Bolza Problems with Discontinuous Lagrangians and Lipschitz-Continuity of 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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Robo-Advising: Learning Investors’ Risk Preferences via Portfolio Choices*

Journal of Financial Econometrics ◽

10.1093/jjfinec/nbz040 ◽

2020 ◽

Author(s):

Humoud Alsabah ◽

Agostino Capponi ◽

Octavio Ruiz Lacedelli ◽

Matt Stern

Keyword(s):

Opportunity Cost ◽

Value Function ◽

Risk Preference ◽

Portfolio Decisions ◽

Learning Framework ◽

Portfolio Choices ◽

Trading Decisions ◽

Exploration Exploitation ◽

The Value Function ◽

Over Time

Abstract We introduce a reinforcement learning framework for retail robo-advising. The robo-advisor does not know the investor’s risk preference but learns it over time by observing her portfolio choices in different market environments. We develop an exploration–exploitation algorithm that trades off costly solicitations of portfolio choices by the investor with autonomous trading decisions based on stale estimates of investor’s risk aversion. We show that the approximate value function constructed by the algorithm converges to the value function of an omniscient robo-advisor over a number of periods that is polynomial in the state and action space. By correcting for the investor’s mistakes, the robo-advisor may outperform a stand-alone investor, regardless of the investor’s opportunity cost for making portfolio decisions.

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Metric for Disassembly and Reuse Decisions: Formulation and Validation

Volume 5: 13th Design for Manufacturability and the Lifecycle Conference; 5th Symposium on International Design and Design Education; 10th International Conference on Advanced Vehicle and Tire Technologies ◽

10.1115/detc2008-49878 ◽

2008 ◽

Cited By ~ 1

Author(s):

Vijitashwa Pandey ◽

Deborah Thurston

Keyword(s):

Maximum Likelihood ◽

Value Function ◽

Choice Theory ◽

Decision Makers ◽

Maximum Likelihood Estimates ◽

Likelihood Method ◽

Maximum Value ◽

Long Range Planning ◽

The Relationship ◽

The Value Function

Design for disassembly and reuse focuses on developing methods to minimize difficulty in disassembly for maintenance or reuse. These methods can gain substantially if the relationship between component attributes (material mix, ease of disassembly etc.) and their likelihood of reuse or disposal is understood. For products already in the marketplace, a feedback approach that evaluates willingness of manufacturers or customers (decision makers) to reuse a component can reveal how attributes of a component affect reuse decisions. This paper introduces some metrics and combines them with ones proposed in literature into a measure that captures the overall value of a decision made by the decision makers. The premise is that the decision makers would choose a decision that has the maximum value. Four decisions are considered regarding a component’s fate after recovery ranging from direct reuse to disposal. A method on the lines of discrete choice theory is utilized that uses maximum likelihood estimates to determine the parameters that define the value function. The maximum likelihood method can take inputs from actual decisions made by the decision makers to assess the value function. This function can be used to determine the likelihood that the component takes a certain path (one of the four decisions), taking as input its attributes, which can facilitate long range planning and also help determine ways reuse decisions can be influenced.

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