Skorohod's Representation Theorem and Optimal Strategies for Markets with Frictions

Chau, Huy N.; Rásonyi, Miklós

doi:10.1137/16m1081336

Cited by 7 publications

(4 citation statements)

References 63 publications

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“…See [13, p. 130] and [22, p. 77] for historical notes, and [5] for the case where μ 0 is not separable. Some other related references are [3,4,[8][9][10][14][15][16]18,20].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A Strong Version of the Skorohod Representation Theorem

Pratelli¹,

Rigo

2022

J Theor Probab

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For each $$n\ge 0$$ n ≥ 0 , let $$\mu _n$$ μ n be a tight probability measure on the Borel $$\sigma $$ σ -field of a metric space S. Let $$(T,{\mathcal {C}})$$ ( T , C ) be a measurable space such that the diagonal $$\bigl \{(t,t):t\in T\bigr \}$$ { ( t , t ) : t ∈ T } belongs to $${\mathcal {C}}\otimes {\mathcal {C}}$$ C ⊗ C . Fix a measurable function $$g:S\rightarrow T$$ g : S → T and suppose $$\mu _n=\mu _0$$ μ n = μ 0 on $$g^{-1}({\mathcal {C}})$$ g - 1 ( C ) for all $$n\ge 0$$ n ≥ 0 . Necessary and sufficient conditions for the existence of S-valued random variables $$X_n$$ X n , defined on the same probability space and satisfying $$\begin{aligned} X_n\overset{\text {a.s.}}{\longrightarrow }X_0,\quad X_n\sim \mu _n\,\text { and } \,g(X_n)=g(X_0)\,\text { for all }n\ge 0, \end{aligned}$$ X n ⟶ a.s. X 0 , X n ∼ μ n and g ( X n ) = g ( X 0 ) for all n ≥ 0 , are given. Such conditions are then applied to several examples. The tightness condition on $$\mu _0$$ μ 0 can be dropped at the price of some assumptions on S and g.

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“…See [13, p. 130] and [22, p. 77] for historical notes, and [5] for the case where μ 0 is not separable. Some other related references are [3,4,[8][9][10][14][15][16]18,20].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A Strong Version of the Skorohod Representation Theorem

Pratelli¹,

Rigo

2022

J Theor Probab

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“…, n the payoffs at time T k are path-independent and given by Y T k = f k (S T k ) and X T k = g k (S T k ) where f k , g k : (0, ∞) → R are measurable functions and 0 ≤ f k ≤ g k . The payoff function H is given by (1). We will assume the following integrability condition…”

Section: Existence Resultsmentioning

confidence: 99%

“…An interesting question which is left for the future, is whether by allowing the investor to randomize from the start (in the spirit of [1]) will provide an existence of an optimal hedging strategy.…”

Section: Remarkmentioning

confidence: 99%

“…The difference t = X t − Y t is the penalty which the seller pays to the buyer for the contract cancellation. In short, if the seller will exercise at a stopping time σ ≤ T and the buyer at a stopping time τ ≤ T then the former pays to the latter the amount H (σ, τ ) where H (σ, τ ) := X σ I σ <τ + Y τ I τ ≤σ (1) and we set I Q = 1 if an event Q occurs and I Q = 0 if not.…”

Section: Introductionmentioning

confidence: 99%

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On shortfall risk minimization for game options

Dolinsky¹

2020

Modern Stochastics: Theory and Applications

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In this paper we study the existence of an optimal hedging strategy for the shortfall risk measure in the game options setup. We consider the continuous time Black-Scholes (BS) model. Our first result says that in the case where the game contingent claim (GCC) can be exercised only on a finite set of times, there exists an optimal strategy. Our second and main result is an example which demonstrates that for the case where the GCC can be stopped on the whole time interval, optimal portfolio strategies need not always exist.

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