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21
Risk vs. ProfitPotential; A Model for Corporate Strategy
 J. Econ. Dynam. Control
, 1996
"... A firm whose net earnings are uncertain, and that is subject to the risk of bankruptcy, must choose between paying dividends and retaining earnings in a liquid reserve. Also, different operating strategies imply different combinations of expected return and variance. We model the firm's cash reserve ..."
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Cited by 27 (0 self)
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A firm whose net earnings are uncertain, and that is subject to the risk of bankruptcy, must choose between paying dividends and retaining earnings in a liquid reserve. Also, different operating strategies imply different combinations of expected return and variance. We model the firm's cash reserve as the difference between the cumulative net earnings and the cumulative dividends. The first is a diffusion (additive), whose drift/volatility pair is chosen dynamically from a finite set, A. The second is an arbitrary nondecreasing process, chosen by the firm. The firm's strategy must be nonclairvoyant. The firm is bankrupt at the first time, T , at which the cash reserve falls to zero (T may be infinite), and the firm's objective is to maximize the expected total discounted dividends from 0 to T , given an initial reserve, x; denote this maximum by V (x). We calculate V explicitly, as a function of the set A and the discount rate. The optimal policy has the form: (1) pay no dividends if ...
Optimal stopping of the maximum process: The maximality principle
, 1998
"... The solution is found to the optimal stopping problem with payoff sup E S 0Z c(Xt) dt 0 where S = (St)t 0 is the maximum process associated with the onedimensional timehomogeneous diffusion X = (Xt)t 0, the function x 7! c(x) is positive and continuous, and the supremum is taken over all stopping ..."
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Cited by 18 (8 self)
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The solution is found to the optimal stopping problem with payoff sup E S 0Z c(Xt) dt 0 where S = (St)t 0 is the maximum process associated with the onedimensional timehomogeneous diffusion X = (Xt)t 0, the function x 7! c(x) is positive and continuous, and the supremum is taken over all stopping times of X for which the integral has finite expectation. It is proved, under no extra conditions, that this problem has a solution, i.e. the payoff is finite and there is an optimal stopping time, if and only if the following maximality principle holds: the firstorder nonlinear differential equation g 0 (s) = 20 g(s)1
Pathwise Inequalities for Local Time: Applications to Skorokhod Embeddings and Optimal Stopping. Annals of Applied Probability
, 2008
"... We develop a class of pathwise inequalities of the form H(Bt) ≥ Mt + F(Lt), whereBt is Brownian motion, Lt its local time at zero and Mt alocal martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to d ..."
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Cited by 12 (2 self)
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We develop a class of pathwise inequalities of the form H(Bt) ≥ Mt + F(Lt), whereBt is Brownian motion, Lt its local time at zero and Mt alocal martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to derive constructions and optimality results of Vallois ’ Skorokhod embeddings. We discuss their financial interpretation in the context of robust pricing and hedging of options written on the local time. In the final part of the paper we use the inequalities to solve a class of optimal stopping problems of the form sup τ E[F(Lτ) − ∫ τ 0 β(Bs)ds]. The solution is given via a minimal solution to a system of differential equations and thus resembles the maximality principle described by Peskir. Throughout, the emphasis is placed on the novelty and simplicity of the techniques. 1. Introduction. The
Inside Information And Stock Fluctuations
, 1999
"... A model of an incomplete market with the incorporation of a new notion of "inside information" is posed. The usual assumption that the stock price is Markovian is modified by adjoining a hidden Markov process to the BlackScholes exponential Brownian motion model for stock fluctuations. The drift ..."
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Cited by 8 (4 self)
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A model of an incomplete market with the incorporation of a new notion of "inside information" is posed. The usual assumption that the stock price is Markovian is modified by adjoining a hidden Markov process to the BlackScholes exponential Brownian motion model for stock fluctuations. The drift and volatility parameters take different values when the hidden Markov process is in different states. For example, it is 0 when there is no subset of the market which has or which believes it has, extra information. However, the hidden process is in state 1 when information is not equally shared by all, and then the behavior of the members in the subset causes increased fluctuations in the stock price. This model
On Waldtype optimal stopping for Brownian motion
 Math. Inst. Aarhus, Preprint Ser
, 1994
"... The solution is presented to all optimal stopping problems of the form: sup E G jB j 0 c where B G0 = (Bt)t jxj1 0 is standard Brownian motion and the supremum is taken over all stopping times for B with finite expectation, while the map G: R+! R satisfies cjxj2 + d for some d 2 R with c> 0 being gi ..."
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Cited by 5 (3 self)
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The solution is presented to all optimal stopping problems of the form: sup E G jB j 0 c where B G0 = (Bt)t jxj1 0 is standard Brownian motion and the supremum is taken over all stopping times for B with finite expectation, while the map G: R+! R satisfies cjxj2 + d for some d 2 R with c> 0 being given and fixed. The optimal stopping time is shown to be the hitting time by the reflecting Brownian motion jBj = (jBtj)t 0 of the set of all ( approximate) maximum points of the map x 7! G(jxj) 0 cx2. The method of proof relies upon Wald’s identity for Brownian motion and simple real analysis arguments. A simple proof of the DubinsJackaSchwarzSheppShiryaev (square root of two) maximal inequality for randomly stopped Brownian motion is given as an application. 1.
Maximum process problems in optimal control theory
, 2001
"... Given a standard Brownian motion (Bt)t 0 and the equation of motion: dXt = vt dt + p 2 dBt ..."
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Cited by 4 (1 self)
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Given a standard Brownian motion (Bt)t 0 and the equation of motion: dXt = vt dt + p 2 dBt
ThreeDimensional Brownian Motion and the Golden Ratio Rule
"... Let X = (Xt)t≥0 be a transient diffusion process in (0, ∞) with the diffusion coefficient σ> 0 and the scale function L such that Xt → ∞ as t → ∞ , let It denote its running minimum for t ≥ 0, and let θ denote the time of its ultimate minimum I ∞. Setting c(i, x) = 1−2L(x)/L(i) we show that the s ..."
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Cited by 3 (1 self)
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Let X = (Xt)t≥0 be a transient diffusion process in (0, ∞) with the diffusion coefficient σ> 0 and the scale function L such that Xt → ∞ as t → ∞ , let It denote its running minimum for t ≥ 0, and let θ denote the time of its ultimate minimum I ∞. Setting c(i, x) = 1−2L(x)/L(i) we show that the stopping time τ ∗ = inf { t ≥ 0  Xt ≥ f∗(It)} minimises E(θ − τ  − θ) over all stopping times τ of X (with finite mean) where the optimal boundary f ∗ can be characterised as the minimal solution to σ 2 (f(i)) L ′ (f(i)) f ′ (i) = − c(i, f(i)) [L(f(i))−L(i)] ∫ f(i) i c ′ i (i, y) [L(y)−L(i)] σ2 (y) L ′ dy (y) staying strictly above the curve h(i) = L −1 (L(i)/2) for i> 0. In particular, when X is the radial part of threedimensional Brownian motion, we find that τ ∗ = inf t ≥ 0 ∣ Xt−It ≥ ϕ
Optimal stopping and maximal inequalities for geometric Brownian motion
 J. Appl. Probab
, 1998
"... Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem: sup E max 0 t Xt 0 c where X = (Xt)t 0 is geometric Brownian motion with drift and volatility> 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, i ..."
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Cited by 2 (2 self)
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Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem: sup E max 0 t Xt 0 c where X = (Xt)t 0 is geometric Brownian motion with drift and volatility> 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if < 0. The optimal stopping time is given by: 3 = inf t> 0 j Xt = g3 max 0 s t Xs where s 7! g3(s) is the maximal solution of the (nonlinear) differential equation:
Controlling the Velocity of Brownian Motion by its Terminal Value
"... Let V = (Vt)t 0 be the OrnsteinUhlenbeck velocity process solving ..."
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Cited by 1 (1 self)
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Let V = (Vt)t 0 be the OrnsteinUhlenbeck velocity process solving
On the Brownian FirstPassage Time Over a OneSided Stochastic Boundary
"... let St = max 0 r t Br be the maximum process associated with B, and let g: R+! R be a (strictly) monotone continuous function satisfying g(s) 0 j Bt g(St)9 G(y) = exp ..."
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Cited by 1 (0 self)
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let St = max 0 r t Br be the maximum process associated with B, and let g: R+! R be a (strictly) monotone continuous function satisfying g(s)<s for all s 0. Let be the firstpassagetime of B over t 7! g(St): Let G be the function defined by: = inf 8 t> 0 j Bt g(St)9 G(y) = exp