125 research outputs found
On an Effective Solution of the Optimal Stopping Problem for Random Walks
We find a solution of the optimal stopping problem for the case when a reward function is an integer function of a random walk on an infinite time interval. It is shown that an optimal stopping time is a first crossing time through a level defined as the largest root of Appell's polynomial associated with the maximum of the random walk. It is also shown that a value function of the optimal stopping problem on the finite interval {0, 1, ? , T} converges with an exponential rate as T approaches infinity to the limit under the assumption that jumps of the random walk are exponentially bounded.optimal stopping; random walk; rate of convergence; Appell polynomials
On a Solution of the Optimal Stopping Problem for Processes with Independent Increments
We discuss a solution of the optimal stopping problem for the case when a reward function is a power function of a process with independent stationary increments (random walks or Levy processes) on an infinite time interval. It is shown that an optimal stopping time is the first crossing time through a level defined as the largest root of the Appell function associated with the maximum of the underlying process.
A quickest detection problem with an observation cost
In the classical quickest detection problem, one must detect as quickly as
possible when a Brownian motion without drift "changes" into a Brownian motion
with positive drift. The change occurs at an unknown "disorder" time with
exponential distribution. There is a penalty for declaring too early that the
change has occurred, and a cost for late detection proportional to the time
between occurrence of the change and the time when the change is declared.
Here, we consider the case where there is also a cost for observing the
process. This stochastic control problem can be formulated using either the
notion of strong solution or of weak solution of the s.d.e. that defines the
observation process. We show that the value function is the same in both cases,
even though no optimal strategy exists in the strong formulation. We determine
the optimal strategy in the weak formulation and show, using a form of the
"principle of smooth fit" and under natural hypotheses on the parameters of the
problem, that the optimal strategy takes the form of a two-threshold policy:
observe only when the posterior probability that the change has already
occurred, given the observations, is larger than a threshold , and
declare that the disorder time has occurred when this posterior probability
exceeds a threshold . The constants and are determined
explicitly from the parameters of the problem.Comment: Published at http://dx.doi.org/10.1214/14-AAP1028 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Esscher transform and the duality principle for multidimensional semimartingales
The duality principle in option pricing aims at simplifying valuation
problems that depend on several variables by associating them to the
corresponding dual option pricing problem. Here, we analyze the duality
principle for options that depend on several assets. The asset price processes
are driven by general semimartingales, and the dual measures are constructed
via an Esscher transformation. As an application, we can relate swap and quanto
options to standard call and put options. Explicit calculations for jump models
are also provided.Comment: Published in at http://dx.doi.org/10.1214/09-AAP600 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On Esscher Transforms in Discrete Finance Models
The object of our study is where each Sn is a m-dimensional stochastic (real valued) vector, i.e. denned on a probability space (Ī©, , P) and adapted to a filtration ( n )0ā¤nā¤N with 0 being the Ļ-algebra consisting of all null sets and their complements. In this paper we interpret as the value of some financial asset k at time n. Remark: If the asset generates dividends or coupon payments, think of as to include these payments (cum dividend process). Think of dividends as being reinvested immediately at the ex-dividend price. Definition 1 (a) A sequence of random vectors where is called a trading strategy. Since our time horizon ends at time N we must always have ĻN ā” 0. The interpretation is obvious: stands for the number of shares of asset k you hold in the time interval [n,n + 1). You must choose Ļn at time n. (b) The sequence of random variables where Sn stands for the payment stream generated by Ļ (set Ļā1 ā” 0
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