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Nonparametric Estimation for I.I.D. Paths of a Martingale Driven Model with Application to Non-Autonomous Financial Models

Nicolas Marie

Abstract

This paper deals with a projection least squares estimator of the function $J_0$ computed from multiple independent observations on $[0,T]$ of the process $Z$ defined by $dZ_t = J_0(t)d\langle M\rangle_t + dM_t$, where $M$ is a continuous and square integrable martingale vanishing at $0$. Risk bounds are established on this estimator, on an associated adaptive estimator and on an associated discrete-time version used in practice. An appropriate transformation allows to rewrite the differential equation $dX_t = V(X_t)(b_0(t)dt +σ(t)dB_t)$, where $B$ is a fractional Brownian motion of Hurst parameter $H\in [1/2,1)$, as a model of the previous type. So, the second part of the paper deals with risk bounds on a nonparametric estimator of $b_0$ derived from the results on the projection least squares estimator of $J_0$. In particular, our results apply to the estimation of the drift function in a non-autonomous Black-Scholes model and to nonparametric estimation in a non-autonomous fractional stochastic volatility model.

Nonparametric Estimation for I.I.D. Paths of a Martingale Driven Model with Application to Non-Autonomous Financial Models

Abstract

This paper deals with a projection least squares estimator of the function computed from multiple independent observations on of the process defined by , where is a continuous and square integrable martingale vanishing at . Risk bounds are established on this estimator, on an associated adaptive estimator and on an associated discrete-time version used in practice. An appropriate transformation allows to rewrite the differential equation , where is a fractional Brownian motion of Hurst parameter , as a model of the previous type. So, the second part of the paper deals with risk bounds on a nonparametric estimator of derived from the results on the projection least squares estimator of . In particular, our results apply to the estimation of the drift function in a non-autonomous Black-Scholes model and to nonparametric estimation in a non-autonomous fractional stochastic volatility model.

Paper Structure

This paper contains 13 sections, 8 theorems, 131 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A projection least squares estimator of the map $J_0$
  3. The objective function
  4. The projection least squares estimator
  5. Risk bounds and model selection
  6. Application to differential equations driven by the fractional Brownian motion
  7. Auxiliary model
  8. An estimator of $Q_0$
  9. Example 1: drift estimation in a non-autonomous Black-Scholes model
  10. Example 2: nonparametric estimation in a non-autonomous fractional stochastic volatility model
  11. Numerical experiments
  12. Experiments in Model \ref{['main_equation']} driven by the Molchan martingale
  13. Experiments in the non-autonomous Black-Scholes model

Key Result

proposition thmcounterproposition

Under Assumption assumption_quadratic_variation,

Figures (7)

  • Figure 1: Plots of $J_{0,1}$ and of $10$ adaptive estimations when $H = 0.6$ ($\overline{\widehat{m}} = 5.4$).
  • Figure 2: Plots of $J_{0,2}$ and of $10$ adaptive estimations when $H = 0.6$ ($\overline{\widehat{m}} = 11.2$).
  • Figure 3: Plots of $J_{0,3}$ and of $10$ adaptive estimations when $H = 0.6$ ($\overline{\widehat{m}} = 8.2$).
  • Figure 4: Plot of one path of the non-autonomous Black-Scholes model with $\sigma = 0.2$.
  • Figure 5: Plots of the paths of $Z^1,\dots,Z^N$ associated to the path of $S$ at Figure \ref{['BS_02']}.
  • ...and 2 more figures

Theorems & Definitions (15)

  • proposition thmcounterproposition
  • proof
  • corollary thmcountercorollary
  • proof
  • theorem 4
  • lemma thmcounterlemma
  • proof
  • proposition thmcounterproposition
  • lemma thmcounterlemma
  • proof
  • ...and 5 more