##
**\(L^p\) solutions of backward stochastic differential equations.**
*(English)*
Zbl 1075.65503

Existence and uniqueness of solutions of the following backward stochastic differential equation are studied:
\[
Y_t=\xi +\int _t^Tf(r,Y_r,Z_r)\,\text dr-\int _t^TZ_r\,\text dB_r, \qquad 0\leq t\leq T.\tag{1}
\]
Here \(0<T<\infty \) is a fixed time, \(B\) a \(d\)-dimensional Brownian motion on a complete probability space (\(\Omega ,\mathcal F,\mathbb P\)) where the natural filtration \((\mathcal F^B_t)_t\) generated by \(B\) is considered, \(\xi \) an \(\mathcal F^B_T\)-measurable random variable, \(f\) maps \([0,T]\times \Omega \times \mathbb R^k\times \mathbb R^{k\times d}\) into \(\mathbb R^k\) and is progressively measurable in first two variables, continuous and uniformly monotone in the third and uniformly Lipschitz in the fourth one. A solution of (1) is a pair \((Y,Z)\) of progressively measurable processes with values in \(\mathbb R^k\times \mathbb R^{k\times d}\) such that paths of \(f(\cdot ,Y_\cdot ,Z_\cdot )\) and \(Z\) are integrable and square integrable respectively almost surely.

If the terminal condition \(\xi \) belongs to \(L^p(\Omega )\) for some \(p>1\) then there exists a unique solution \((Y,Z)\) of (1) in the class of progressively measurable processes in \(L^p(\Omega ;L^\infty (0,T))\times L^p(\Omega ;L^2(0,T))\). This result weakens the assumptions imposed on \(f\) in the analogous Theorem 2.2 in E.Pardoux [Math. Phys. Sci. 528, 503–549 (1999; Zbl 0959.60049)]. This result (Theorem 2.2) is the starting point in the proof of the present theorem: \(f\) is approximated by a suitable sequence of more regular \(f_n\)’s for which existence and uniqueness of solutions \((Y_n,Z_n)\) with the terminal condition \(\xi \) is known by Theorem 2.2 in the mentioned Pardoux’s paper, and this sequence \((Y_n,Z_n)\) is Cauchy, hence convergent to a solution of the equation (1). To show that \((Y_n,Z_n)\) are Cauchy, the authors lean upon apriori estimates which they have developed in the preliminary-part of this reviewed paper, and which yield, on the same hand, uniqueness of the constructed solution.

The second part of the paper deals with existence and uniqueness of solutions of the equation (1) where \(T\) is a stopping time (including the possibility \(T=\infty \)) and \(\xi \) is \(\mathcal F^B_T\) measurable. In this case, the equation (1) has a form \[ Y_{t\land T}=Y_{u\land T}+\int _{t\land T}^{u\land T}f(r,Y_r,Z_r)\,\text dr-\int _{t\land T}^{u\land T}Z_r\,\text dB_r,\qquad 0\leq t\leq u\tag{2} \] and \(Y=\xi \) and \(Z=0\) on the set \([T,\infty )\) almost surely. Existence and uniqueness of the equation (2) is shown under additional, yet fairly general set of assumptions upon integrability of \(f\) and form and integrability of \(\xi \).

The third part of the paper concerns existence and uniqueness of solutions of the equation (1) when the terminal time \(T\) is fixed, positive and finite, and the terminal condition \(\xi \) belongs to \(L^1(\Omega )\). Growth and integrability assumptions on \(f\) additional to those assumed in the first part of the paper are shown to yield a solution \((Y,Z)\) of (1) unique in a certain class of processes.

If the terminal condition \(\xi \) belongs to \(L^p(\Omega )\) for some \(p>1\) then there exists a unique solution \((Y,Z)\) of (1) in the class of progressively measurable processes in \(L^p(\Omega ;L^\infty (0,T))\times L^p(\Omega ;L^2(0,T))\). This result weakens the assumptions imposed on \(f\) in the analogous Theorem 2.2 in E.Pardoux [Math. Phys. Sci. 528, 503–549 (1999; Zbl 0959.60049)]. This result (Theorem 2.2) is the starting point in the proof of the present theorem: \(f\) is approximated by a suitable sequence of more regular \(f_n\)’s for which existence and uniqueness of solutions \((Y_n,Z_n)\) with the terminal condition \(\xi \) is known by Theorem 2.2 in the mentioned Pardoux’s paper, and this sequence \((Y_n,Z_n)\) is Cauchy, hence convergent to a solution of the equation (1). To show that \((Y_n,Z_n)\) are Cauchy, the authors lean upon apriori estimates which they have developed in the preliminary-part of this reviewed paper, and which yield, on the same hand, uniqueness of the constructed solution.

The second part of the paper deals with existence and uniqueness of solutions of the equation (1) where \(T\) is a stopping time (including the possibility \(T=\infty \)) and \(\xi \) is \(\mathcal F^B_T\) measurable. In this case, the equation (1) has a form \[ Y_{t\land T}=Y_{u\land T}+\int _{t\land T}^{u\land T}f(r,Y_r,Z_r)\,\text dr-\int _{t\land T}^{u\land T}Z_r\,\text dB_r,\qquad 0\leq t\leq u\tag{2} \] and \(Y=\xi \) and \(Z=0\) on the set \([T,\infty )\) almost surely. Existence and uniqueness of the equation (2) is shown under additional, yet fairly general set of assumptions upon integrability of \(f\) and form and integrability of \(\xi \).

The third part of the paper concerns existence and uniqueness of solutions of the equation (1) when the terminal time \(T\) is fixed, positive and finite, and the terminal condition \(\xi \) belongs to \(L^1(\Omega )\). Growth and integrability assumptions on \(f\) additional to those assumed in the first part of the paper are shown to yield a solution \((Y,Z)\) of (1) unique in a certain class of processes.

Reviewer: Martin Ondreját (Praha)

### MSC:

65C30 | Numerical solutions to stochastic differential and integral equations |

### Citations:

Zbl 0959.60049
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\textit{Ph. Briand} et al., Stochastic Processes Appl. 108, No. 1, 109--129 (2003; Zbl 1075.65503)

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### References:

[1] | Briand, Ph.; Carmona, R., BSDEs with polynomial growth generators, J. Appl. Math. Stochastic Anal., 13, 3, 207-238 (2000) · Zbl 0979.60046 |

[2] | Dellacherie, C.; Meyer, P.-A., Probabilités et Potentiel. Théorie des Martingales (1980), Hermann: Hermann Paris, (Chapitres V à VIII) · Zbl 0464.60001 |

[3] | El Karoui, N.; Peng, S.; Quenez, M.-C., Backward stochastic differential equations in finance, Math. Finance, 7, 1, 1-71 (1997) · Zbl 0884.90035 |

[4] | Kobylanski, M., Résultats d’existence et d’unicité pour des équations différentielles stochastiques rétrogrades avec des générateurs à croissance quadratique, C. R. Acad. Sci. Paris Sér. I Math., 324, 1, 81-86 (1997) · Zbl 0880.60061 |

[5] | Lepeltier, J.-P.; San Martin, J., Existence for BSDE with superlinear-quadratic coefficient, Stochastics Stochastics Rep., 63, 3,4, 227-240 (1998) · Zbl 0910.60046 |

[7] | Pardoux, E.; Peng, S., Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14, 1, 55-61 (1990) · Zbl 0692.93064 |

[8] | Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., 37, 1,2, 61-74 (1991) · Zbl 0739.60060 |

[9] | Peng, S., Backward SDE and related \(g\)-expectation, (El Karoui, N.; Mazliak, L., Backward Stochastic Differential Equations. Backward Stochastic Differential Equations, Pitman Research Notes Mathematical Series, Vol. 364 (1997), Longman: Longman Harlow), 141-159 · Zbl 0892.60066 |

[10] | Revuz, D.; Yor, M., Continuous martingales and Brownian motion, Grundlehren Math. Wiss., Vol. 293 (1991), Springer: Springer Berlin, Heidelberg, New York · Zbl 0731.60002 |

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