A new kind of augmentation of filtrations suitable for a change of probability measure by a strict local martingale

Stochastic Processes and their Applications

2015

3 A strict local martingale is a local martingale which is not a martingale. There are few explicit examples 4 of "naturally occurring" strict local martingales with jumps available in the literature. The purpose of 5 this paper is to provide such examples, and to illustrate how they might arise via filtration shrinkage, a 6 phenomenon we would contend is common in applications such as filtering, control, and especially in 7 mathematical finance. We give a method for constructing such examples and analyze one particular method 8 in detail.9

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confidence: 98%

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confidence: 99%

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confidence: 99%

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confidence: 99%

“…where N are the P null sets of G. 28 Assume that Λ contains a sequence of points α n with lim α n = ∞. 29 Proposition 5.…”

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confidence: 99%

See 3 more Smart Citations

Strict local martingales with jumps

Stochastic Processes and their Applications

2015

Progressive Filtration Expansions via a Process, With Applications to Insider Trading

Kchia¹,

Int. J. Theor. Appl. Finan.

2015

In this paper we study progressive filtration expansions with càdlàg processes. Using results from the theory of the weak convergence of σ-fields, we first establish a semimartingale convergence theorem. Then we apply it in a filtration expansion with a process setting and provide sufficient conditions for a semimartingale of the base filtration to remain a semimartingale in the expanded filtration. Applications to the expansion of a Brownian filtration are given. The paper concludes with applications to models of insider trading in financial mathematics. * Let D be the space of càdlàg 1 functions from [0, T ] into R. Let Λ be the set of time changes from [0, T ] into [0, T ], i.e. the set of all continuous strictly increasing functions λ : [0, T ] → [0, T ] such that λ(0) = 0 and λ(T ) = T . We define the Skorohod distance as follows d S (x, y) = inf λ∈Λ ||λ − Id|| ∞ ∨ ||x − y • λ|| ∞for each x and y in D. Let (X n ) n≥1 and X be càdlàg processes (i.e. whose paths are in D), indexed by [0, T ] and defined on (Ω, H, P ). We will write X n P → X when (X n ) n≥1 converges in probability under the Skorohod J 1 topology to X i.e. when the sequence of 1 French acronym for right-continuous with left limits

Strict Local Martingales via Filtration Enlargement

Dandapani

Int. J. Theor. Appl. Finan.

2019

A strict local martingale is a local martingale that is not a martingale. We investigate how such a process might arise from a true martingale as a result of an enlargement of the filtration. We study and implement a particular type of enlargement, initial expansion of filtration, for various stochastic differential equations and provide sufficient conditions in each of these cases such that initial expansion can create a strict local martingale.