Abstract
The one-dimensional SDE with non Lipschitz diffusion coefficient
is widely studied in mathematical finance. Several works have proposed asymptotic analysis of densities and implied volatilities in models involving instances of (1), based on a careful implementation of saddle-point methods and (essentially) the explicit knowledge of Fourier transforms. Recent research on tail asymptotics for heat kernels (Deuschel et al. Comm. in Pure and Applied Math., 67(1):40–82, 2014, [11]) suggests to work with the rescaled variable \( X^{\varepsilon }:=\varepsilon ^{1/(1-\gamma )} X\): while allowing to turn a space asymptotic problem into a small-\(\varepsilon \) problem, the process \(X^{\varepsilon }\) satisfies a SDE in Wentzell–Freidlin form (i.e. with driving noise \(\varepsilon \textit{dB}\)). We prove a pathwise large deviation principle for the process \(X^{\varepsilon }\) as \(\varepsilon \rightarrow 0\). As it will be seen, the limiting ODE governing the large deviations admits infinitely many solutions, a non-standard situation in the Wentzell–Freidlin theory. As for applications, the \(\varepsilon \)-scaling allows to derive leading order asymptotics for path functionals: while on the one hand the resulting formulae are confirmed by the CIR-CEV benchmarks, on the other hand the large deviation approach (i) applies to equations with a more general drift term and (ii) potentially opens the way to heat kernel analysis for higher-dimensional diffusions involving (1) as a component.
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Notes
- 1.
The precise statement here is .
- 2.
When \(\beta =0\), \(\gamma =1/2\) and \(\dot{h}\equiv 1\), one retrieves the textbook example of ODE for which uniqueness fails, \(\dot{\varphi }_{t} = \sigma \sqrt{|\varphi _t|}\), whose solutions from \(\varphi _0=0\) are given by the one-parameter family \(\varphi ^{(\theta )}_t = \frac{\sigma ^2}{4} (t-\theta )^2 1_{\{t \ge \theta \}}\).
- 3.
When \(\gamma \in [1/2,1)\), the law of \(X_T\) also possesses an atom at zero, \(\mathbb {P}(X_T=0)=m_T>0\), and an explicit formula for the mass \(m_T\) is available (see again [16, Chap. 6]). From our point of view, this only means that the density \(f_{X_T}\) does not integrate to 1 on \((0,\infty )\), without affecting our analysis of the tail asymptotics at \(\infty \).
- 4.
The second derivative reads \(e^{a \varepsilon ^{-2} (1+y)^{2(1-\gamma )}} \times 2a\varepsilon ^{-2}(1-\gamma )(1+y)^{-2\gamma } \times [1-2\gamma +\frac{2a}{\varepsilon ^2}(1-\gamma )\) \((1+y)^{2(1-\gamma )}]\).
- 5.
By perturbing the initial condition and the drift in (1.2), one can retrieve the trajectory \(\varphi ^*\) in (3.29) as the limit as \(\rho \rightarrow 0\) of the solution of the equation \(d\varphi _{t} = \rho \,+\,\beta \varphi _t \textit{dt}\,+\,\sigma \varphi _t^{\gamma }dh, \varphi _{0} = \rho \), for which existence and uniqueness hold.
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Acknowledgments
We would like to thank an anonymous referee for the careful reading of the paper and for several valuable comments which helped to improve the presentation. We thank Peter Friz for stimulating discussions and Antoine Jacquier for useful references on integrated CIR processes. SDM (affiliated with TU-Berlin when this work was started) acknowledges partial financial support from Matheon. GC acknowledges financial support from Berlin Mathematical School. SDM and GC acknowledge financial support for travel expenses from the research program ‘Chaire Risques Financiers’ of the Fondation du Risque.
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Appendix A
Appendix A
We complete the proof of Proposition 3.2 here.
Proof of Proposition 3.2 Let us define an auxiliary process \(\overline{X}\) by
after a simple application of the product rule, one has that the process \(Z_{t} := \exp (|\beta | t)\overline{X}_{t} \) is a solution to
Since \(|\alpha |_{\infty } \exp (|\beta | t) \ge |\alpha |_{\infty }\), an application of the comparison principle for SDE’s [17, Proposition 5.2.18] yields \(Z_{t} \ge \tilde{X}_{t}\), for all \(t \ge 0\). Therefore, if \(\overline{X}^{2(1-\gamma )}\) admits (some) exponential moments, so does \(Z^{2(1-\gamma )}_{t}\) and by comparison \(\tilde{X}^{2(1-\gamma )}_{t}\). In this sense, the process \(\overline{X}\) is not covered by Proposition 3.3 in [9], since the latter deals with the case of a diffusion coefficient that does not depend on time (see [9, Eq. (3.1)]); nonetheless, the essential condition that [9, Proposition 3.3] relies on is the presence of a non-strictly positive slope coefficient, say b in the drift term \(a+bX\) (cf. [9, Eq. (3.3)]). Since this is the case for the process \(\overline{X}\) (which has zero slope coefficient b), it is straightforward to extend the proof to the present setting: in particular, in the spirit of Lamperti’s change-of-variable argument, one still defines the function \(\varphi (x)=\int _0^{x}\frac{1}{\sigma x^{\gamma }}=\frac{1}{\sigma (1-\gamma )}x^{1-\gamma }\) and studies the process \(\tilde{\varphi }(X_t)\), where the function \(\tilde{\varphi }\) is a modification of \(\varphi \) identically null around zero. Itô’s formula shows that \(\tilde{\varphi }(X_t)\) is an Itô process with bounded quadratic variation and a bounded drift term; the existence of quadratic exponential moments for \(\tilde{\varphi }(X_t)\), then, is a consequence of Dubins–Schwarz time-change argument and Fernique’s theorem. As a consequence, there exist \(c',C > 0\) such that \(\sup _{t \le T} \mathbb {E}[\exp (c' \overline{X}^{2(1-\gamma )}_{t})] \le C\); it follows \(\sup _{t \le T} \mathbb {E}[\exp (c \tilde{X}^{2(1-\gamma )}_{t})] \le \sup _{t \le T} \mathbb {E}[\exp (c Z^{2(1-\gamma )}_{t})] \le C\) with \(c:=c' \exp (-2|\beta |(1-\gamma )T)\), and the claim is proved.\(\blacksquare \)
We report the statement given in [26, Chap. 2, Theorem 2.13].
Lemma A.1
(Garsia-Rodemich-Rumsey’s Lemma) Let p and \(\Psi \) be continuous, strictly increasing functions on \([0, +\infty )\) such that \(p(0) = \Psi (0)=0\) and \(\lim _{t \rightarrow + \infty } \Psi (t)= + \infty \). If \(\omega \in \Omega \) is such that:
then
Lemma A.1 allows us to prove Proposition 3.3:
Proof of Proposition 3.3 Assume that (3.3) holds true with the left hand side replaced by \(K>0\). Applying Lemma A.1 with the choice of functions \(\Psi (y)=\exp (\varepsilon ^{-2} y)-1, p(y) = \sqrt{y}\), one has for all s, t
Dividing on both sides by \((t-s)^{\eta }\) and taking suprema we obtain
Since the right hand side in the last estimate is \(K^{-1}_{\varepsilon , \eta } \left( K \right) \), (3.3) yields (3.4).\(\blacksquare \)
Finally, we prove Lemma 3.10.
Proof of Lemma 3.10 Denote \(T^{\varepsilon }\) the stopping time
We can apply Itô formula to the function \(f(x) = x^{1-\gamma }\) up to time \(T^{\varepsilon }(Y^{\varepsilon ,h})\), and obtain
where \(\tilde{b}^{\varepsilon }\) is given by
We need to prove
In order to simplify the notation, there is no ambiguity in writing Y instead of \(Y^{\varepsilon ,h}\) inside this proof.
Step 1. We first prove (A.6) under the assumption
Let us fist show that
A direct computation shows that there exist a constant \(c>0\) depending on \(x,\sigma ,\alpha (\cdot )\) such that:
Define \((Z_{t})_{t \in [0,T]}\) by
Using (A.9), it follows from the comparison principle for SDEs that
We claim that
holds true. Since \(W \left( T^{\varepsilon } \left( Y \right) \le T \right) \le W \left( T^{\varepsilon } \left( Z\right) \le T \right) \) by (A.11), then (A.8) holds. We prove (A.12) later on. Now, it follows from the definition of \(S_0(h)_t\) and an application of Gronwall’s Lemma that
therefore, for any \(R >0\) and \(\varepsilon \) small enough
Since both the events in the right hand side of the last inequality have probability converging to 1, (A.6) follows, and Lemma 3.10 is proved under condition (A.7).
Step 2. We assume that (A.7) holds only on the time interval \([0,\rho ]\), that is \(\dot{h}_{t} \ge k\) for every \(t \le \rho \), for some \(k, \rho >0\). Repeating the argument of Step 1 with \(T= \rho \), we have
We apply estimate (A.13) together with a localization argument. Define a time-shift operator \(\tau _{\rho } \omega \), for every \(\omega \in \Omega \), by \( (\tau _{ \rho } \omega )_{t} = \omega _{\rho +t}\) for all \(t \in [0, T-\rho ]\). For any fixed \(y>0\), denote \(X^{y,\rho }\) the strong solution of the SDE:
and set
Note that \(Y^{y,\rho }\) is well defined since \(X^{y,\rho } \ge 0\) for all \(t \in [0,T]\), W-almost surely. If \(h=0\) the non negativity of the trajectories of \(X^{y,\rho }\) follows from an application Proposition 3.1 in [9] and extends to \(h \in H \) by an application of the Girsanov theorem. By definition of Y and \(Y^{y,\rho }\), the Markov property yields
By the continuity of the map \((h,y) \mapsto \mathcal {S}_{y}(h)\) we can choose \(R'>0\) such that
Therefore, using (A.14) the following inclusion of events holds (assume w.lo.g \(R' \le \frac{R}{2}\)):
Applying the Markov property
We want to show that
It follows from the hypothesis \(\mathcal {S}_{0}(h)_{t} >0 \ \forall t>0\) and the continuity of the map \((y,h) \mapsto \mathcal {S}_{y}(h)\) that, if \(R',R\) are small enough
Define \(U^{y,\rho }\) as the unique strong solution of the SDE:
where
Then one has
Now observing that \(\tilde{b}^{u}_{\varepsilon } \) is globally Lipschitz continuous \(\forall \varepsilon >0\) and \(C^{\varepsilon } := \sup _{y \in \mathbb {R}}|\tilde{b}^{\varepsilon }_{u} (y) -\beta (1-\gamma )y| \rightarrow 0\), an application of Gronwall’s lemma gives
By letting \(\varepsilon \rightarrow 0\) and applying the Markov inequality, observing that the right hand side of (A.19) does not depend on y, we have proven (A.16). By letting \(\varepsilon \rightarrow 0\) in (A.15) and applying (A.13) and (A.16), the proof of Lemma 3.10 is complete.\(\blacksquare \)
Proof of (A.12). Observe that \(\tilde{Z}:=\frac{1}{\varepsilon }Z\) is an Ornstein-Uhlenbeck process,
where \(\mu _{\varepsilon }:= \frac{1}{\varepsilon }(-c \varepsilon \,+\,\sigma (1\,-\,\gamma )k) = -c\,+\,\frac{\sigma (1-\gamma )k}{\varepsilon }\). It is immediate by the definition of \(\tilde{Z}\) that \(W \left( T^{\varepsilon }(Z) \le T \right) = W \left( \inf _{t \in [0,T]} \tilde{Z} \le \frac{x^{1-\gamma }}{2} \right) \). The explicit representation of \(\tilde{Z}\) reads
with \(f_{\varepsilon }(t) = -\frac{\mu _{\varepsilon } (1-\exp (\beta (1-\gamma ) t) )}{\beta (1-\gamma )}\). Consider a deterministic time \(\tau _{\varepsilon }\) with \(\tau _{\varepsilon } \rightarrow 0\) as \(\varepsilon \rightarrow 0\), to be chosen precisely later on. Noting that \(f_{\varepsilon }\) is a decreasing function, for \(\tau _{\varepsilon } \le t \le T\) one has
hence, using Markov’s inequality and Doob’s inequality
Now, the choice \(\tau _{\varepsilon } = \sqrt{\varepsilon }\) gives \(f_{\varepsilon }(\tau _{\varepsilon }) \sim \mu _{\varepsilon } \tau _{\varepsilon } \rightarrow \infty \) as \(\varepsilon \rightarrow 0\), so that \(\left( f_{\varepsilon }(\tau _{\varepsilon }) - x^{1-\gamma }/2 \right) ^{-1} \rightarrow 0\). On the other hand, \(\inf _{t \in [0,\tau _{\varepsilon }]} \tilde{Z}_t \rightarrow x^{1-\gamma }\) a.s. as \(\varepsilon \rightarrow 0\), hence \(W \Bigl ( \inf _{t \in [0,\tau _{\varepsilon }]} \tilde{Z}_t \le x/2 \Bigr ) \rightarrow 0\) as \(\varepsilon \rightarrow 0\), and the claim is proven.
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Conforti, G., De Marco, S., Deuschel, JD. (2015). On Small-Noise Equations with Degenerate Limiting System Arising from Volatility Models. In: Friz, P., Gatheral, J., Gulisashvili, A., Jacquier, A., Teichmann, J. (eds) Large Deviations and Asymptotic Methods in Finance. Springer Proceedings in Mathematics & Statistics, vol 110. Springer, Cham. https://doi.org/10.1007/978-3-319-11605-1_17
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