arXiv subjects

Ari-Pekka Perkkiö

Publications and source records attributed to Ari-Pekka Perkkiö.

At least 19 recordsLinked to original sources

arXiv2025

Optimal Operation and Valuation of Electricity Storages in Intraday Markets

This paper applies computational techniques of convex stochastic optimization to optimal operation and valuation of electricity storages in the face of uncertain electricity prices. Our valuations are based on the indifference pricing principle, which builds on optimal trading strategies and calibrates to the user's financial position, market views and risk preferences. The underlying optimization problem is solved with the Stochastic Dual Dynamic Programming algorithm which is applicable to various specifications of storages, and it allows for e.g. hard constraints on storage capacity and charging speed. We illustrate the approach in intraday trading where the agent charges or discharges a battery over a finite number of delivery periods, and the electricity prices are subject to bid-ask spreads and significant uncertainty. Optimal strategies are found in a matter of minutes on a regular PC. We find that the corresponding trading strategies and battery valuations vary consistently with respect to the agent's risk preferences as well as the physical characteristics of the battery.

math.OC↗

arXiv2025

Dynamic programming and dimensionality in convex stochastic optimization and control

This paper studies stochastic optimization problems and associated Bellman equations in formats that allow for reduced dimensionality of the cost-to-go functions. In particular, we study stochastic control problems in the ``decision-hazard-decision'' form where at each stage, the system state is controlled both by predictable as well as adapted controls. Such an information structure may result in a lower dimensional system state than what is required in more traditional ``decision-hazard'' or ``hazard-decision'' formulations. The dimension is critical for the complexity of numerical dynamic programming algorithms and, in particular, for cutting plane schemes such as the stochastic dual dynamic programming algorithm. Our main result characterizes optimal solutions and optimum values in terms of solutions to generalized Bellman equations. Existence of solutions to the Bellman equations is established under general conditions that do not require compactness. We allow for general randomness but show that, in the Markovian case, the dimensionality of the Bellman equations reduces with respect to randomness just like in more traditional control formulations.

math.OC↗

arXiv2025

Optimal Operation and Valuation of Electricity Storages

This paper applies computational techniques of convex stochastic optimization to optimal operation and valuation of electricity storages in the face of uncertain electricity prices. Our approach is applicable to various specifications of storages, and it allows for e.g.\ hard constraints on storage capacity and charging speed. Our valuations are based on the indifference pricing principle, which builds on optimal trading strategies and calibrates to the user's initial position, market views and risk preferences. We illustrate the effects of storage capacity and charging speed by numerically computing the valuations using stochastic dual dynamic programming.

math.OC↗

arXiv2022

Dual solutions in convex stochastic optimization

This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex space of random variables. It implies, in particular, the necessity of scenario-wise optimality conditions that are behind many fundamental results in operations research, stochastic optimal control and financial mathematics. Our analysis builds on the theory of Fréchet spaces of random variables whose topological dual can be identified with the direct sum of another space of random variables and a space of singular functionals. The results are illustrated by deriving sufficient and necessary optimality conditions for several more specific problem classes. We obtain significant extensions to earlier models e.g.\ on stochastic optimal control, portfolio optimization and mathematical programming.

math.OC↗

arXiv2022

Duality in convex stochastic optimization

This paper studies duality and optimality conditions in general convex stochastic optimization problems introduced by Rockafellar and Wets in 1976. We derive an explicit dual problem in terms of two dual variables, one of which is the shadow price of information while the other one gives the marginal cost of a perturbation much like in classical Lagrangian duality. Existence of primal solutions and the absence of duality gap are obtained without compactness or boundedness assumptions. In the context of financial mathematics, the relaxed assumptions are satisfied under the well-known no-arbitrage condition and the reasonable asymptotic elasticity condition of the utility function. We extend classical portfolio optimization duality theory to problems of optimal semi-static hedging. Besides financial mathematics, we obtain several new frameworks in stochastic programming and stochastic optimal control.

math.OC↗

arXiv2022

Michael selections and Castaing representations with cadlag functions

Michael's selection theorem implies that a closed convex nonempty-valued mapping from the Sorgenfrey line to a euclidean space is inner semicontinuous if and only if the mapping can be represented as the image closure of right-continuous selections of the mapping. This article gives necessary and sufficient conditions for the representation to hold for cadlag selections, i.e., for selections that are right-continuous and have left limits. The characterization is motivated by continuous time stochastic optimization problems over cadlag processes. Here, an application to integral functionals of cadlag functions is given.

math.OC↗

arXiv2022

Dynamic programming in convex stochastic optimization

This paper studies the dynamic programming principle for general convex stochastic optimization problems introduced by Rockafellar and Wets in [30]. We extend the applicability of the theory by relaxing compactness and boundedness assumptions. In the context of financial mathematics, the relaxed assumption are satisfied under the well-known no-arbitrage condition and the reasonable asymptotic elasticity condition of the utility function. Besides financial mathematics, we obtain several new results in linear and nonlinear stochastic programming and stochastic optimal control.

math.OC↗

arXiv2020

Dual spaces of cadlag processes

This article characterizes topological duals of spaces of cadlag processes. We obtain extensions of functional analytic results of Dellacherie and Meyer that underlie many fundamental results in stochastic analysis. In particular, we obtain a characterization of the dual of cadlag processes of class $(D)$ in terms of optional measures of essentially bounded variation. When specialized to regular processes, we find extensions of the main result of Bismut \cite{bis78} on projections of continuous processes. The dual characterizations yield existence results and optimality conditions for many fundamental problems from optimal stopping to singular stochastic control well beyond classical formulations.

math.PR↗

arXiv2020

Topological duals of Banach function spaces

This paper studies topological duals of Banach function spaces (BFS). We assume a finite measure but our arguments extend to general locally convex function spaces whose topology is generated by seminorms that satisfy the usual BFS axioms. The dual is identified with the direct sum of another space of random variables (Köthe dual), a space of purely finitely additive measures and the annihilator of $L^\infty$. In the special case of rearrangement invariant spaces, the second component in the dual vanishes and we obtain various classical as well as new duality results e.g. on Lebesgue, Orlicz, Lorentz-Orlicz spaces and spaces of finite moments. Beyond rearrangement invariant spaces, we find the topological duals of Musielak-Orlicz spaces and those associated with general convex risk measures.

math.PR↗

arXiv2019

Optimal stopping without Snell envelopes

This paper proves the existence of optimal stopping times via elementary functional analytic arguments. The problem is first relaxed into a convex optimization problem over a closed convex subset of the unit ball of the dual of a Banach space. The existence of optimal solutions then follows from the Banach--Alaoglu compactness theorem and the Krein--Millman theorem on extreme points of convex sets. This approach seems to give the most general existence results known to date. Applying convex duality to the relaxed problem gives a dual problem and optimality conditions in terms of martingales that dominate the reward process.

math.OC↗

arXiv2018

Convex integral functionals of cadlag processes

This article characterizes conjugates and subdifferentials of convex integral functionals over linear spaces of cadlag stochastic processes. The approach is based on new measurability results on the Skorokhod space and new interchange rules of integral functionals that are developed in the article. The main results provide a general approach to apply convex duality in a variety of optimization problems ranging from optimal stopping to singular stochastic control and mathematical finance.

math.OC↗

arXiv2017

Convex duality in nonlinear optimal transport

This article studies problems of optimal transport, by embedding them in a general functional analytic framework of convex optimization. This provides a unified treatment of a large class of related problems in probability theory and allows for generalizations of the classical problem formulations. General results on convex duality yield dual problems and optimality conditions for these problems. When the objective takes the form of a convex integral functional, we obtain more explicit optimality conditions and establish the existence of solutions for a relaxed formulation of the problem. This covers, in particular, the mass transportation problem and its nonlinear generalizations.

math.PR↗

arXiv2017

Convex integral functionals of regular processes

This article gives dual representations for convex integral functionals on the linear space of regular processes. This space turns out to be a Banach space containing many more familiar classes of stochastic processes and its dual can be identified with the space of optional Radon measures with essentially bounded variation. Combined with classical Banach space techniques, our results allow for a systematic treatment of stochastic optimization problems over BV processes and, in particular, yields a maximum principle for a general class of singular stochastic control problems.

math.PR↗

arXiv2017

Conjugates of integral functionals on continuous functions

This article gives necessary and sufficient conditions for the dual representation of Rockafellar in (Integrals which are convex functionals. II, Pacific J. Math., 39:439--469, 1971) for integral functionals on the space of continuous functions.

math.FA↗

arXiv2016

Optional and predictable projections of normal integrands and convex-valued processes

This article studies optional and predictable projections of integrands and convex-valued stochastic processes. The existence and uniqueness are shown under general conditions that are analogous to those for conditional expectations of integrands and random sets. In the convex case, duality correspondences between the projections and projections of epigraphs are given. These results are used to study projections of set-valued integrands. Consistently with the general theory of stochastic processes, projections are not constructed using reference measures on the optional and predictable sigma-algebras.

math.PR↗

arXiv2016

Convex integral functionals of processes of bounded variation

This article characterizes conjugates and subdifferentials of convex integral functionals over the linear space $\mathcal N^\infty$ of stochastic processes of essentially bounded variation (BV) when $\mathcal N^\infty$ is identified with the Banach dual of the space of regular processes. Our proofs are based on new results on the interchange of integration and minimization of integral functionals over BV processes. Under mild conditions, the domain of the conjugate is shown to be contained in the space of semimartingales which leads to several applications in the duality theory in stochastic control and mathematical finance.

math.OC↗

arXiv2016

Convex duality in optimal investment and contingent claim valuation in illiquid markets

This paper studies convex duality in optimal investment and contingent claim valuation in markets where traded assets may be subject to nonlinear trading costs and portfolio constraints. Under fairly general conditions, the dual expressions decompose into tree terms, corresponding to the agent's risk preferences, trading costs and portfolio constraints, respectively. The dual representations are shown to be valid when the market model satisfies an appropriate generalization of the no-arbitrage condition and the agent's utility function satisfies an appropriate generalization of asymptotic elasticity conditions. When applied to classical liquid market models or models with bid-ask spreads, we recover well-known pricing formulas in terms of martingale measures and consistent price systems. Building on the general theory of convex stochastic optimization, we also derive optimality conditions in terms of an extended notion of a "shadow price".

q-fin.MF↗

arXiv2016

Shadow price of information in discrete time stochastic optimization

The shadow price of information has played a central role in stochastic optimization ever since its introduction by Rockafellar and Wets in the mid-seventies. This article studies the concept in an extended formulation of the problem and gives relaxed sufficient conditions for its existence. We allow for general adapted decision strategies, which enables one to establish the existence of solutions and the absence of a duality gap e.g. in various problems of financial mathematics where the usual boundedness assumptions fail. As applications, we calculate conjugates and subdifferentials of integral functionals and conditional expectations of normal integrands. We also give a dual form of the general dynamic programming recursion that characterizes shadow prices of information.

math.OC↗