A Nonlinear PDE in Mathematical Finance

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1 A Nonlinear PDE in Mathematical Finance Sergio Polidoro Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, Bologna (Italy) Summary. We study a non linear degenerate Cauchy problem arising in mathematical finance. We prove the existence of a local strong solution and we study its regularity in the framework of subelliptic operators on nilpotent Lie groups. Moreover we give some conditions for the existence of global solutions. 1 Introduction This talk aims to present some regularity results obtained in a joint work in collaboration with G. Citti and A. Pascucci, from the University of Bologna. We consider the following partial differential equation in the variables z = (x, y, t) R 3 Lu xx u + u y u t u = f, (1) that arises in a work by Antonelli, Barucci and Mancino [1], concerning a financial problem. In their paper, Antonelli, Barucci and Mancino study the problem of the agents decisions under risk in the framework of the theory of utility functions and propose a model where the utility functional is described by a stochastic process, which is related to a suitable backward-forward stochastic differential equations. In [1] it is proved that there exists a unique utility functional for the financial problem, which is the unique viscosity solution (in the sense of the User s guide [6]) of a Cauchy problem related to the partial differential equation (1) and to a suitable initial data u(x, y, 0) = g(x, y). (2) Moreover, in [1] it is proved that the solution u, of the Cauchy problem (1) (2) is defined in a suitably small interval of time [0, T [ and satisfies ( ) u(x, y, t) u(ξ, η, τ) C x ξ + y η + t τ 1 2 (3) for every (x, y, t), (ξ, η, τ) R 2 [0, T [, under the assumption that f and g are uniformly Lipschitz continuous functions. We recall that other existence results for weak solutions of the Cauchy problem for a more general class of equations, that contains (1), have been Partially supported by the University of Bologna, Funds for Selected Research Topics.

2 2 Sergio Polidoro obtained by Vol pert and Hudjaev in [12] and by Escobedo, Vazquez and Zuazua in [7], in a different functional setting. This kind of solutions, however, may be discontinuous, thus do not satisfy condition (3). We are interested on the interior regularity of the solution u found in [1]. We first note that the standard regularity theory for distributional solutions, as well as the regularity results for viscosity solutions, is not applicable to our problem. On the contrary, in some cases the operator L behaves as a hyperbolic one. For example, when f 0, any solution u independent of the variable x satisfies the Burgers equation u y u t u = 0 and its regularity stated in (3) cannot be improved. However, under some suitable conditions, we are able to prove that the solution u is smooth. Our results have been obtained in the framework of the analysis on Lie groups. 2 Analysis on Lie groups The operator L defined in (1) is a second order operator such that the matrix of the coefficients of the second order derivatives is only semi-definite positive. As one can expect, the solution u of the equation Lu = f, with f C, is smooth in the directions in which the matrix is non-degenerate, but not necessarily in other directions. Consider for example the operator L 2 x + 2 y in the variables (x, y, t) R 3 : every solution u of Lu = 0 is smooth with respect to the variables x and y, but is not regular in the variable t. However, as Hörmander pointed out in [8], there are other regularity directions for the solution u of an equation Lu = f, and these directions are the directions of the commutators. For instance, let s consider the Kohn-Laplace operator in R 3 : L ( x + 2y t ) 2 + ( y 2x t ) 2. As before, only 2 directional derivatives occur in the operator, while the dimension of the space is 3, but in this case every solution of Lu = 0 is smooth, not only in the directions of the derivatives X = x + 2y t and Y = y 2x t, but also in the direction of the commutator [X, Y ] = XY Y X = 4 t. The above operator is the simplest meaningful example of the class studied by Hörmander: let X 0,..., X p be a set of directional derivatives (vector fields) defined as

3 X j = A Nonlinear PDE in Mathematical Finance 3 n a ij (x) xi i=1 j = 0,..., p where a ij are C (Ω) functions, for some open set Ω R n and let f C (Ω). The result proved by Hörmander is the following one: if u is a solution of the equation X i X j u + X 0 u = f, in Ω, (4) i,j and the vector fields X 0,..., X p together with their commutators span R n, then u C (Ω). The Hörmander s result was the starting point of an extensive research aiming to investigate the regularity properties of the operators in (4) and their links with some suitable Lie group structures on R n. We refer to the paper by Rothschild and Stein [10] and to the book by Varopoulos, Saloff- Coste and Coulhon [11] for a general regularity theory. In the study of the regularity problem for the non linear equation (1), we aim to use the above linear theory, thus we consider the linearized operator L u 2 x + u y t, where u is considered as a coefficient. We immediately find that the smoothness of the coefficients a ij is a fundamental assumption in the previous papers and, in our problem, we cannot assume that the coefficient u of the equation is C, since the smoothness of the solution u is exactly the goal of our study. We stress that a general theory for operators with non-smooth coefficients is not available, however we can employ a technique introduced in some recent works by Citti (see [2] and [3]), where the regularity problem for the prescribed Levi curvature equation is considered. The main idea introduced in [2] is a modification of the classical freezing method and we developed the same method in our study concerning our operator L defined in (1). We first represent L as in (4) by letting Lu = X 2 u + Y u (5) X = x and Y = u y t, (6) then, for any ζ = (ξ, η, τ) Ω we consider the approximating operator where L ζ u = X 2 u + Y ζ u, Y ζ = ( u(ζ) + (x ξ)u x (ζ) ) y t. Note that L ζ is a linear Hörmander s operator, moreover it is a good approximation of L in the sense that, under condition (3), we have Lu(z) L ζ u(z) = u(z) u(ζ) + (x ξ)u x (ζ) y u(z) C z ζ. Our regularity results have been obtained by using some representation formulas for the solution u in terms of the fundamental solution of L ζ.

4 4 Sergio Polidoro 3 Main results In order to study the regularity problem in by the analysis on Lie groups we have to consider the commutator of X and Y, that is [X, Y ] = u x y and the Hörmander condition is satisfied if u x (z) 0 for every z. (7) Note that u satisfies condition (3), then u x is defined almost everywhere and the above condition has to be considered only formally. Our main results are contained in the following statements: Theorem 1. Let u be a viscosity solution of the differential equation Lu = f, for some Lipschitz continuous function f, and suppose that u satisfies condition (3), then u is a classical solution of Lu = f. We say that u is a classical solution if u xx and Y u, defined as the directional derivative with respect to the vector ν z = (0, u(z), 1): u u(z + hν z ) u(z) (z) = lim ν z h 0 h (8) are defined everywhere, if u xx and z u ν z (z) are continuous functions and the equation Lu = f is satisfied at every point. The above fact is proved in [5]. The result is reasonable: without assuming the Hörmander condition we are able to prove the regularity of u only in the directions of the derivatives. Note that the Hörmander condition u x 0 is now meaningful since u x is defined as a continuous function. We next show that, under some natural assumptions, the classical solution of the Cauchy problem satisfies condition (7). Theorem 2. R 2 ]0, T [, Let u be a classical solution of the Cauchy problem Lu = f in u(x, y, 0) = g(x, y). Suppose that f x and g x are defined as continuous functions and that f x 0 in R 2 ]0, T [ and g x > 0 in R 2. Then u x > 0 in R 2 ]0, T [. The above result is contained in [5]. The condition on f x is the natural condition in order to apply a minimum principle, while the condition on g x is suggested by the financial problem studied in [1]. Our next theorem is proved in [4]. Theorem 3. Let u be a classical solution of the differential equation Lu = f, where f C, and suppose that u x > 0, then u is C.

5 A Nonlinear PDE in Mathematical Finance 5 We end this talk by a result obtained in collaboration with Pascucci in [9], concerning the existence for large times. Theorem 4. Suppose that f y and g y are defined as continuous functions and that f y 0 in R 2 R + and g y 0 in R 2. Then u is defined in R 2 R +. As said before, our regularity results have been obtained by using a representation formula for the solution u of equation (1). In our problem, the main difficulty is due to the fact that a representation formula for viscosity solutions does not hold; however we are able to show that the solution u found in [1] is the limit of a sequence of smooth functions for which the representation formula does hold, thus the formula also apply to the solution u. The proof is rather technique and we refer to the references [4], [5] and [9] for the details. References 1. Antonelli, F.; Barucci, E.; Mancino, M.E.: Backward-forward stochastic differential utility: existence, optimal consumption and equilibrium analysis. Preprint. 2. Citti, G. (1996): C regularity of solutions of a quasilinear equation related to the Levi operator. Ann. Scuola Norm. Sup. Pisa Cl. Sci., Serie IV XXIII, Citti, G.; Lanconelli, E.; Montanari, A.: Smoothness of Lipschitz continuous graphs, with non vanishing Levi curvature. To appear on Acta Math. 4. Citti, G.; Pascucci, A.; Polidoro, S. (2001): On the regularity of solutions to a nonlinear ultraparabolic equation arising in mathematical finance. Diff. Int. Eq. 14 (6), Citti, G.; Pascucci, A.; Polidoro, S.: Regularity properties of viscosity solutions of a non-hörmander degenerate equation. To appear on J. Math. Pures Appl. 6. Crandall, M.G.; Ishii, H.; Lions, P.-L. (1992): User s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc., New Ser. 27 (1), Escobedo, M.; Vazquez, J.L.; Zuazua, E (1994): Entropy solutions for diffusionconvection equations with partial diffusivity. Trans. Am. Math. Soc. 343 (2), Hörmander, L. (1967): Hypoelliptic second order differential equations. Acta Math. 1119, Pascucci, A.; Polidoro, S.: On the Cauchy problem for a non linear Kolmogorov equation. To appear. 10. Rothschild, L.P.; Stein, E.M. (1977): Hypoelliptic differential operators on nilpotent groups. Acta Math. 137, Varopoulos, N.Th.; Saloff-Coste, L.; Coulhon, T. (1992): Analysis and geometry on groups. Cambridge Tracts in Mathematics, Vol. 100, Cambridge Univ. Press, Cambridge. 12. Vol pert, A.I.; Hudjaev, S.I. Cauchy s problem for degenerate second order quasilinear parabolic equations. Math. USSR, Sb. 1970, 7 (3),