65 research outputs found
Dynkin Game of Stochastic Differential Equations with Random Coefficients, and Associated Backward Stochastic Partial Differential Variational Inequality
A Dynkin game is considered for stochastic differential equations with random
coefficients. We first apply Qiu and Tang's maximum principle for backward
stochastic partial differential equations to generalize Krylov estimate for the
distribution of a Markov process to that of a non-Markov process, and establish
a generalized It\^o-Kunita-Wentzell's formula allowing the test function to be
a random field of It\^o's type which takes values in a suitable Sobolev space.
We then prove the verification theorem that the Nash equilibrium point and the
value of the Dynkin game are characterized by the strong solution of the
associated Hamilton-Jacobi-Bellman-Isaacs equation, which is currently a
backward stochastic partial differential variational inequality (BSPDVI, for
short) with two obstacles. We obtain the existence and uniqueness result and a
comparison theorem for strong solution of the BSPDVI. Moreover, we study the
monotonicity on the strong solution of the BSPDVI by the comparison theorem for
BSPDVI and define the free boundaries. Finally, we identify the counterparts
for an optimal stopping time problem as a special Dynkin game.Comment: 40 page
Dynkin Game of Convertible Bonds and Their Optimal Strategy
This paper studies the valuation and optimal strategy of convertible bonds as
a Dynkin game by using the reflected backward stochastic differential equation
method and the variational inequality method. We first reduce such a Dynkin
game to an optimal stopping time problem with state constraint, and then in a
Markovian setting, we investigate the optimal strategy by analyzing the
properties of the corresponding free boundary, including its position,
asymptotics, monotonicity and regularity. We identify situations when call
precedes conversion, and vice versa. Moreover, we show that the irregular
payoff results in the possibly non-monotonic conversion boundary. Surprisingly,
the price of the convertible bond is not necessarily monotonic in time: it may
even increase when time approaches maturity.Comment: 28 pages, 9 figures in Journal of Mathematical Analysis and
Application, 201
Stochastic differential games involving impulse controls and double-obstacle quasi-variational inequalities
We study a two-player zero-sum stochastic differential game with both players
adopting impulse controls, on a finite time horizon. The
Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equation of the game
turns out to be a double-obstacle quasi-variational inequality, therefore the
two obstacles are implicitly given. We prove that the upper and lower value
functions coincide, indeed we show, by means of the dynamic programming
principle for the stochastic differential game, that they are the unique
viscosity solution to the HJBI equation, therefore proving that the game admits
a value
Optimal stopping problems in mathematical finance
This thesis is concerned with the pricing of American-type contingent claims. First, the explicit solutions to the perpetual American compound option pricing problems in the Black-Merton-Scholes model for financial markets are presented. Compound options are financial contracts which give their holders the right (but not the obligation) to buy or sell some other options at certain times in the future by the strike prices given. The method of proof
is based on the reduction of the initial two-step optimal stopping problems for the underlying geometric Brownian motion to appropriate sequences of ordinary one-step problems. The latter are solved through their associated one-sided free-boundary problems and the subsequent martingale verification for ordinary differential operators. The closed form solution to the perpetual
American chooser option pricing problem is also obtained, by means of the analysis of the equivalent two-sided free-boundary problem. Second, an extension of the Black-Merton-Scholes model with piecewise-constant dividend
and volatility rates is considered. The optimal stopping problems related to the pricing of the perpetual American standard put and call options are solved in closed form. The method of proof is based on the reduction of the initial optimal stopping problems to the associated free-boundary problems and the subsequent martingale verification using a local time-space formula. As a result, the explicit algorithms determining the constant hitting thresholds for the underlying asset price process, which provide the optimal exercise boundaries for the options,
are presented. Third, the optimal stopping games associated with perpetual convertible bonds in an extension of the Black-Merton-Scholes model with random dividends under different information flows are studied. In this type of contracts, the writers have a right to withdraw the bonds
before the holders can exercise them, by converting the bonds into assets. The value functions and the stopping boundaries' expressions are derived in closed-form in the case of observable dividend rate policy, which is modelled by a continuous-time Markov chain. The analysis of the associated parabolic-type free-boundary problem, in the case of unobservable dividend rate policy, is also presented and the optimal exercise times are proved to be the first times at which the asset price process hits boundaries depending on the running state of the filtering dividend rate estimate. Moreover, the explicit estimates for the value function and the optimal exercise boundaries, in the case in which the dividend rate is observable by the writers but unobservable by the holders of the bonds, are presented. Finally, the optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model, in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and its maximum
drawdown, are studied. The latter process represents the difference between the running maximum and the current asset value. The optimal stopping times for exercising are shown to be the first times, at which the price of the underlying asset exits some regions restricted by
certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. The closed-form solutions to the equivalent free-boundary problems for the value functions are obtained with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. The optimal exercise boundaries of the perpetual American call, put and strangle options are obtained as solutions of arithmetic equations and first-order nonlinear ordinary differential equations
Weak Solution for a Class of Fully Nonlinear Stochastic Hamilton-Jacobi-Bellman Equations
This paper is concerned with the stochastic Hamilton-Jacobi-Bellman equation
with controlled leading coefficients, which is a type of fully nonlinear
backward stochastic partial differential equation (BSPDE for short). In order
to formulate the weak solution for such kind of BSPDEs, the classical potential
theory is generalized in the backward stochastic framework. The existence and
uniqueness of the weak solution is proved, and for the partially non-Markovian
case, we obtain the associated gradient estimate. As a byproduct, the existence
and uniqueness of solution for a class of degenerate reflected BSPDEs is
discussed as well.Comment: 29 page
Controlled diffusion processes
This article gives an overview of the developments in controlled diffusion
processes, emphasizing key results regarding existence of optimal controls and
their characterization via dynamic programming for a variety of cost criteria
and structural assumptions. Stochastic maximum principle and control under
partial observations (equivalently, control of nonlinear filters) are also
discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The obstacle problem for semilinear parabolic partial integro-differential equations
This paper presents a probabilistic interpretation for the weak Sobolev
solution of the obstacle problem for semilinear parabolic partial
integro-differential equations (PIDEs).
The results of Leandre (1985) concerning the homeomorphic property for the
solution of SDEs with jumps are used to construct random test functions for the
variational equation for such PIDEs. This results in the natural connection
with the associated Reflected Backward Stochastic Differential Equations with
jumps (RBSDEs), namely Feynman Kac's formula for the solution of the PIDEs.
Moreover it gives an application to the pricing and hedging of contingent
claims with constraints in the wealth or portfolio processes in financial
markets including jumps.Comment: 31 page
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