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On some Brownian functionals and their applications to moments in the lognormal stochastic volatility model

Jacek Jakubowski, Maciej Wiśniewolski (2013)

Studia Mathematica

We find a probabilistic representation of the Laplace transform of some special functional of geometric Brownian motion using squared Bessel and radial Ornstein-Uhlenbeck processes. Knowing the transition density functions of these processes, we obtain closed formulas for certain expectations of the relevant functional. Among other things we compute the Laplace transform of the exponent of the T transforms of Brownian motion with drift used by Donati-Martin, Matsumoto, and Yor in a variety of identities...

Optimal investment under stochastic volatility and power type utility function

Benchaabane, Abbes, Benchettah, Azzedine (2011)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 37F21, 70H20, 37L40, 37C40, 91G80, 93E20.In this work we will study a problem of optimal investment in financial markets with stochastic volatility with small parameter. We used the averaging method of Bogoliubov for limited development for the optimal strategies when the small parameter of the model tends to zero and the limit for the optimal strategy and demonstrated the convergence of these optimal strategies.

Optimal position targeting with stochastic linear-quadratic costs

Stefan Ankirchner, Thomas Kruse (2015)

Banach Center Publications

We consider the dynamic control problem of attaining a target position at a finite time T, while minimizing a linear-quadratic cost functional depending on the position and speed. We assume that the coefficients of the linear-quadratic cost functional are stochastic processes adapted to a Brownian filtration. We provide a probabilistic solution in terms of two coupled backward stochastic differential equations possessing a singularity at the terminal time T. We verify optimality of the candidate...

Option pricing in a CEV model with liquidity costs

Krzysztof Turek (2016)

Applicationes Mathematicae

The goal of this paper is to make an attempt to generalise the model of pricing European options with an illiquid underlying asset considered by Rogers and Singh (2010). We assume that an investor's decisions have only a temporary effect on the price, which is proportional to the square of the change of the number of asset units in the investor's portfolio. We also assume that the underlying asset price follows a CEV model. To prove existence and uniqueness of the solution, we use techniques similar...

Option valuation under the VG process by a DG method

Jiří Hozman, Tomáš Tichý (2021)

Applications of Mathematics

The paper presents a discontinuous Galerkin method for solving partial integro-differential equations arising from the European as well as American option pricing when the underlying asset follows an exponential variance gamma process. For practical purposes of numerical solving we introduce the modified option pricing problem resulting from a localization to a bounded domain and an approximation of small jumps, and we discuss the related error estimates. Then we employ a robust numerical procedure...

Parabolic variational inequalities with generalized reflecting directions

Eduard Rotenstein (2015)

Open Mathematics

We study, in a Hilbert framework, some abstract parabolic variational inequalities, governed by reflecting subgradients with multiplicative perturbation, of the following type: y´(t)+ Ay(t)+0.t Θ(t,y(t)) ∂φ(y(t))∋f(t,y(t)),y(0) = y0,t ∈[0,T] where A is a linear self-adjoint operator, ∂φ is the subdifferential operator of a proper lower semicontinuous convex function φ defined on a suitable Hilbert space, and Θ is the perturbing term which acts on the set of reflecting directions, destroying the...

Pricing bonds and CDS in the model with rating migration induced by a Cox process

Jacek Jakubowski, Mariusz Niewęgłowski (2008)

Banach Center Publications

We investigate the properties of a rating migration process assuming that it is given by subordination of a discrete time Markov chain and a Cox process. The problem of pricing of defaultable bonds with fractional recovery of par value with rating migration and credit default swaps is considered. As an example of applications of our results, we give an explicit solution to the pricing problem in a model with short rate and intensity processes given by the solution of a two-dimensional Ornstein-Uhlenbeck...

Probabilistic properties of the continuous double auction

Martin Šmíd (2012)

Kybernetika

In this paper we formulate a general model of the continuous double auction. We (recursively) describe the distribution of the model. As a useful by-product, we give a (recursive) analytic description of the distribution of the process of the best quotes (bid and ask).

Real-valued conditional convex risk measures in Lp(ℱ, R)

Treviño-Aguilar Erick (2011)

ESAIM: Proceedings

The numerical representation of convex risk measures beyond essentially bounded financial positions is an important topic which has been the theme of recent literature. In other direction, it has been discussed the assessment of essentially bounded risks taking explicitly new information into account, i.e., conditional convex risk measures. In this paper we combine these two lines of research. We discuss the numerical representation of conditional...

Representation of Itô integrals by Lebesgue/Bochner integrals

Qi Lü, Jiongmin Yong, Xu Zhang (2012)

Journal of the European Mathematical Society

In [Yong 2004], it was proved that as long as the integrand has certain properties, the corresponding Itô integral can be written as a (parameterized) Lebesgue integral (or a Bochner integral). In this paper, we show that such a question can be answered in a more positive and refined way. To do this, we need to characterize the dual of the Banach space of some vector-valued stochastic processes having different integrability with respect to the time variable and the probability measure. The later...

Robust estimates of certain large deviation probabilities for controlled semi-martingales

Hideo Nagai (2015)

Banach Center Publications

Motivated by downside risk minimization on the wealth process in an incomplete market model, we have studied in the recent work the asymptotic behavior as time horizon T → ∞ of the minimizing probability that the empirical mean of a controlled semi-martingale falls below a certain level on the time horizon T. This asymptotic behavior relates to a risk-sensitive stochastic control problem in the risk-averse case. Indeed, we obtained an expression of the decay rate of the probability by the Legendre...

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