Generalisation of Hajek’s Stochastic Comparison Results to Stochastic Sums

Kampen, Joerg

doi:10.1155/2016/1018509

Cited by 3 publications

(3 citation statements)

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“…Our interest in this paper is to combine these results with Green's identity and related properties of the adjoint of fundamental solutions in order to obtain comparison results for multivariate pure diffusions with univariate convex data. This extends generalizations of Hajek's univariate comparison results obtained in [11]. The Hörmander condition seems natural in this context, because a) stronger conditions of degeneracy may lead to regularity constraints or even non-existence of densities, b) comparison of jump diffusions does not hold in general since it does not hold for simple Poisson processes as is shown in [19].…”

Section: Introductionsupporting

confidence: 68%

Comparison results for highly degenerate parabolic equations with univariate convex data and optimal strategies for options on trading accounts

Kampen¹,

Večeř²

2017

Preprint

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For linear multivariate purely second order highly degenerated parabolic equations with univariate convex data, monotonicity of coefficient matrices implies monotonicity of the related value functions. For multivariate data, comparison holds only for trivial coefficients, where we recover and extend existing results for uniformly elliptic equations to highly degenerate equations by a different method of proof based on Green's identity. The results extend to multivariate parabolic equations with first order terms and monotonically increasing univariate data. Related convexity criteria are derived from this new perspective. The univariate data are assumed to have some upper bound on exponential growth. Extensions of the argument to lognormal coordinates allow for applications beyond power options. Representation formulas of Greeks are implied. Multivariate comparison with univariate data is used in order to determine optimal strategies for multivariate passport options. These optimal strategies reveal new features of multivariate passport options compared to univariate passport options due to correlation effects. Passport option values are determined by HJB-Cauchy problems with control spaces of measurable bounded functions. Especially the values of optimal strategy functions of passport options, where the control space of strategy values forms a hypercube, are located on the inverse images of the vertices of that cube under the rotation matrix mapping which defines the diagonalization of the correlation matrix of the underlying assets. Especially, multivariate passport options cannot be reduced to lookback options as in the univariate case. However multivariate passport options inherit from univariate passport options the feature that optimal strategies prescribe switching between long and short limit positions on a high frequency basis. This corresponds to control spaces of measurable functions, where more than Hölder regularity cannot be expected. As this is often not feasible, it is interesting to introduce a new product of symmetric passport options with positive trading position constraints. The comparison result is applied to this new product in the case of one underlying share (next to a money account), where optimal strategies prescribe a maximal limit position in the lower asset at each time. Hence, coefficients related to optimal strategies are not continuous in general, but in a more regular class which is more interesting from the trading perspective and from the point of view of regularity theory than the case of classical passport options.

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Section: Introductionsupporting

confidence: 68%

Comparison results for highly degenerate parabolic equations with univariate convex data and optimal strategies for options on trading accounts

Kampen¹,

Večeř²

2017

Preprint

Self Cite

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“…for some constant c > 0. The purpose of this paper is to provide a different elementary proof of the main result in [10]. In the list of minimal assumptions for comparison we need the existence of continuous solutions of the stochastic differential equations describing the stochastic processes involved, which is ensured by bounded Lipschitz continuous volatility matrices.…”

Section: Introductionmentioning

confidence: 99%

“…) is strictly monoton with respect to time in the whole time interval [0, T ]. This theorem is considered from a different perspective in [10]. Possibilities of generalisations are limited in the sense that (n, n) is not a strong partial convolution pair as is shown in [17].…”

Section: Introductionmentioning

confidence: 99%

On partial convolution and mean comparison

Kampen

2018

Preprint

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We prove mean comparison results from a different perspective, where we introduce the concept of partial convolutions. For a parabolic initial data problem on the whole domain of dimension n we consider data functions which live on a subspace of lower dimension k and coefficient functions which live on the whole space or some subspace of dimension l. The pair of natural numbers (k, l) is called a strong partial convolution pair if for some coordinate transformations the coefficients functions of the equation and the initial data functions live in complementary linear spaces of dimension k and l respectively. Here the qualification 'strong' indicates that this transformation exists without any further restriction concerning the intersection of the original linear subspaces involved, and that refined concepts are possible in this respect. For purely second order parabolic equations we show that (1, n) for any n ≥ 2 is a strong partial convolution pair. As a consequence new criteria for convexity preservation for classes of initial value problems are obtained. A further consequence is mean comparison for univariate convex functions of a considerable class of sums of locally continuous martingales. We leave the problem of determination of all partial convolution pairs as an open problem. In the literature it is observed that (n, n) is not a strong partial convolution pair.

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