Assume we are given market with m risky assets. Denote by S_i(t) the price of i-th asset at time t. We shall assume that the prices of assets depend on k economical factors x_i(n), i=1,...,k, with dynamics changes in discrete time moments denoted for simplicity by n=0,1,...., in the following way:
for t belonging to the interval [n,n+1),

dS_i(t)/S_i(t)=a_i(x(n))dt+Σ_j=1^k+mσ_ij(x(n))dw_j(t), (1)

where (w(t)=(w₁(t),w₂(t),..., w_k+m(t)) is a k+m dimensional Brownian motion defined on a given probability space (Ω,(F_t),F). Economical factors x(n)=(x₁(n),...,x_k(n)), satisfy the equation:

x_i(n+1)=x_i(n)+b_i(x(n))+Σ_j=1^k+md_ij(x(n))(w_j(n+1)-w_j(n))=g(x(n),W(n)), (2)

where W(n):=(w₁(n+1)-w₁(n),...,w_k+m(n+1)-w_k+m(n)).

We assume that 'a', 'b' are bounded continuous vector functions, and 'σ ', 'd' are bounded continuous matrix functions of suitable dimensions. Additionally we shall assume that the matrix dd^T (the superscript 'T' stands for transponse) is nondegenerate. Notice that equation (2) corresponds to discretization of a diffusion process. The set of factors may include divident yields, price - earning rations, short term interest rates, the rate of inflation see e.g. [1]. The dynamics of such factors is usually modeled using diffusion, frequently linear equations eg. in the case when we assume following [1] that 'a' is a function of spot interest rate governed by the Vasicek process. Our assumptions concerning boundedness of vector functions 'a' and 'b' may be relaxed allowing linear growth, however in such case we shall need other more complicated assumptions.

Assume that starting with an initial capital V(0) we invest in assets. Let h_i(n) be the part of the wealth process located in the i-th asset at time n, which is assumed to be nonnegative. The choice of h_i(n) depends on our observation of the asset prices and economical factors up to time 'n'. Denoting by V(n) the wealth process at time 'n' and by h(n)=(h₁(n), ...,h_m(n)) our investment strategy at time 'n', we have that h(n) belongs to U={(h₁,...,h_m), h_i >=0, Σ_i=1^m h_i=1} and

V(n+1)/V(n) = Σ_i=1^m h_i(n)ξ_i(x(n),W(n)), (3)

where

ξ_i(x(n),W(n))=exp{a_i(x(n))-σ²_ij(x(n))/2 + Σ_j=1^k+mσ_ij(x(n))(w_j(n+1)-w_j(n))}.

We are interested in the following investment problems:
maximize risk neutral cost functional

J⁰_x(h(n))= lim inf_n->oo {{E_x{lnV(n)}}/n}, (4)

and maximize risk sensitive cost functional

J⁰_x(h(n))= {lim sup_n->oo {{E_x{V(n)^γ}}/n}}/γ , (5)

with γ <0. Using (3) we can write the cost functionals (4) and (5) in the more convenient forms. Namely,

J⁰_x(h(n))= lim inf_n->oo {{E_x{Σ_t=0^n-1 ln (Σ_i=1^m h_i(t)ξ_i(x(t),h(t))}}/n}
=lim inf_n->\infty {{E_x{Σ_t=0^n-1 c(x(t),h(t))}}/n}, (6),

with c(x,h)=E{ln(Σ_i=1^mh_iξ_i(x,W(0)))}. It is clear that risk neutral cost functional J⁰ depends on uncontrolled Markov process (x(n)) and we practically maximize the cost function c itself. Consequently an optional control is of the form control (u'(x(n)), where sup_h c(x,h)=c(x, u'(x)) and function Borel measurable u': R^k -> U exists by continuity of c for fixed x belonging to R^k. This control does not depend on asset prices and is a time independent function of current values of the factors x only. The Bellman equation corresponding to the risk neutral control problem is of the form

w(x) + λ = sup_h (c(x,h) + Pw(x)), (7)

where Pf(x):= E_x{f(x(1))} for f belonging to bB(R^k) - the space of bounded Borel measurable functions on R^k, is a transition operator corresponding to (x(n)). We shall show that there are solutions w and λ to the equation (7) and λ is an optimal value of the cost functional J⁰. Letting

ξ^h,γ _n(ω):=
Π_t=0^n-1exp{γ ln (Σ_i=1^m h_i(t)ξ_i(x(t),W(t)))}(E{exp{γ ln (Σ_i=1^m h_i(t)ξ_i(x(t),W(t))}|F_t})^-1

consider a probability measure P^h,γ defined by its restrictions P^h,γ to the first n time moments given by the formula

P_|n^h,γ=ξ^h,γ_n(ω)=P_|n(dω ).

Then

J_x^γ(h(n))={lim sup _n->oo{ln E_x{exp{γ Σ_t=0^n-1ln(Σ_i=1^m h_i(t)ξ_i(x(t), W(t)))}}/n}}/γ
={lim sup_n->oo {ln E^h,γ_x{exp{Σ_t=0^n-1 c_γ(x(t),h(t))}}/n}}/γ, (8)

with

c_γ(x,h):=ln(E{(Σ_i=1^m h_iξ_i(x,W(0)))^γ}). (9)

Risk sensitive Bellman equation corresponding to the cost functional J^γ is of the form

exp(w_γ(x))=inf_h {exp(c_γ(x,h)-λ_γ)\int_E exp(w_γ(y))P^h,γ(x,dy)}, (10)

where for f belonging to bB(R^k)

P^h,γ=E{(Σ_i=1^m h_iξ(x,W(0)))^γexp{-c_γ(x,h)f(g(x,W(0)))}, (11)

and where λ_γ/γ corresponds to optimal value of the cost functional (8). Notice that under measure P^h,γ the process (x(n)) is still Markov but with controlled transition operator P^h,γ(x,dy). Following [5] we shall show that

λ_γ/γ -> 0

whenever γ ->0.

The study of risk sensitive portfolio optimization has been originated in [1] and then continued in a number of papers in particular in [12]. Risk sensitive cost functional was studied in papers [9], [5], [6], [3], [4], [8], [2], [7] and references therein. Using splitting of Markov processes arguments (see [11]) we study Poisson equation for additive cost functional the solution of which is also a solution to risk neutral Bellman equation. We consider then risk sensitive portfolio optimization with risk factor close to 0. We generalize the result of [12], where uniform ergodicity of factors was required and using [7] show the existence of Bellman equation for small risk in a more general ergodic case. The proof of that nearly optimal continuous risk neutral control function is also nearly optimal for risk sensitive cost functional with risk factor close to 0 is based on modification of the arguments of [5] using some results from the theory of large deviations.

References

[1] T.R. Bielecki, S. Pliska, "Risk sensitive dynamic asset management", JAMO, 39 (1999), pp.337-360.

[2] V.S. Borkar, S.P. Meyn, "Risk-Sensitive Optimal Control for Markov Decision Processes with Monotone Cost", Math. Meth. Oper. Res., 27 (2002), pp. 192-209.

[3] R. Cavazos-Cadena, "Solution to the risk-sensitive average cost optimality in a class of Markov decision processes with finite state space", Math. Meth. Oper. Res., 57 (2003), pp. 263-285.

[4] R. Cavazos-Cadena, D. Hernandez-Hernandez, "Solution to the risk-sensitive average optimally equation in communicating Markov decision chains with finite state space: An alternative approach", Math. Meth. per. Res., 56 (2002), pp.473-479.

[5] G.B. Di. Masi, L. Stettner, "Risk sensitive control of discrete time Markov processes with infinite horizon", SIAM J. Control Optimiz., 38 (2000), pp. 61-78.

[6] G.B. Di Masi, L. Stettner, "Infinite horizon risk sensitive control of discrete time Markov processes with small risk, Sys. Control Lett., 40 (2000), pp. 305-321.

[7] G.B. Di Masi, L. Stettner, "Infinite horizon risk sensitive control of discrete time Markov processes under minorization property", submitted.

[8] W.H. Fleming, D. Hernandez-Hernandez, "Risk control of finite state machines on an infinite horizon", SIAM J. Control Optimiz., 35 (1997), pp. 1790-1810.

[9] D. Hernandez-Hernandez, S.J. Marcus, "Risk sensitive control of Markov processes in countable state space", Sys. Control Letters, 29 (1996), pp. 147-155.

[10] I. Kontoyiannis, S.P. Meyn, "Spectral Theory and Limit Theorems for Geometrically Ergodic Markov Process", Ann. Appl. Prob. 13 (2003), 304-362.

[11] S.P. Myen, R.L. Tweedie, "Markov Chains and Stochastic Stability", Springer 1996.

[12] L. Stettner, "Risk sensitive portfolio optimization", Math. Meth. Oper. Res. 50 (1999), 463-474.

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