Abstract
In this article, we revisit in a didactic manner the forward and backward approaches of the parametrix method for one-dimensional reflected stochastic differential equations on the half line. We give probabilistic expressions for the expectation of functionals of its solution and we also discuss properties of the associated density.
Aurélien Alfonsi—This research benefited from the support of the “Chaire Risques Financiers”, Fondation du Risque and of Labex Bézout.
Masafumi Hayashi—This research was supported by KAKENHI grant 26800061.
Arturo Kohatsu-Higa—This research was supported by KAKENHI grant 2434002.
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Notes
- 1.
We consider the reflected SDE without drift just to simplify the discussion. For a general case scenario, see [20].
- 2.
See also, the typo-corrected version with comments on the webpage of the second author.
- 3.
The reason for the use of “backward” on this terminology is that one uses an Euler scheme which runs backward in time. Researchers in parabolic partial differential equation prefer the terminology forward because the method corresponds to the application of the forward Kolmogorov equation to the density of X. As the current article deals with approximations using the Euler scheme, we will keep using the former terminology. The forward method was known as early as [11]. See [10] for a translation (see also the historical references there). The backward method appeared in [19]. Thanks to Valentin for these references. The essential idea for the method was first introduced for elliptic equations and is due to [17].
- 4.
By these properties, it may appear at first that the symmetry of the density \(\bar{\pi }^{(z)}_{t}(x,x') \) with respect to the variables \( (x,x') \) is important, but in fact this is not the case. This can be seen if one considers the general case including a drift coefficient like in [20].
- 5.
That is, through a proper renormalization one may say that \( \bar{\pi }^{(x')}_t(x,x') \) is proportional to the density of a reflected Brownian motion at the point \( x\ge 0 \) which starts at \( x' \) with diffusion coefficient \( \sigma (x') \).
- 6.
When f is not a density function, one has to draw the initial point according to some density funtion \(q_0\) and then multiply by the well-defined weight \(\frac{f(y_2)}{q_0(y_2)}\).That is, we apply an importance sampling method.
- 7.
Note that the operator \( {{\mathscr {L}}}-{{\bar{\mathscr {L} } }}\) is applied to the variable x.
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Alfonsi, A., Hayashi, M., Kohatsu-Higa, A. (2017). Parametrix Methods for One-Dimensional Reflected SDEs. In: Panov, V. (eds) Modern Problems of Stochastic Analysis and Statistics. MPSAS 2016. Springer Proceedings in Mathematics & Statistics, vol 208. Springer, Cham. https://doi.org/10.1007/978-3-319-65313-6_3
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