A STOCHASTIC MODEL FOR THE FINANCIAL MARKET WITH DISCONTINUOUS PRICES LEDA

This paper models some situations occurring in the financial market. The asset prices evolve according to a stochastic integral equation driven by a Gaussian martingale. A portfolio process is constrained in such a way that the wealth process covers some obligation. A solution to a linear stochastic integral equation is obtained in a class of cadlag stochastic processes.


Introduction
In the present paper we model investments of an economic agent whose decisions cannot affect market prices (a "small investor").
Karatzas and Shreve in [7] considered a market model in which prices evolve according to a stochastic differential equation, driven by Brownian motion.Aase [1] and M. Picqu and M. Pontier [9] studied a more general model in which the evolution of asset prices is a combination of a continuous process based on Brownian motion (a semimartingale) and a Poisson point process.
The security price model that we use is a linear stochastic equation driven by a Gaussian martingale.This is a natural generalization, because the market is not continuous and the Brownian motion cannot model jump processes.Moreover, the instants of jumps of a Gaussian martingale are nonrandom.
The techniques we use include the martingale representation theorem and the Girsanov's type theorem.We also find a solution to a linear stochastic integral equation.

The Model
We consider a model of a security market where an economic agent is allowed to trade contin- uously up to some fixed planning horizon 0 <_ T < oc.We shall denote by X the wealth of this 1This work was supported by the National Science Foundation of Bulgaria, Grant No. MM-440/94.agent at time t.Let the process M-(Mt, Ft, O <_ <_ T) be a Gaussian martingale on a fixed probability space (f, F, P) and the filtration F {Ft, 0 <_ t <_ T} be the augmentation under P of a natural filtration Ft M=r(Ms,0<_s<_t), O<_t<c.F o contains the null sets of P and F is right continuous.(M)t EM2t, t E + [0, c) is the square characteristic of U.
Let us suppose that the agent invests in two assets (or "securities").bond, has a finite variation on [0, T], and its price model is One of the assets, called f Po(t) / Po(s-)r(s-)d(M)s P0(0)-P0, 0 _< t <_ T.
Here the interest rate process r(t) > 0, 0 <_ t < oc of the bond, the appreciation rate process A(t) of the stock, and volatility process r(t) > 0, 0 _< t < c will all be nonrandom, F-predictable pro- er(s-)d(M)s < cxa, P-a.s. (0,) In addition, A(t-)A(M)t + r(t-)AM > 1, t e (0, T], to ensure a limited liability of the stock. Let re(t) denote the number of stocks held at time t.Then the amount invested in the stocks is II(,) m(t)P(t).
The process (H(t),Ft) 0 <_t<_ T describes the investment policy and will be called a portfolio process.It is assumed to be measurable, Ft-predictable and H2(s_)d(M)s (o,T] < c, P-a.s. (2) for every finite number T > 0.
stock short.
Note that H(t) can be negative, which amounts to selling the On the other hand, C(t), 0 <_ t _< T is a non-negative consumption process, assumed to be nondecreasing and Ft-predictable such that /C(s-)d</)s < oo, (3) P-a.s.

(o,T]
for every finite number T > 0. The quantity no(*) x,is invested in the bond at any particular time and may also become negative.
interpreted as borrowing at the interest rate r(t).

Characterization of the Portfolio Process
If A(t) r(t) for every t e [0, c), the drift vanishes from the right-hand side of (4).When A(t) r(t) we introduce a new probability measure P which removes this drift.
Let us denote by Ct the solution of the equation where Ct-1-] Cs-O(s-)dMs, O<_t <_T, (0,t] O(t) A(t)-r(t) From our assumptions on A, r, and r, it follows that O(t) is bounded, measurable and adapted to {F t-}.Then the exponential supermartingale is actually a martingale, where a(t-) for0<t<T.
AMt Mt-Mt-A(t-r(t- Here M and (MClt are the continuous parts of the processes M and (M}t respectively, for t E +. We define the new probability measure P" P (A) E(TIA) A F T on (, F).
The probability measures P and P are mutually absolutely continuous on F T.
With respect to a new probability measure, equation (4) can be rewritten as X II(s )r(s )dM s + [X s r(s x(0)and the solution (see Section 5) for 0 _< t _< T, leads to X C(s-) d(M) x / (s )A(M)s ] where is a unique strong solution of the homogeneous equation corresponding to ( 7)" (P(t) 1 + / ((s-)r(s-)d(M)s.(11) Then there exists a portfolio process II such that the pair (H,C) is admissible for the initial en- dowment x and the terminal wealth X T is at least B T.
Proof: It is obvious that we can assume equality to hold in (11).
If we suppose that 1 + r(s-)A(M}s < 0 for some s e S, then A(M)s < 1 But this is (s-)" impossible if r(s) is nonnegative.Consequently, 1 + r(s-)A(M)s > 0 for every s e +.
Let us notice also that inf I(I)(t)l >0.

t[+
The right-hand side of ( 8) is a P-local martingale.If (H, ) is an admissible pair (i.e., X >_ 0, 0 _< t _< T a.s.), the left-hand side is nonnegative, consequently it is a nonnegative supermartin- gale under P. From the supermartingale property we obtain that IXT / C(s-) d ( / 1 7 / > s l < x , (10) E (I)(T) + ((s-)[1 + r(s-)A(M)s (0,T] where E denotes the expectation operator under measure P.This condition is also sufficient for the admissibility in the sense of the following theorem.
Suppose that x >_ 0 and B T is a nonnegative FT-measurable random variable, which is a P-martingale and has "cadlag" paths.

(o,t]
The process (13) can be represented as From equalities ( 6), ( 14), and (15)it follows that is a well-defined portfolio process.

Valuation of Contingent Claim
Definition: A contingent claim is a nonnegative FT-measurable random variable B that satis- fies 0<E <x.
Theorem 2: The value of the contingent claim is attained and Proof: Let us suppose that XF'c''>_ B a.s.for some value of x > 0 and a suitable pair (II, C).Then from (10)it follows that T) -< -< Consequently, z-E _< U.
Let us define the nonnegative random process where ffh-E F is a P-Gaussian martingale, such that ff0-E Analogously to the proof of the Theorem 1, we can apply the generalized Girsanov's theorem and the martingale representation theorem.
By comparing the processes and x''x we obtain that Consequently, z >_ U. (21) Remark 1: Let us note that (21) yields Xo()-x ''z-, .s., i.e., the contingent claim is attained with the initial capital U, portfolio II, and zero consumption.This fact could be used as a starting point for solving appropriate optimal problems.empty.Then Theorem 1 and Theorem 2 reduce to the results of Karatzas and Shreve [7] and Cvitani and Karatzas [2].

P(T)
By Ito's rule it can be proved that (i.T) is a P-Gaussian martingale and is a unique solution of the following stochastic equation: T is a P-Gaussian martin- gale and it yields representation (ii). []