scholarly journals On properties of one functional used in software constructions for solving differential games

2021 ◽  
Vol 31 (4) ◽  
pp. 668-696
Author(s):  
A.G. Chentsov
Keyword(s):  
Value Function ◽  
Sufficient Conditions ◽  
Game Problem ◽  
Limit Function ◽  
Phase Constraints ◽  
Nonlinear Differential Game ◽  
Target Set ◽  
Iteration Number ◽  
Game Position ◽  
The Value Function

Nonlinear differential game (DG) is investigated; relaxations of the game problem of guidance are investigated also. The variant of the program iterations method realized in the space of position functions and delivering in limit the value function of the minimax-maximin DG for special functionals of a trajectory is considered. For every game position, this limit function realizes the least size of the target set neighborhood for which, under proportional weakening of phase constraints, the player interested in a guidance yet guarantees its realization. Properties of above-mentioned functionals and limit function are investigated. In particular, sufficient conditions for realization of values of given function under fulfilment of finite iteration number are obtained.

scholarly journals Optimal Harvesting When the Exchange Rate Is a Semimartingale

2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
E. R. Offen ◽  
E. M. Lungu
Keyword(s):  
Exchange Rate ◽  
General Solution ◽  
Value Function ◽  
Sufficient Conditions ◽  
Optimal Harvesting ◽  
Singular Stochastic Control ◽  
Total Income ◽  
Black Scholes ◽  
The Exchange Rate ◽  
The Value Function

We consider harvesting in the Black-Scholes Quanto Market when the exchange rate is being modeled by the process Et=E0exp⁡{Xt}, where Xt is a semimartingale, and we ask the following question: What harvesting strategy γ* and the value function Φ maximize the expected total income of an investment? We formulate a singular stochastic control problem and give sufficient conditions for the existence of an optimal strategy. We found that, if the value function is not too sensitive to changes in the prices of the investments, the problem reduces to that of Lungu and Øksendal. However, the general solution of this problem still remains elusive.

Relaxation of the game problem of guidance connected with alternative in guidance-evasion differential game

2020 ◽  
pp. 196-244
Author(s):  
Aleksandr G. Chentsov
Keyword(s):  
Fixed Point ◽  
Differential Game ◽  
Game Problem ◽  
Time Interval ◽  
Finite Time Interval ◽  
Alternative Theorem ◽  
Phase Constraints ◽  
Target Set ◽  
Program Iteration ◽  
Solvability Set

Differential game (DG) of guidance-evasion for a finite time interval is considered;as parameters, the target set (TS) and the set defining phase constraints (PC) are used.Player I interested in realization of guidance with TS under validity PC uses set-valuedquasistrategies (nonanticipating strategies) and Player II having opposite target uses strategieswith nonanticipating choice of correction instants and finite numbers of such instants.On informative level, the setting corresponds to alternative theorem of N. N. Krasovskii andA. I. Subbotin. For position not belonging to solvability set of Player I, determination ofthe least size of neighborhoods for set-parameters under that Player I guarantees guidance(under weakened conditions) is interested. In article, this scheme is supplemented by priorityelements in questions of TS attainment and PC validity; this is realized by special parameterdefining relation for sizes of corresponding neighborhoods. Under these conditions, a functionof the least size of TS neighborhood is defined by procedure used program iteration methodfor two variants. The above-mentioned function is fixed point for one of two used “program”operators. Special type of the quality functional for which values of the above-mentionedfunction coincide with values of the minimax-maximin games is established.

scholarly journals Differentiability of the Efficient Frontier when Commitment to Risk Sharing is Limited

2006 ◽  
Vol 6 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Thorsten V. Koeppl
Keyword(s):  
Risk Sharing ◽  
Value Function ◽  
Sufficient Conditions ◽  
Efficient Frontier ◽  
Limited Commitment ◽  
History Dependence ◽  
Efficient Allocations ◽  
The Value Function

This paper shows that the value function describing efficient risk sharing with limited commitment is not necessarily differentiable everywhere. We link differentiability of the value function to history dependence of efficient allocations and provide sufficient conditions for both properties.

The value function for time-related decisions

2011 ◽  
Author(s):  
Anouk Festjens ◽  
Siegfried Dewitte ◽  
Enrico Diecidue ◽  
Sabrina Bruyneel
Keyword(s):  
Value Function ◽  
The Value Function

scholarly journals Pricing Perpetual American Put Options with Asset-Dependent Discounting

2021 ◽  
Vol 14 (3) ◽  
pp. 130
Author(s):  
Jonas Al-Hadad ◽  
Zbigniew Palmowski
Keyword(s):  
Value Function ◽  
Asset Price ◽  
Stopping Times ◽  
Martingale Measure ◽  
Put Options ◽  
American Put Options ◽  
Exact Calculations ◽  
Negative Exponential ◽  
American Put ◽  
The Value Function

The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as VAPutω(s)=supτ∈TEs[e−∫0τω(Sw)dw(K−Sτ)+], where T is a family of stopping times, ω is a discount function and E is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process St is a geometric Lévy process with negative exponential jumps, i.e., St=seζt+σBt−∑i=1NtYi. The asset-dependent discounting is reflected in the ω function, so this approach is a generalisation of the classic case when ω is constant. It turns out that under certain conditions on the ω function, the value function VAPutω(s) is convex and can be represented in a closed form. We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of ω such that VAPutω(s) takes a simplified form.

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