A Change-of-Variable Formula with Local Time on Curves

2005 ◽  
Vol 18 (3) ◽  
pp. 499-535 ◽  
Author(s):  
Goran Peskir
Keyword(s):  
Local Time ◽  
Variable Formula ◽  
Change Of Variable

scholarly journals A Fourier Approach for the Level Crossings of Shot Noise Processes with Jumps

2012 ◽  
Vol 49 (01) ◽  
pp. 100-113 ◽  
Author(s):  
Hermine Biermé ◽  
Agnès Desolneux
Keyword(s):  
Shot Noise ◽  
Poisson Point Process ◽  
Functions Of Bounded Variation ◽  
Level Crossing ◽  
Level Crossings ◽  
Variable Formula ◽  
Shot Noise Process ◽  
Change Of Variable ◽  
The Mean ◽  
The Fourier Transform

We use a change-of-variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossing function of shot noise processes with jumps. We illustrate the result in some examples and give some applications. In particular, it allows us to study the asymptotic behavior of the mean number of level crossings as the intensity of the Poisson point process of the shot noise process goes to infinity.

scholarly journals MEAN REVERSION TRADING WITH SEQUENTIAL DEADLINES AND TRANSACTION COSTS

2018 ◽  
Vol 21 (01) ◽  
pp. 1850004 ◽  
Author(s):  
YERKIN KITAPBAYEV ◽  
TIM LEUNG
Keyword(s):  
Integral Equations ◽  
Local Time ◽  
Optimal Timing ◽  
Nonlinear Integral Equations ◽  
Optimal Trading ◽  
Price Process ◽  
Time Space ◽  
Change Of Variable ◽  
Timing Strategies ◽  
Solution Methodology

We study the optimal timing strategies for trading a mean-reverting price process with a finite deadline to enter and a separate finite deadline to exit the market. The price process is modeled by a diffusion with an affine drift that encapsulates a number of well-known models, including the Ornstein–Uhlenbeck (OU) model, Cox–Ingersoll–Ross (CIR) model, Jacobi model, and inhomogeneous geometric Brownian motion (IGBM) model. We analyze three types of trading strategies: (i) the long–short (long to open, short to close) strategy; (ii) the short–long (short to open, long to close) strategy, and (iii) the chooser strategy whereby the trader has the added flexibility to enter the market by taking either a long or short position, and subsequently close the position. For each strategy, we solve an optimal double stopping problem with sequential deadlines, and determine the optimal timing of trades. Our solution methodology utilizes the local time-space calculus of [Peskir (2005) A change-of-variable formula with local time on curves, Journal of Theoretical Probability 18, 499–535] to derive nonlinear integral equations of Volterra-type that uniquely characterize the trading boundaries. The numerical implementation of the integral equations provides examples of the optimal trading boundaries.

scholarly journals Generalized Integral Transforms via the Series Expressions

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 539 ◽  
Author(s):  
Hyun Soo Chung
Keyword(s):  
Integral Transform ◽  
Integral Transforms ◽  
Wiener Space ◽  
New Method ◽  
Variable Formula ◽  
Change Of Variable ◽  
Conventional Methods

From the change of variable formula on the Wiener space, we calculate various integral transforms for functionals on the Wiener space. However, not all functionals can be obtained by using this formula. In the process of calculating the integral transform introduced by Lee, this formula is also used, but it is also not possible to calculate for all the functionals. In this paper, we define a generalized integral transform. We then introduce a new method to evaluate the generalized integral transform for functionals using series expressions. Our method can be used to evaluate various functionals that cannot be calculated by conventional methods.

scholarly journals Perpetual American Defaultable Options in Models with Random Dividends and Partial Information

Risks ◽  
2018 ◽  
Vol 6 (4) ◽  
pp. 127 ◽  
Author(s):  
Pavel V. Gapeev ◽  
Hessah Al Motairi
Keyword(s):  
Free Boundary Problems ◽  
Partial Information ◽  
Asset Price ◽  
Risky Asset ◽  
Boundary Problems ◽  
Variable Formula ◽  
Closed Form Solutions ◽  
Price Process ◽  
Pricing Problems ◽  
Change Of Variable

We present closed-form solutions to the perpetual American dividend-paying put and call option pricing problems in two extensions of the Black–Merton–Scholes model with random dividends under full and partial information. We assume that the dividend rate of the underlying asset price changes its value at a certain random time which has an exponential distribution and is independent of the standard Brownian motion driving the price of the underlying risky asset. In the full information version of the model, it is assumed that this time is observable to the option holder, while in the partial information version of the model, it is assumed that this time is unobservable to the option holder. The optimal exercise times are shown to be the first times at which the underlying risky asset price process hits certain constant levels. The proof is based on the solutions of the associated free-boundary problems and the applications of the change-of-variable formula.

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