1,721,154 research outputs found
On optimal stopping problems with positive discounting rates and related Laplace transforms of first hitting times in models with geometric Brownian motions
No full textWe derive closed-form solutions to some optimal stopping problems for one-dimensional geometric Brownian motions with positive discounting rates. It is assumed that the original processes can be trapped or reflected or sticky at some fixed lower levels and the conditions on the gain functions imply that the the optimal stopping times turn out to be the first times at which the processes hit some upper level which are to be determined. The proof is based on the reduction of the original optimal stopping problems to the to the equivalent free-boundary problems and the solutions of the latter problems by means of the instantaneous stopping and smooth-fit conditions for the value functions at the optimal stopping boundaries. We also obtain explicit expressions for the Laplace transforms or moment generating functions (with positive exponents or parameters) of the first hitting times for the geometric Brownian motion of given upper levels under various conditions on the parameters of the model. In particular, we determine the upper bounds for the hitting levels and given positive exponents or parameters of the Laplace transforms for which the resulting expectations are finite under various relations between the parameters of the model. Moreover, we determine the upper bounds for the positive exponents or parameters of the Laplace transforms and given hitting levels for which the resulting expectations are finite under various relations between the parameters of the model. The main aim of this short article is to derive closed-form solutions to the optimal stopping problem of (2) for the geometric Brownian motion X defined in (1) with a positive exponential discounting rate λ > 0. We assume that the process X can be trapped or reflected or sticky at some level a > 0 and the gain function G(x) is a twice continuously differentiable positive and strictly increasing concave function on (0, ∞). Optimal stopping problems for one-dimensional diffusion processes with negative exponential discounting rates have been studied after Dynkin (1963) by many authors in the literature including Fakeev (1970), Mucci (1978), Salminen (1985), Øksendal and Reikvam (1998), Alvarez (2001), Dayanik and Karatzas (2003), and Lamberton and Zervos (2013) among others (we refer to Øksendal (1998, Chapter X), Peskir and Shiryaev (2006) and Gapeev and Lerche (2011) for further references). The consideration of optimal stopping problems for diffusions with positive discounting rates was initiated by Shepp and Shiryaev (1996) and then has been continued by other authors in the literature (we refer to Gapeev (2019) and Gapeev (2020) for further references). In this short article, we also present explicit expressions for the Laplace transforms (with positive exponents or parameters) of the first hitting times of given upper levels under various conditions on the parameters of the model (see Borodin and Salminen (2002, Part II) for other computations of the Laplace transforms of first hitting times)
The Gapeev-Shiryaev Conjecture
No full textThe Gapeev-Shiryaev conjecture (originating in Gapeev and Shiryaev (2011) and Gapeev and Shiryaev (2013)) can be broadly stated as follows: Monotonicity of the signal-to-noise ratio implies monotonicity of the optimal stopping boundaries. The conjecture was originally formulated both within (i) sequential testing problems for diffusion processes (where one needs to decide which of the two drifts is being indirectly observed) and (ii) quickest detection problems for diffusion processes (where one needs to detect when the initial drift changes to a new drift). In this paper we present proofs of the Gapeev-Shiryaev conjecture both in (i) the sequential testing setting (under Lipschitz/Holder coefficients of the underlying SDEs) and (ii) the quickest detection setting (under analytic coefficients of the underlying SDEs). The method of proof in the sequential testing setting relies upon a stochastic time change and pathwise comparison arguments. Both arguments break down in the quickest detection setting and get replaced by arguments arising from a stochastic maximum principle for hypoelliptic equations (satisfying Hormander's condition) that is of independent interest. Verification of the Gapeev-Shiryaev conjecture establishes the fact that sequential testing and quickest detection problems with monotone signal-to-noise ratios are amenable to known methods of solution.<br/
The Gapeev-Shiryaev Conjecture
No full textThe Gapeev-Shiryaev conjecture (originating in Gapeev and Shiryaev (2011) and
Gapeev and Shiryaev (2013)) can be broadly stated as follows: Monotonicity of
the signal-to-noise ratio implies monotonicity of the optimal stopping
boundaries. The conjecture was originally formulated both within (i) sequential
testing problems for diffusion processes (where one needs to decide which of
the two drifts is being indirectly observed) and (ii) quickest detection
problems for diffusion processes (where one needs to detect when the initial
drift changes to a new drift). In this paper we present proofs of the
Gapeev-Shiryaev conjecture both in (i) the sequential testing setting (under
Lipschitz/Holder coefficients of the underlying SDEs) and (ii) the quickest
detection setting (under analytic coefficients of the underlying SDEs). The
method of proof in the sequential testing setting relies upon a stochastic time
change and pathwise comparison arguments. Both arguments break down in the
quickest detection setting and get replaced by arguments arising from a
stochastic maximum principle for hypoelliptic equations (satisfying Hormander's
condition) that is of independent interest. Verification of the Gapeev-Shiryaev
conjecture establishes the fact that sequential testing and quickest detection
problems with monotone signal-to-noise ratios are amenable to known methods of
solution.Comment: 24 page
Discounted Optimal Stopping for Maxima in Diffusion Models with Finite Horizon
No full textWe present a solution to some discounted optimal stopping problem for the maximum of a geometric Brownian motion on a finite time interval. The method of proof is based on reducing the initial optimal stopping problem with the continuation region determined by an increasing continuous boundary surface to a parabolic free-boundary problem. Using the change-of-variable formula with local time on surfaces we show that the optimal boundary can be characterized as a unique solution of a nonlinear integral equation. The result can be interpreted as pricing American fixed-strike lookback option in a diffusion model with finite time horizon.Discounted optimal stopping problem, finite horizon, geometric Brownian motion, maximum process, parabolic free-boundary problem, smooth fit, normal reflection, a nonlinear Volterra integral equation of the second kind, boundary surface, a change-of-variable formula with local time on surfaces, American lookback option problem
The Gapeev–Kühn stochastic game driven by a spectrally positive Lévy process
No full textAbstractIn Gapeev and Kühn (2005) [8], the Dynkin game corresponding to perpetual convertible bonds was considered, when driven by a Brownian motion and a compound Poisson process with exponential jumps. We consider the same stochastic game but driven by a spectrally positive Lévy process. We establish a complete solution to the game indicating four principle parameter regimes as well as characterizing the occurrence of continuous and smooth fit. In Gapeev and Kühn (2005) [8], the method of proof was mainly based on solving a free boundary value problem. In this paper, we instead use fluctuation theory and an auxiliary optimal stopping problem to find a solution to the game
Multiple Disorder Problems for Wiener and Compound Poisson Processes With Exponential Jumps
No full textThe multiple disorder problem consists of finding a sequence of stopping times which are as close as possible to the (unknown) times of "disorder" when the distribution of an observed process changes its probability characteristics. We present a formulation and solution of the multiple disorder problem for a Wiener and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial optimal switching problems to the corresponding coupled optimal stopping problems and solving the equivalent coupled free-boundary problems by means of the smooth- and continuous-fit conditions.Multiple disorder problem, Wiener process, compound Poisson process, optimal switching, coupled optimal stopping problem, (integro-differential) coupled free-boundary problem, smooth and continuous fit, Ito-Tanaka-Meyer formula.
The Gapeev-Kühn stochastic game driven by a spectrally positive Lévy process
No full textIn Gapeev and Kühn (2005) [8], the Dynkin game corresponding to perpetual convertible bonds was considered, when driven by a Brownian motion and a compound Poisson process with exponential jumps. We consider the same stochastic game but driven by a spectrally positive Lévy process. We establish a complete solution to the game indicating four principle parameter regimes as well as characterizing the occurrence of continuous and smooth fit. In Gapeev and Kühn (2005) [8], the method of proof was mainly based on solving a free boundary value problem. In this paper, we instead use fluctuation theory and an auxiliary optimal stopping problem to find a solution to the game.Stochastic games Optimal stopping Pasting principles Fluctuation theory Levy processes
Integral Options in Models with Jumps
No full textWe present an explicit solution to the formulated in [17] optimal stopping problem for a geometric compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problem to an integro-differential free-boundary problem where the smooth fit may break down and then be replaced by the continuous fit. The result can be interpreted as pricing perpetual integral options in a model with jumps.Jump process, stochastic differential equation, optimal stopping problem, integral American option, compound Poisson process, Shiryaev´s process, Girsanov´s theorem, Ito´s formula, integrodifferential free-boundary problem, smooth and continuous fit, hypergeometric functions
On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes
No full textWe obtain closed-form expressions for the value of the joint Laplace transform of therunning maximum and minimum of a diffusion-type process stopped at the first time at which theassociated drawdown or drawup process hits a constant level before an independent exponentialrandom time. It is assumed that the coefficients of the diffusion-type process are regular functionsof the current values of its running maximum and minimum. The proof is based on the solution tothe equivalent inhomogeneous ordinary differential boundary-value problem and the applicationof the normal-reflection conditions for the value function at the edges of the state space of theresulting three-dimensional Markov process. The result is related to the computation of probabilitycharacteristics of the take-profit and stop-loss values of a market trader during a given time period.</jats:p
On Maximal Inequalities for some Jump Processes
No full textWe present a solution to the considered in [5] and [22] optimal stopping problem for some jump processes. The method of proof is based on reducing the initial problem to an integro-differential free-boundary problem where the normal reflection and smooth fit may break down and the latter then be replaced by the continuous fit. The derived result is applied for determining the best constants in maximal inequalities for a compound Poisson process with linear drift and exponential jumps.Jump process, stochastic differential equation, maximum process, optimal stopping problem, compound Poisson process, Ito’s formula, integro-differential free-boundary problem, normal reflection, continuous and smooth fit, maximality principle, maximal inequalities
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