic triplet

of M, extending that known in the purely temporal case A major problem solved in

that regard has been to merge the time and space aspects in a general and tractablefashion Armed with that notion they determine the conditions for existence of the

apply An important property here is that now predictable integrands are allowed

Ambit Stochastics generally, and in particular to superposition of stochastic volatilitymodels, is discussed

Below we briefly discuss how the metatime change is incorporated in the

L ([0, 1] d+1) is equal in law to μ Let (γ, Σ, ν) be the characteristic triplet of μ.

Thusγ ∈ R, Σ ≥ 0 and ν is a Lévy measure on R.

B(R d+1)} That is the sets T(A) and T(B) are disjoint whenever A, B ∈ B(R d+1)

n=0A n ) = ∪∞

n=0T(A n ) whenever A n , ∪∞

T (A) = Leb d+1(T(A)) for all A ∈ B(R d+1) Here and in what follows, Leb k

meta-times cf [11] Suppose also thatλ ∈ I D(R) is the law associated to T and that

λ ∼ I D(β, 0, ρ) Thus β ≥ 0 and ρ is a Lévy measure such that ρ(R−) = 0 and



R(1 ∧ x) ρ(dx) < ∞.

whereμ sis given by μ s = μ s for any s≥ 0

discontinuity (see [25]) By rewriting the stochastic integral in the right-hand side of(3) as

X (x, t) =



Rd+1 H (x, ξ, t, s) L T (dξds), with H (x, ξ, t, s) = 1 A (x,t) (ξ, s)g(x, ξ, t, s)σ(ξ, s) we can use [25, Theorem 4.1]

g (x, ξ, t, s) are deterministic This gives us that X is well defined for all (x, t) if the

following hold almost surely for all(x, t) ∈ R d+1:

8 O.E Barndorff-Nielsen et al.

are termed Brownian semistationary processes—or BSS for short Here the setting

chronome-ter T (that is, an increasing, càdlàg and stochastically continuous process ranging

the volatility/intermittency If T has stationary increments then the process Y is

(1/2, 1) and (1, 3/2] Note also that the sample path behaviour is drastically different

whenν ∈ (1, 3/2] Further, the sample paths are purely discontinuous if ν ∈ (1/2, 1)

The cases whereν ∈ (1/2, 1) have a particular bearing in the context of turbulence,

theory of statistical turbulence, cf [3,33]

Lévy process is referred to as the class of Lévy semistationary processes—or LSSprocesses for short Such processes are discussed in [8,24, Sect 3.7] and referencestherein

3.2 Trawl Processes

The simplest non-trivial kind of ambit field is perhaps the trawl process, introduced

in [2] In a trawl process, the kernel function and the volatility field are constantand equal to 1, and so the process is given entirely by the ambit set and the Lévy

The process is by construction stationary Depending on the purpose of the modelling,

the time component of the ambit set A may or may not be supported on the negative real axis When the time component of A is supported on the negative real axis,

we obtain a causal model Despite their apparent simplicity, trawl processes possess

function (i.e the distinguished logarithm of the characteristic function) of Y is given

by

autocovariance and autocorrelation it follows that

E[Y (t)] = |A| E[L ],

var(Y (t)) = |A| var(L ),

r(t) := cov(Y (t), Y (0)) = |A ∩ A(t)| var(L ), (11)

ρ(t) := cov(Y (t), Y (0))

var(Y (0)) =

|A ∩ A(t)|

From this we conclude the following The one-dimensional marginal distribution

is determined entirely in terms of the size (not shape) of the ambit set and thedistribution of the Lévy seed; given any infinitely divisible distribution there existstrawl processes having this distribution as the one-dimensional marginal; and theautocorrelation is determined entirely by the size of the overlap of the ambit sets,

that is, by the shape of the ambit set A Thus we can specify the autocorrelation

and marginal distribution independently of each other It is, for example, easy to

1The Lévy seed L (x) at x of a Lévy basis L with control measure ν is a random variable with the

property that C {ζ ‡L(A)} =A C {ζ ‡L (x)} ν(dx) For a homogeneous Lévy basis, the distribution

of the seed is independent of x.

10 O.E Barndorff-Nielsen et al.

trawl processes are obtained These processes are studied in detail in [5] and applied

to high frequency stock market data

processes to trawl fields It has the same simple properties as the trawl process.Trawl processes can be used to directly model an object of interest, for example,the exponential of the trawl process has been used to model the energy dissipation,see the next section, or they can be used as a component in a composite model, forexample to model the volatility/intermittency in a Brownian semistationary process

4 Modelling of Volatility/Intermittency/Energy

Dissipation

A very general approach to specifying volatility/intermittency fields for inclusion in

an ambit field, as in (1), is to takeτ = σ2as being given by a Lévy-driven Volterrafield, either directly as

When the goal is to have stationary volatility/intermittency fields, such as in

mod-elling homogeneous turbulence, that can be achieved by choosing L to be

which are by far the most common, particularly in turbulence studies Inhomogeneity

can be expressed both by not having f of translation type and by taking the Lévy basis L inhomogeneous.

In the following we discuss two aspects of the volatility/intermittency elling issue Trawl processes have proved to be a useful tool for the modelling ofvolatility/intermittency and in particular for the modelling of the energy dissipation,

the construction of volatility/intermittency fields

4.1 The Energy Dissipation

repro-duce the main stylized features of the (surrogate) energy dissipation observed for awide range of datasets Those stylized features include the one-dimensional marginaldistributions and the scaling and self-scaling of the correlators

c p ,q (s) = E[ε(t) p ε(t + s) q]

The correlator is a natural analogue to the autocorrelation when one considers a

that the correlator of the surrogate energy dissipation displays a scaling behaviourfor a certain range of lags,

intermit-tency exponent Typical values are in the range 0.1 to 0.2 The intermittency exponent

quantifies the deviation from Kolmogorov’s 1941 theory and emphasizes the role ofintermittency (i.e volatility) in turbulence In some cases, however, the scaling range

of the correlators can be quite small and therefore it can be difficult to determine the

value of the scaling exponents, especially when p and q are large Therefore one also

considers the correlator of one order as a function of a correlator of another order Inthis case, self-scaling is observed, i.e., the one correlator is proportional to a power

of the other correlator,

c p ,q (s) ∝ c p ,q (s) τ(p,q;p ,q ) , (16)whereτ(p, q; p ...

F.E Benth and G Di Nunno (eds.), Stochastics of Environmental< /small>

and Financial Economics, Springer Proceedings in Mathematics and Statistics 138,... in the case of the analysed assets from S&P500, a family of distributionsexists such that all distributions of log-returns are members of this family and suchthat the variance of the log-returns... predicted from the shape, not location and scale, of theone-point distribution of the energy dissipation alone

that the one-dimensional marginal of the logarithm of the energy dissipation is well