Process-based risk measures and risk-averse control of discrete-time systems

Fan, Jingnan; Ruszczyński, Andrzej

doi:10.1007/s10107-018-1349-2

Cited by 14 publications

(12 citation statements)

References 52 publications

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“…Dentcheva and Ruszczyński (2018) consider Markov risk measures for a countable state space, see also Fan and Ruszczyński (2018a), Fan and Ruszczyński (2018b), and Ruszczyński (2010) for the discrete‐time case. Here, the focus lies on time‐consistent risk measurement related to a fixed reference continuous‐time Markov chain

X = {(X_{t})}_{t \geq 0}

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…However, regarding the existence of Markov chains under convex expectations and their connection to nonlinear ordinary differential equations (ODEs), this restriction could easily be overcome with a slight modification of the construction of the transition operators. Dentcheva and Ruszczyński (2018) consider Markov risk measures for a countable state space, see also Fan and Ruszczyński (2018a), Fan and Ruszczyński (2018b), and Ruszczyński (2010) for the discrete-time case. Here, the focus lies on time-consistent risk measurement related to a fixed reference continuous-time Markov chain = ( ) ≥0 .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Markov chains under nonlinear expectation

Nendel

2020

Mathematical Finance

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In this paper, we consider continuous‐time Markov chains with a finite state space under nonlinear expectations. We define so‐called Q‐operators as an extension of Q‐matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q‐operators in terms of a positive maximum principle, a dual representation by means of Q‐matrices, time‐homogeneous Markov chains under convex expectations, and a class of nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to the case of model uncertainty in the generator. We further derive an explicit primal and dual representation of convex semigroups arising from Markov chains under convex expectations via the Fenchel–Legendre transformation of the generator. We illustrate the results with several numerical examples, where we compute price bounds for European contingent claims under model uncertainty in terms of the rate matrix.

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X = {(X_{t})}_{t \geq 0}

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Markov chains under nonlinear expectation

Nendel

2020

Mathematical Finance

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“…This is nothing else, but the conditional risk form defined in (11). As a function of x, we obtain the conditional risk operator ρ Y |X c, P Y |X .…”

Section: Composite Disintegrationmentioning

confidence: 99%

Risk forms: representation, disintegration, and application to partially observable two-stage systems

Dentcheva

Ruszczyński

2019

Math. Program.

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We introduce the concept of a risk form, which is a real functional of two arguments: a measurable function on a Polish space and a measure on that space. We generalize the duality theory and the Kusuoka representation to this setting. For a risk form acting on a product of Polish spaces, we define marginal and conditional forms and we prove a disintegration formula, which represents a risk form as a composition of its marginal and conditional forms. We apply the proposed approach to two-stage stochastic programming problems with partial information and decision-dependent observation distribution.Proof. Using the support property twice, the translation property, and the normalization property, we obtain the chain of equations:This property was called state-consistency in [9]. Essential role in our analysis will be played by the increasing convex order (the counterpart of the second order stochastic dominance, when smaller outcomes are preferred).Here, [a] + = max(0, a).This concept allows us to consider risk forms consistent with the increasing order.Definition 2.4. A risk form ρ : (X ) × P(X ) → Ê is consistent with the increasing convex order, ifEvidently, consistency with the increasing convex order implies monotonicity and law invariance. We call two functions Z,V ∈ (X ) comonotonic, ifDefinition 2.5. A risk form ρ : (X ) × P(X ) → Ê is comonotonically convex, if for all comonotonic functions Z,V ∈ (X ), all P ∈ P(X ), and all λ ∈ [0, 1], ρ[λ Z + (1 − λ )V, P] ≤ λ ρ[Z, P] + (1 − λ )ρ[V, P].3

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“…In [16], so called Markov risk measures are introduced and an optimization problem is solved in a controlled Markov decision framework both in finite and discounted infinite horizon, where the cost functions are assumed to be bounded. This idea is extended to transient models in [26,27] and to unbounded costs with w-weighted bounds in [28,29,30] and to so called process-based measures in [31] and to partially observable Markov chain frameworks in [32].…”

Section: Introductionmentioning

confidence: 99%

Robust optimal control using conditional risk mappings in infinite horizon

Uğurlu

2018

Journal of Computational and Applied Mathematics

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We use one-step conditional risk mappings to formulate a risk averse version of a total cost problem on a controlled Markov process in discrete time infinite horizon. The nonnegative one step costs are assumed to be lower semi-continuous but not necessarily bounded. We derive the conditions for the existence of the optimal strategies and solve the problem explicitly by giving the robust dynamic programming equations under very mild conditions. We further give an ǫ-optimal approximation to the solution and illustrate our algorithm in two examples of optimal investment and LQ regulator problems.

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Process-based risk measures and risk-averse control of discrete-time systems

Cited by 14 publications

References 52 publications

Markov chains under nonlinear expectation

Markov chains under nonlinear expectation

Risk forms: representation, disintegration, and application to partially observable two-stage systems

Robust optimal control using conditional risk mappings in infinite horizon

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