Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: We consider the utility portfolio optimization problem of an investor whose
activities are influenced by an exogenous financial risk (like bad weather or
energy shortage) in an incomplete financial market. We work with a fairly
general non-Markovian model, allowing stochastic correlations between the
underlying assets. This important problem in finance and insurance is tackled
by means of backward stochastic differential equations (BSDEs), which have
been shown to be powerful tools in stochastic control. To lay stress on the
importance and the omnipresence of BSDEs in stochastic control, we present
three methods to transform the control problem into a BSDEs. Namely, the
martingale optimality principle introduced by Davis, the martingale representation
and a method based on Itô-Ventzell’s formula. These approaches enable
us to work with portfolio constraints described by closed, not necessarily convex
sets and to get around the classical duality theory of convex analysis. The
solution of the optimization problem can then be simply read from the solution
of the BSDE. An interesting feature of each of the different approaches is that
the generator of the BSDE characterizing the control problem has a quadratic
growth and depends on the form of the set of constraints. We review some
recent advances on the theory of quadratic BSDEs and its applications. There
is no general existence result for multidimensional quadratic BSDEs. In the
one-dimensional case, existence and uniqueness strongly depend on the form
of the terminal condition. Other topics of investigation are measure solutions
of BSDEs, notably measure solutions of BSDE with jumps and numerical approximations.
We extend the equivalence result of Ankirchner et al. (2009)
between existence of classical solutions and existence of measure solutions to
the case of BSDEs driven by a Poisson process with a bounded terminal condition.
We obtain a numerical scheme to approximate measure solutions. In
fact, the existing self-contained construction of measure solutions gives rise
to a numerical scheme for some classes of Lipschitz BSDEs. Two numerical
schemes for quadratic BSDEs introduced in Imkeller et al. (2010) and based,
respectively, on the Cole-Hopf transformation and the truncation procedure
are implemented and the results are compared.
Keywords: BSDE, quadratic growth, measure solutions, martingale theory,
numerical scheme, indifference pricing and hedging, non-tradable underlying,
defaultable claim, utility maximization. / AFRIKAANSE OPSOMMING: Ons beskou die nuts portefeulje optimalisering probleem van ’n belegger wat
se aktiwiteite beïnvloed word deur ’n eksterne finansiele risiko (soos onweer of
’n energie tekort) in ’n onvolledige finansiële mark. Ons werk met ’n redelik
algemene nie-Markoviaanse model, wat stogastiese korrelasies tussen die onderliggende
bates toelaat. Hierdie belangrike probleem in finansies en versekering
is aangepak deur middel van terugwaartse stogastiese differensiaalvergelykings
(TSDEs), wat blyk om ’n onderskeidende metode in stogastiese beheer
te wees. Om klem te lê op die belangrikheid en alomteenwoordigheid van TSDEs
in stogastiese beheer, bespreek ons drie metodes om die beheer probleem
te transformeer na ’n TSDE. Naamlik, die martingale optimaliteits beginsel
van Davis, die martingale voorstelling en ’n metode wat gebaseer is op ’n
formule van Itô-Ventzell. Hierdie benaderings stel ons in staat om te werk
met portefeulje beperkinge wat beskryf word deur geslote, nie noodwendig
konvekse versamelings, en die klassieke dualiteit teorie van konvekse analise te
oorkom. Die oplossing van die optimaliserings probleem kan dan bloot afgelees
word van die oplossing van die TSDE. ’n Interessante kenmerk van elkeen van
die verskillende benaderings is dat die voortbringer van die TSDE wat die
beheer probleem beshryf, kwadratiese groei en afhanglik is van die vorm van
die versameling beperkings. Ons herlei ’n paar onlangse vooruitgange in die
teorie van kwadratiese TSDEs en gepaartgaande toepassings. Daar is geen algemene
bestaanstelling vir multidimensionele kwadratiese TSDEs nie. In die
een-dimensionele geval is bestaan ââen uniekheid sterk afhanklik van die vorm
van die terminale voorwaardes. Ander ondersoek onderwerpe is maatoplossings
van TSDEs, veral maatoplossings van TSDEs met spronge en numeriese
benaderings. Ons brei uit op die ekwivalensie resultate van Ankirchner et al.
(2009) tussen die bestaan van klassieke oplossings en die bestaan van maatoplossings
vir die geval van TSDEs wat gedryf word deur ’n Poisson proses
met begrensde terminale voorwaardes. Ons verkry ’n numeriese skema om
oplossings te benader. Trouens, die bestaande self-vervatte konstruksie van
maatoplossings gee aanleiding tot ’n numeriese skema vir sekere klasse van
Lipschitz TSDEs. Twee numeriese skemas vir kwadratiese TSDEs, bekendgestel
in Imkeller et al. (2010), en gebaseer is, onderskeidelik, op die Cole-Hopf
transformasie en die afknot proses is geïmplementeer en die resultate word
vergelyk.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/17950 |
| Date | 12 1900 |
| Creators | Ndounkeu, Ludovic Tangpi |
| Contributors | Ghomrasni, Raouf, Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 108 p. : ill. |
| Rights | Stellenbosch University |
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