Local Hölder Continuity Property of the Densities of Solutions of SDEs with Singular Coefficients

Abstract

We prove that the weak solution of a uniformly elliptic stochastic differential equation with locally smooth diffusion coefficient and Hölder continuous drift has a Hölder continuous density function. This result complements recent results of Fournier–Printems (Bernoulli 16(2):343–360, 2010), where the density is shown to exist if both coefficients are Hölder continuous, and exemplifies the role of the drift coefficient in the regularity of the density of a diffusion.

Appendix

1.1 7.1 Estimate of (4) on the Event C

Lemma 3

Under (H1) and (H2), we have the following estimate:

$$ \bigl|E_Q\bigl[e^{i\theta X_t}\phi_{\varepsilon}(X_t-y_0)1_C \bigr]\bigr| \le \varepsilon^{-2n}K_n \bigl(M_n\| \bar{\sigma}\|^{2n}_{\infty}\delta^{n}+ \delta^{2n}\|\bar{b}\|_{\infty}^{2n} \bigr), $$

(8)

where K _n and M _n are constants depend only on n.

Proof

Using Markov’s inequality, we have

where K _n is a constant which depends only on n.

Since $\bar{\sigma}$ and $\bar{b}$ are bounded, by Doob’s inequality and Burkholder–Davis–Gundy inequality, we have

for any n∈ℕ, where M _n is a constant depends only on n. Therefore (8) follows. □

1.2 7.2 Estimate of (4) on the Event A

Now we turn to estimate the second term of (4).

Lemma 4

Under (H1), (H2), and (H3), we have the following estimate:

Proof

By the definition of $\bar{X}$, on the event A,

Hence, we obtain

Since

we have

(9)

By the definitions of ν and τ, we have

So, as in Lemma 3 we obtain

$$ Q(\nu=t-\delta;\tau\le t)\le \varepsilon^{-2n}K_n \bigl(M_n\|\bar{\sigma}\|^{2n}_{\infty} \delta^{n} +\delta^{2n}\|\bar{b}\|_{\infty}^{2n} \bigr). $$

Therefore, we have the following upper bound for the second term in (9):

(10)

For the first term in (9), we change the probability measure from Q to P defined by (5). That is,

Then we have

(11)

Since $1_{\{X_{t-\delta}\in\overline{B_{3\varepsilon}(y_{0})}\}}$ is $\mathcal{F}_{t-\delta}$-measurable, using conditional expectation and the Markov property for $\bar{X}$, we have

As in Proposition 1, the integration by parts formula of Malliavin calculus in the interval [t−δ,t], implies that for any n ₂∈ℕ and $y\in\overline{B_{3\varepsilon}(y_{0})}$, there exists a random variable $H_{n_{2}}(\bar{X}_{t}(t-\delta,y),\phi_{\varepsilon}(\bar{X}_{t}(t-\delta,y) - y_{0}))\in\mathbb{D}^{\infty}$ such that

Furthermore, by Theorem 2.3. and Corollary 1 of [9] (which are consequences of the application of Proposition 1 to our situation), there exists a constant $C_{\varepsilon,n_{2}}$ which depends on ε, n ₂ and derivatives of $\bar{\sigma}$ up to the order n ₂ such that for any $y\in\overline{B_{3\varepsilon}(y_{0})}$,

$$ \bigl\|H_{n_2}\bigl(\bar{X}_t(t-\delta,y), \phi_{\varepsilon}\bigl(\bar{X}_t(t-\delta,y) - y_0\bigr) \bigr)\bigr\|_{L^2(P)} \le C_{\varepsilon,n_2}\delta^{-\frac{n_2}{2}}. $$

(12)

In fact, Theorem 2.3 of [9] tells us that there exists some constant $C^{\star}_{\varepsilon ,n}$ such that

On the other hand, thanks to (H1), Corollary 1 of [9] implies that there exists some constant $C^{\dagger}_{\varepsilon ,n_{2}}$ such that

$$ \bigl\|\phi_{\varepsilon}\bigl(\bar{X}_t(t-\delta,y) - y_0 \bigr)\bigr\|_{n_2,2^{n_2+1}} \le C^{\dagger}_{\varepsilon ,n_2}. $$

The above constant $C_{\varepsilon ,n_{2}}$ is the product of these constants $C^{\star}_{\varepsilon ,n_{2}}$ and $C^{\dagger}_{\varepsilon ,n_{2}}$.

By (12) and recalling that Z is a non-negative martingale with mean one, for any n ₂∈ℕ, we obtain the following inequality:

(13)

However, since Z _t −1 is not $\mathcal{F}_{t-\delta}$-measurable and we do not assume the smoothness of the coefficient b, we cannot apply the integration by parts formula for the first term in (11). Instead, we rewrite

Thus we obtain

(14)

For the first term, by the Hölder continuity of $\bar{\sigma}^{-1}\bar{b}$, (6) and Hölder’s inequality, we have

(15)

where

$$ C_{\alpha}:= \biggl(\frac {2}{\sqrt{1+\alpha}}\|Z_t\|_{L^{\frac{2}{(1-\alpha)}}(P)} \|\bar{\sigma}\|_{\infty}^{\alpha} \biggr) \vee \bigl(\bigl\|\bar{\sigma}^{-1}\bar{b}\bigr\|_{\infty}^2\|Z_t \|_{L^2(P)} \bigr). $$

For the second term of (14), we proceed as in (13). Since $(\bar{\sigma}^{-1}\bar{b})(X_{t-\delta})$ is bounded and $\mathcal{F}_{t-\delta}$-measurable, we have

Now we can apply the integration by parts formula which implies that for any n ₂∈ℕ and $y\in\overline{B_{3\varepsilon}(y_{0})}$ there exists a random variable

$H_{n_{2}}(\bar{X}_{t}(t-\delta,y),\phi_{\varepsilon}(\bar{X}_{t}(t-\delta,y) - y_{0})(W_{t}-W_{t-\delta}))\in\mathbb{D}^{\infty}$ such that

and by the Hölder inequality for the stochastic Sobolev norms (see Proposition 1.5.6 of [11]), its L ²(P)-norm is bounded by $C_{\varepsilon,n_{2}}\delta^{-\frac{n_{2}}{2}} c_{n_{2}}\|(W_{t}-W_{t-\delta})\|_{n_{2},2^{n_{2}+1}}$, where $c_{n_{2}}$ is a constant depends only on n ₂.

However, the kth order H-derivatives of W _t −W _t−δ vanish when k≥2. Therefore, there exists a positive constant C (independent of n ₂) such that

$$ \bigl\|(W_t-W_{t-\delta})\bigr\|_{n_2,2^{n_2+1}} = \bigl\|(W_t-W_{t-\delta})\bigr\|_{1,2^{n_2+1}} \le C $$

(16)

and hence, we have

(17)

where $\tilde{C}_{\varepsilon,n_{2}}:=2C_{\varepsilon,n_{2}}c_{n_{2}}$.

Substituting (17) and (13) into (11), we have

(18)

As a result, we have

by substituting (18) and (10) into (9). □

Remark 7

The above estimate (16) for the Sobolev norm of the Wiener process is clearly non-optimal. However, as the term appearing in (15) decreases slowly, improving the estimate in (16) will not change the final result. The same comment applies to other terms such as (10).