Stochastic Automatic Differentiation: Automatic Differentiation for Monte-Carlo Simulations

Fries, Christian

doi:10.2139/ssrn.2995695

Cited by 12 publications

(7 citation statements)

References 18 publications

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“…A stochastic algorithmic differentiation is an algorithmic differentiation (formally) applied to random variables, [6]. The approach allows us to inspect the stochastic nature of the random variables and their derivatives, e.g., the variance of the argument of the indicator function.…”

Section: Algorithmic Differentiation and Stochastic Algorithmic Diffe...mentioning

confidence: 99%

“…We consider a stochastic algorithmic differentiation, see [6], where the algorithmic differentiation is applied to random variables. Let X denote a random variable, θ an arbitrary model parameter, 1 the indicator function and f a operator on random variables.…”

Section: Derivative Of the Indicator As Conditional Expectationmentioning

confidence: 99%

“…• An adjoint automatic differentiation (AAD) implementation ( [6]), where the differentiation of the indicator function is replaced by a discretized delta function, i.e.,…”

Section: Test Setupmentioning

confidence: 99%

“…sensitivity) of expectations of functions involving an indicator function by combining a stochastic algorithmic differentiation and a regression. The method is an improvement of the approach presented in [6,4].The finite difference approximation of a partial derivative of a Monte-Carlo integral of a discontinuous function is known to exhibit a high Monte-Carlo error. The issue is obvious, since the Monte-Carlo approximation of a discontinuous function is just a finite sum of discontinuous functions and as such not even differentiable.The algorithmic differentiation of a discontinuous pay-off is problematic.…”

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confidence: 99%

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confidence: 99%

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Stochastic Algorithmic Differentiation of (Expectations of) Discontinuous Functions (Indicator Functions)

Fries

2018

SSRN Journal

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In this paper we present a method for the accurate estimation of the derivative (aka. sensitivity) of expectations of functions involving an indicator function by combining a stochastic algorithmic differentiation and a regression. The method is an improvement of the approach presented in [6,4].The finite difference approximation of a partial derivative of a Monte-Carlo integral of a discontinuous function is known to exhibit a high Monte-Carlo error. The issue is obvious, since the Monte-Carlo approximation of a discontinuous function is just a finite sum of discontinuous functions and as such not even differentiable.The algorithmic differentiation of a discontinuous pay-off is problematic. A natural approach is to replace the discontinuity by continuous functions (aka. pay-off smoothing). This is equivalent to replacing a path-wise automatic differentiation by a (local) finite difference approximation.We show that this local finite difference approximation can be seen as a linear regression with the simplest regression basis function (a single indicator). Investigating the expression, we observe that we can separate the expectation of the indicator function (the density) and the regression of the size and speed of the discontinuity. With this formulation, we then replace the regression(s) by more accurate estimators.

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Section: Algorithmic Differentiation and Stochastic Algorithmic Diffe...mentioning

confidence: 99%

Section: Derivative Of the Indicator As Conditional Expectationmentioning

confidence: 99%

“…• An adjoint automatic differentiation (AAD) implementation ( [6]), where the differentiation of the indicator function is replaced by a discretized delta function, i.e.,…”

Section: Test Setupmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Stochastic Algorithmic Differentiation of (Expectations of) Discontinuous Functions (Indicator Functions)

Fries

2018

SSRN Journal

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Automatic adjoint differentiation for gradient descent and model calibration

Goloubentsev¹,

Lakshtanov

2020

Int. J. Wavelets Multiresolut Inf. Process.

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In this work, we discuss the Automatic Adjoint Differentiation (AAD) for functions of the form [Formula: see text], which often appear in the calibration of stochastic models. We demonstrate that it allows a perfect SIMD a parallelization and provides its relative computational cost. In addition, we demonstrate that this theoretical result is in concordance with numerical experiments. a Single Input Multiple Data.

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