2,970 research outputs found
Conditioning an additive functional of a markov chain to stay non-negative. I, Survival for a long time
Let (X-t)(t >= 0) be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E -> R \ {0}, and let (phi(t))(t >= 0) be an additive functional defined by phi(t) = integral(0)(t)(X-s) ds. We consider the case in which the process (phi(t))(t >= 0) is oscillating and that in which (phi(t))(t >= 0) has a negative drift. In each of these cases, we condition the process (X-t, phi(t))(t >= 0) on the event that (phi(t))(t >= 0) is nonnegative until time T and prove weak convergence of the conditioned process as T -> infinity
Game-theoretic approach to risk-sensitive benchmarked asset management
In this article we consider a game theoretic approach to the Risk-Sensitive
Benchmarked Asset Management problem (RSBAM) of Davis and Lleo \cite{DL}. In
particular, we consider a stochastic differential game between two players,
namely, the investor who has a power utility while the second player represents
the market which tries to minimize the expected payoff of the investor. The
market does this by modulating a stochastic benchmark that the investor needs
to outperform. We obtain an explicit expression for the optimal pair of
strategies as for both the players.Comment: Forthcoming in Risk and Decision Analysis. arXiv admin note: text
overlap with arXiv:0905.4740 by other author
On the regularity of American options with regime-switching uncertainty
We study the regularity of the stochastic representation of the solution of a
class of initial-boundary value problems related to a regime-switching
diffusion. This representation is related to the value function of a
finite-horizon optimal stopping problem such as the price of an American-style
option in finance. We show continuity and smoothness of the value function
using coupling and time-change techniques. As an application, we find the
minimal payoff scenario for the holder of an American-style option in the
presence of regime-switching uncertainty under the assumption that the
transition rates are known to lie within level-dependent compact sets.Comment: 22 pages, to appear in Stochastic Processes and their Application
Monotonicity of the value function for a two-dimensional optimal stopping problem
We consider a pair of stochastic processes satisfying the equation
driven by a Brownian motion and study the monotonicity and
continuity in of the value function
, where the supremum is taken
over stopping times with respect to the filtration generated by . Our
results can successfully be applied to pricing American options where is
the discounted price of an asset while is given by a stochastic volatility
model such as those proposed by Heston or Hull and White. The main method of
proof is based on time-change and coupling.Comment: Published in at http://dx.doi.org/10.1214/13-AAP956 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Markov chain approximations to scale functions of L\'evy processes
We introduce a general algorithm for the computation of the scale functions
of a spectrally negative L\'evy process , based on a natural weak
approximation of via upwards skip-free continuous-time Markov chains with
stationary independent increments. The algorithm consists of evaluating a
finite linear recursion with its (nonnegative) coefficients given explicitly in
terms of the L\'evy triplet of . Thus it is easy to implement and
numerically stable. Our main result establishes sharp rates of convergence of
this algorithm providing an explicit link between the semimartingale
characteristics of and its scale functions, not unlike the one-dimensional
It\^o diffusion setting, where scale functions are expressed in terms of
certain integrals of the coefficients of the governing SDE.Comment: 46 pages, 4 figure
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