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This paper has been motivated by general considerations on the topic of Risk Measures, which essentially are convex monotone maps defined on spaces of random variables, possibly with the so-called Fatou property. We show first that the celebrated Namioka-Klee theorem for linear, positive functionals holds also for convex monotone maps π on Frechet lattices. It is well-known among the...
We study some properties of a continuous local martingale stopped at the first time when its range (the difference between the running maximum and minimum) reaches a certain threshold. The laws and the conditional laws of its value, maximum, and minimum at this time are simple and do not depend on the local martingale in question. As a consequence, the price and hedge of options which mature when...
We investigate differentiability properties of monetary utility functions. At the same time we give a counter-example—important in finance—to automatic continuity for concave functions.
We study the exponential utility indifference valuation of a contingent claim H when asset prices are given by a general semimartingale S. Under mild assumptions on H and S, we prove that a no-arbitrage type condition is fulfilled if and only if H has a certain representation. In this case, the indifference value can be written in terms of processes from that representation, which is useful in two...
According to the well-known Doob’s lemma, the expected number of crossings of every fixed interval (a,b) by trajectories of a bounded martingale (Xn) is finite on the infinite time interval. For such a random sequence (r.s.) with an extra condition that Xn takes no more than N, N<∞, values at each moment n≥1, this result was...
The goal of this paper is to study the immersion property through its links with credit risk modelling. The construction of a credit model by the enlargement of a reference filtration with the progressive knowledge of a credit event occurrence has become a standard for reduced form modelling. It is known that such a construction rises mathematical difficulties, mainly relied to the properties of the...
We investigate optimal consumption and investment problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall. We formulate various utility maximisation problems, which can be solved explicitly. We compare the optimal solutions in form of optimal value, optimal control and optimal wealth to analogous problems under additional uniform risk bounds. Our proofs...
This paper studies a comparison theorem for solutions of stochastic differential equations and its generalization to the multi-dimensional case. We show, that even though the proof of the generalized theorem follows that of the one-dimensional comparison theorem, the multi-dimensional case requires a different condition on the drift coefficient, known in the theory of differential equations as Kamke-Wazewski...
We consider a diffusion type process being a weak solution of Itô’s equation $$dX^\varepsilon_t=b(\omega,X^\varepsilon_t/\varepsilon)dt+\sigma(\omega,X^\varepsilon_t/\varepsilon)dB_t$$ relative to fixed initial condition , Brownian motion (Bt)t≥0, and ergodic stationary random process (b(ω,u),σ(ω,u))u∈ℝ...
Let , be correlated geometric Brownian motions. We consider the following problem: find the stopping time τ*≤T such that $$\sup_{\tau\in[0,T]}\mathsf{E}[S_\tau^1-S_\tau^2]=\mathsf{E}[S_{\tau^*}^1-S_{\tau^*}^2]$$ where the supremum is taken over all stopping times from [0,T]. A similar problem, but on infinite interval, was studied by MacDonald and Siegel (Int...
We present a novel arbitrage-related notion for markets with transaction costs in discrete time and characterize it in terms of price systems. Pertinence of this concept is demonstrated. A discussion of the case with one risky asset and an outlook on continuous-time models complement the main result.
This paper considers the generalized Bayesian disorder problem in the discrete time case with two types of the penalty function—the linear and the nonlinear ones. The main results for these cases are given in Theorems 1 and 2, respectively.
Discrete and continuous growth optimal portfolio optimization over long time horizon is studied. Proportional transaction costs consisting of fixed proportional plus proportional to the volume of transaction are considered. An obligatory diversification is imposed, which allows the process of portions of capital invested in assets to be ergodic. Existence of solutions to suitable Bellman equations...
We give an approximation to geometric fractional Brownian motion. The approximation is a simple corollary to a ‘teletraffic’ functional central limit theorem by Gaigalas and Kaj in (Bernoulli 9:671–703, 2003). We analyze the central limit theorem of Gaigalas and Kaj from the point of view of semimartingale limit theorems to have a better understanding of the arbitrage in the limit model. With this...
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