scholarly journals Skew-Unfolding the Skorokhod Reflection of a Continuous Semimartingale

2014 ◽  
pp. 349-376
Author(s):  
Tomoyuki Ichiba ◽  
Ioannis Karatzas
Keyword(s):  
Continuous Semimartingale ◽  
Skorokhod Reflection

scholarly journals On an optimal extraction problem with regime switching

2018 ◽  
Vol 50 (3) ◽  
pp. 671-705 ◽  
Author(s):  
Giorgio Ferrari ◽  
Shuzhen Yang
Keyword(s):  
Optimal Control ◽  
Regime Switching ◽  
Spot Price ◽  
Discounted Cash Flow ◽  
State Process ◽  
Running Cost ◽  
The One ◽  
A Company ◽  
Skorokhod Reflection ◽  
The Value Function

AbstractIn this paper we study a finite-fuel two-dimensional degenerate singular stochastic control problem under regime switching motivated by the optimal irreversible extraction problem of an exhaustible commodity. A company extracts a natural resource from a reserve with finite capacity and sells it in the market at a spot price that evolves according to a Brownian motion with volatility modulated by a two-state Markov chain. In this setting, the company aims at finding the extraction rule that maximizes its expected discounted cash flow, net of the costs of extraction and maintenance of the reserve. We provide expressions for both the value function and the optimal control. On the one hand, if the running cost for the maintenance of the reserve is a convex function of the reserve level, the optimal extraction rule prescribes a Skorokhod reflection of the (optimally) controlled state process at a certain state and price-dependent threshold. On the other hand, in the presence of a concave running cost function, it is optimal to instantaneously deplete the reserve at the time at which the commodity's price exceeds an endogenously determined critical level. In both cases, the threshold triggering the optimal control is given in terms of the optimal stopping boundary of an auxiliary family of perpetual optimal selling problems with regime switching.

Backward SDEs with two barriers and continuous coefficient: an existence result

2004 ◽  
Vol 41 (1) ◽  
pp. 162-175 ◽  
Author(s):  
Jean-Pierre Lepeltier ◽  
Jaime San Martín
Keyword(s):  
Existence Result ◽  
Existence Of A Solution ◽  
Continuous Coefficient ◽  
Continuous Semimartingale ◽  
Backward Sde ◽  
Backward Sdes ◽  
Two Barriers

In this work we prove the existence of a solution for a doubly reflected backward SDE with a continuous linearly increasing coefficient in the case where the barriers L and U are such that L < U on [0,T) and there exists a continuous semimartingale between L and U.

KUNITA-TYPE STOCHASTIC FLOWS OF HOMEOMORPHISMS IN EUCLIDEAN SPACE

2007 ◽  
Vol 10 (04) ◽  
pp. 523-538 ◽  
Author(s):  
ZONGXIA LIANG
Keyword(s):  
Differential Equation ◽  
Euclidean Space ◽  
Stochastic Differential Equation ◽  
Local Characteristic ◽  
Stochastic Flow ◽  
Stochastic Flows ◽  
Continuous Semimartingale ◽  
Lipschitz Conditions

In this paper we prove Kunita-type stochastic differential equation (SDE) [Formula: see text], t > s, driven by a spatial continuous semimartingale F (x, t) =(F1 (x, t), …, Fm (x, t)), x ∈ ℜm, with local characteristic (a, b), in the sense of Kunita (Chap. 3 of Ref. 10), can produce a stochastic flow of homeomorphisms of ℜm into itself almost surely under the (a, b) satisfies non-Lipschitz conditions. This result bases on recent works12 by the author.

A note on integral representations of the Skorokhod map

2016 ◽  
Vol 53 (1) ◽  
pp. 293-298 ◽  
Author(s):  
Patrick Buckingham ◽  
Brian Fralix ◽  
Offer Kella
Keyword(s):  
Continuous Function ◽  
Integral Representation ◽  
Bounded Variation ◽  
Integral Representations ◽  
The One ◽  
Skorokhod Reflection

Abstract We present a very short derivation of the integral representation of the two-sided Skorokhod reflection Z of a continuous function X of bounded variation, which is a generalization of the integral representation of the one-sided map featured in Anantharam and Konstantopoulos (2011) and Konstantopoulos et al. (1996). We also show that Z satisfies a simpler integral representation when additional conditions are imposed on X.

scholarly journals A Note on Asymptotic Exponential Arbitrage with Exponentially Decaying Failure Probability

2013 ◽  
Vol 50 (03) ◽  
pp. 801-809 ◽  
Author(s):  
Kai Du ◽  
Ariel David Neufeld
Keyword(s):  
Large Deviations ◽  
Growth Condition ◽  
Failure Probability ◽  
Quantitative Form ◽  
Continuous Semimartingale ◽  
The Mean ◽  
Mean Variance

The goal of this paper is to prove a result conjectured in Föllmer and Schachermayer (2007) in a slightly more general form. Suppose that S is a continuous semimartingale and satisfies a large deviations estimate; this is a particular growth condition on the mean-variance tradeoff process of S. We show that S then allows asymptotic exponential arbitrage with exponentially decaying failure probability, which is a strong and quantitative form of long-term arbitrage. In contrast to Föllmer and Schachermayer (2007), our result does not assume that S is a diffusion, nor does it need any ergodicity assumption.

scholarly journals A Note on Asymptotic Exponential Arbitrage with Exponentially Decaying Failure Probability

2013 ◽  
Vol 50 (3) ◽  
pp. 801-809 ◽  
Author(s):  
Kai Du ◽  
Ariel David Neufeld
Keyword(s):  
Large Deviations ◽  
Growth Condition ◽  
Failure Probability ◽  
Quantitative Form ◽  
Continuous Semimartingale ◽  
The Mean ◽  
Mean Variance

The goal of this paper is to prove a result conjectured in Föllmer and Schachermayer (2007) in a slightly more general form. Suppose that S is a continuous semimartingale and satisfies a large deviations estimate; this is a particular growth condition on the mean-variance tradeoff process of S. We show that S then allows asymptotic exponential arbitrage with exponentially decaying failure probability, which is a strong and quantitative form of long-term arbitrage. In contrast to Föllmer and Schachermayer (2007), our result does not assume that S is a diffusion, nor does it need any ergodicity assumption.

scholarly journals The functional Itō formula under the family of continuous semimartingale measures

2016 ◽  
Vol 16 (04) ◽  
pp. 1650010 ◽  
Author(s):  
Harald Oberhauser
Keyword(s):  
Continuous Dependence ◽  
Quadratic Variation ◽  
Stochastic Integrals ◽  
Itô’S Formula ◽  
Itô Formula ◽  
Ito Formula ◽  
Underlying Process ◽  
Continuous Semimartingale ◽  
The Family ◽  
Nonlinear Functionals

Dupire [16] introduced a notion of smoothness for functionals of paths and arrived at a generalization of Itō’s formula that applies to functionals with a continuous dependence on the trajectories of the underlying process. In this paper, we study nonlinear functionals that do not have such continuity. By revisiting old work of Bichteler and Karandikar we show that one can construct pathwise versions of complex functionals like the quadratic variation, stochastic integrals or Itō processes that are still regular enough such that a functional Itō-formula applies.

scholarly journals Nonexplosion and Pathwise Uniqueness of Stochastic Differential Equation Driven by Continuous Semimartingale with Non-Lipschitz Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Jinxia Wang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Sufficient Conditions ◽  
Pathwise Uniqueness ◽  
Strong Uniqueness ◽  
Continuous Semimartingale

We study a class of stochastic differential equations driven by semimartingale with non-Lipschitz coefficients. New sufficient conditions on the strong uniqueness and the nonexplosion are derived ford-dimensional stochastic differential equations onRd(d>2)with non-Lipschitz coefficients, which extend and improve Fei’s results.

Backward Stochastic PDE and Imperfect Hedging

2003 ◽  
Vol 06 (07) ◽  
pp. 663-692 ◽  
Author(s):  
M. Mania ◽  
R. Tevzadze
Keyword(s):  
Financial Market ◽  
Asset Prices ◽  
Random Function ◽  
Value Function ◽  
Market Model ◽  
Stochastic Pde ◽  
Continuous Semimartingale ◽  
Incomplete Financial Market ◽  
Mean Variance ◽  
The Value Function

We consider a problem of minimization of a hedging error, measured by a positive convex random function, in an incomplete financial market model, where the dynamics of asset prices is given by an Rd-valued continuous semimartingale. Under some regularity assumptions we derive a backward stochastic PDE for the value function of the problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As an example the case of mean-variance hedging is considered.

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