A Parallel between Brownian Bridges and Gamma Bridges

Émery, Michel; Yor, Marc

doi:10.2977/prims/1145475488

Cited by 33 publications

(27 citation statements)

References 11 publications

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“…Suppose we replace by the approximate problem,

v^{n} (t, x, γ) = \underset{u}{trueprefixinf} E [κ \int_{t}^{T} u^{2} (s) normald s + λ σ^{2} \int_{t}^{T} {(γ (s) - X^{u} (s))}^{2} normald s + n {(X^{u} (T) - 1)}^{2}],

where the infimum is over all adapted and integrable u with

d X^{u} (s) = u (s) d s, X^{u} (t) = x .

Observe now that

X^{u}

has no fixed terminal condition. Using that

γ (. / m)

is a gamma bridge on

[t m, T m]

with underlying mean growth rate equal to 1, it follows from corollary 1 of Émery and Yor () that the infinitesimal generator of the gamma bridge γ is given by

m \int_{0}^{1} (f (γ (t) + (1 - γ (t)) z) - f (γ (t))) {(1 - z)}^{T m - t m - 1} \frac{1}{z} d z,

where f is a function on

R_{+}

with bounded variation on compacts. If we assume that

v^{n}

is sufficiently regular, it should satisfy

\begin{matrix} true \\ left v_{t}^{n} \end{matrix}

…”

Section: Resultsmentioning

confidence: 99%

“…Under a gamma bridge

{(γ (s))}_{t \leq s \leq T}

with

γ (t) = γ

, we understand a process of the form

γ (s) = γ + \frac{L (s) - L (t)}{L (T) - L (t)} (1 - γ)

for a gamma process L so that starting with γ at t , we take the remaining part

1 - γ

proportional to the remaining relative portion of L . Note that

\frac{L (s) - L (t)}{L (T) - L (t)}

t \leq s \leq T

, is again a gamma bridge and independent of

\frac{L (t)}{L (T)}

(Émery and Yor , p. 673). The reader will note that we have not considered auctions in the current model.…”

Section: A Framework For Using a Vwap Benchmarkmentioning

confidence: 99%

See 1 more Smart Citation

Optimal Execution of a Vwap Order: A Stochastic Control Approach

Frei

Westray

2013

Mathematical Finance

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We consider the optimal liquidation of a position of stock (long or short) where trading has a temporary market impact on the price. The aim is to minimize both the mean and variance of the order slippage with respect to a benchmark given by the market volume-weighted average price (VWAP). In this setting, we introduce a new model for the relative volume curve which allows simultaneously for accurate data fit, economic justification, and mathematical tractability. Tackling the resulting optimization problem using a stochastic control approach, we derive and solve the corresponding Hamilton-Jacobi-Bellman equation to give an explicit characterization of the optimal trading rate and liquidation trajectory.

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“…Suppose we replace by the approximate problem,

v^{n} (t, x, γ) = \underset{u}{trueprefixinf} E [κ \int_{t}^{T} u^{2} (s) normald s + λ σ^{2} \int_{t}^{T} {(γ (s) - X^{u} (s))}^{2} normald s + n {(X^{u} (T) - 1)}^{2}],

where the infimum is over all adapted and integrable u with

d X^{u} (s) = u (s) d s, X^{u} (t) = x .

Observe now that

X^{u}

has no fixed terminal condition. Using that

γ (. / m)

is a gamma bridge on

[t m, T m]

with underlying mean growth rate equal to 1, it follows from corollary 1 of Émery and Yor () that the infinitesimal generator of the gamma bridge γ is given by

m \int_{0}^{1} (f (γ (t) + (1 - γ (t)) z) - f (γ (t))) {(1 - z)}^{T m - t m - 1} \frac{1}{z} d z,

where f is a function on

R_{+}

with bounded variation on compacts. If we assume that

v^{n}

is sufficiently regular, it should satisfy

\begin{matrix} true \\ left v_{t}^{n} \end{matrix}

…”

Section: Resultsmentioning

confidence: 99%

“…Under a gamma bridge

{(γ (s))}_{t \leq s \leq T}

with

γ (t) = γ

, we understand a process of the form

γ (s) = γ + \frac{L (s) - L (t)}{L (T) - L (t)} (1 - γ)

for a gamma process L so that starting with γ at t , we take the remaining part

1 - γ

proportional to the remaining relative portion of L . Note that

\frac{L (s) - L (t)}{L (T) - L (t)}

t \leq s \leq T

, is again a gamma bridge and independent of

\frac{L (t)}{L (T)}

(Émery and Yor , p. 673). The reader will note that we have not considered auctions in the current model.…”

Section: A Framework For Using a Vwap Benchmarkmentioning

confidence: 99%

Optimal Execution of a Vwap Order: A Stochastic Control Approach

Frei

Westray

2013

Mathematical Finance

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“…(1.f ) As an end to this introduction, let us point out that, the subordinators S α,β we represent here, and their symmetric counterparts, are arguably the most studied and used among Lévy processes, and this, for the following reasons: for α > 0, S α,β is obtained by Esscher transform (see [7,26]) from the fundamental stable (α) subordinator, hence it "retains" some scaling property, while the Gamma process ( [6], [27], [28], [29], [30]) and the variance-gamma processes ( [15], [16], [17]) have some fundamental quasi-invariance properties, which make them comparable, in some respect, to Brownian motion with drift.…”

mentioning

confidence: 99%

Some Explicit Krein Representations of Certain Subordinators, Including the Gamma Process

Donati-Martin¹,

Yor²

2006

Publ. Res. Inst. Math. Sci.

Self Cite

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“…The gamma bridge has some nice independence properties as discussed by Emery and Yor (2004). We propose the following construction of bridges.…”

Section: The Gamma Bridges and Their Derived Estimatorsmentioning

confidence: 98%

Incorporating the Stochastic Process Setup in Parameter Estimation

Sant

Caruana

2014

Methodol Comput Appl Probab

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Estimation problems within the context of stochastic processes are usually studied with the help of statistical asymptotic theory and proposed estimators are tested with the use of simulated data. For processes with stationary increments it is customary to use differenced time series, treating them as selections from the increments' distribution. Though distributionally correct, this approach throws away most information related to the stochastic process setup. In this paper we consider the above problems with reference to parameter estimation of a gamma process. Using the derived bridge processes we propose estimators whose properties we investigate in contrast to the gamma-increments MLE. The proposed estimators have a smaller bias, comparable variance and offer a look at the time-evolution of the parameter estimation. Empirical results are presented.

show abstract

A Parallel between Brownian Bridges and Gamma Bridges

Abstract: Some properties of the Gamma bridges (obtained by conditioning the Gamma subordinator to take a given value at a given time) are investigated; similarities with the Brownian bridges are emphasized.

Cited by 33 publications

References 11 publications

Optimal Execution of a Vwap Order: A Stochastic Control Approach

Optimal Execution of a Vwap Order: A Stochastic Control Approach

Some Explicit Krein Representations of Certain Subordinators, Including the Gamma Process

Incorporating the Stochastic Process Setup in Parameter Estimation

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