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Some properties of solutions of stochastic differential equations driven by semi-martingales. (English) Zbl 1020.60034

Let \[ Z(t)=\int_0^tu(s,\omega) dB_s+\int_0^tv(s,\omega) ds \] be an Itô process with values in \(\mathbb R^d\), where \(B_t\) is a standard \(d\)-dimensional Brownian motion. N. V. Krylov [“Nonlinear elliptic and parabolic equations of the second order” (1987; Zbl 0619.35004) and Math. USSR, Sb. 58, 207–221 (1987); translation from Mat. Sb., Nov. Ser. 130(172), No. 2, 207–221 (1986; Zbl 0625.35041)] proposed the following estimate \[ E\int_0^{\min(t,\tau_R)}f(s,Z(s)) ds\leq N\|f\|_{L^{(d+1)}([0,t]\times D_R)} \] for every Borel positive function \(f\colon \mathbb R_+\times\mathbb R^d\), where \(D_R\) denotes the ball \(\mathbb R^d\), \(\tau_R\) is the first exit time of \(Z(t)\) from \(D_R\) and the constant \(N\) depends on \(t,R,d\) and the constant of ellipticity of the matrix \(u(s,\omega)\cdot u^*(s,\omega)\). This inequality plays an important role in the theory of stochastic integrals, stochastic differential equations and in stochastic control. A. V. Mel’nikov [Stochastics 10, 81–102 (1983; Zbl 0539.60059)] extended the inequality to continuous semi-martingales given by \(Z_t=Z_0+m_t+L_t\), where \(m_t\) is a continuous local martingale and \(L_t\) is a continuous process of finite variation. He obtained the following estimate \[ E\int_0^{\min(t,\tau_R)}(\det Q_s)^{1/d} f(Z(s)) d \langle m\rangle_s \leq N\|f\|_{L^{d}(D_R)}, \] where \(Q_s\) is the matrix of densities of the finite variation process \(\langle m^i,m^i\rangle_t\).
An extension of the last inequality is proposed. Under the condition of absolute continuity of the quadratic variation process associated to the martingale part it is proved that Krylov’s inequality remains valid for functions which depend explicitly on the time variable. This last inequality is then applied to derive various properties such as pathwise uniqueness, non confluent property and continuity with respect to initial data for stochastic differential equations driven by continuous semi-martingales with Sobolev space valued coefficients. Exponential martingales will be the main tool in the proofs.

MSC:

60G48 Generalizations of martingales
35K55 Nonlinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35B50 Maximum principles in context of PDEs
60H20 Stochastic integral equations
60G44 Martingales with continuous parameter
35J60 Nonlinear elliptic equations
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