SOME ESTIMATES ON EXPONENTIALS OF SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS

Exponential of functionals of solutions to certain stochastic
differential equations (SDEs) plays an interesting role in some
mathematical finance problems. The purpose of this paper is to
establish some estimates for these exponentials.

We return to (1.2).Formally, the solution of (1. (1.4) In studying some mathematical finance problems such as contingent claim pricing, optimal investment, and so on, one hopes that the wealth process Y (•) is a well-defined process belonging to, say, L 1 Ᏺ (Ω;C([0,T];R)), the set of all continuous {Ᏺ t } t≥0 -adapted processes ϕ(•) such that E[sup t∈ [0,T] for some λ ∈ (0,1), at least.On the other hand, in almost all relevant studies, we should at least be allowed to take π(•) = 0 (which means that the investor puts all the wealth in the bond and does not hold any stocks).Then, in order that Y (•; y,0)∈L 1  Ᏺ (Ω;C([0,T];R)) or E[Y (T; y,0) λ ]<∞, one should at least have E e λ T 0 r(u)du < ∞, (1.5) for some λ > 0. In the case that r(•) is bounded, (1.5) holds automatically.However, when r(•) is a strong solution of (1.3), it is by no means clear whether (1.5) holds.As a matter of fact, unfortunately, we will show the following result.
The above result tells us that in studying problems that involve E[Y (T; y,π(•)) λ ], the interest rate model (1.3) with δ > 1/2 is not very suitable, and the CIR model could be used but one has to restrict the time duration T.
Next, we look at another interesting problem.We still take market model (1.1) and the wealth process equation (1.2).We assume that n = d and σ(t) −1 exists for all t ∈ [0,T].Then one can define which is referred to as the risk premium of the market.It is known that if the so-called Novikov's condition (see [10,14]) holds, Jiongmin Yong 289 or the so-called Kazamaki's condition (see [12]) (which is weaker than (1.7)) holds, then the process is a uniformly integrable {Ᏺ t } t≥0 -martingale on [0, T], and is an {Ᏺ t } t≥0 -standard Brownian motion on (Ω,Ᏺ T , P T ) with P T being a probability measure on (Ω,Ᏺ T ) equivalent to P| ᏲT , defined by Moreover, one calls P T an equivalent martingale measure of the market, and every discounted stock price process is an {Ᏺ t } t≥0 -martingale under P T .In this case, the market is arbitrage-free.One usually refers to (1.9), (1.10), and (1.11) as Girsanov's transformation (see [10]).
In mathematical finance, the existence of equivalent martingale measure plays a very important role because it is (almost) equivalent to the market being arbitrage-free (see [4]).Thus, people hope that either (1.7) or (1.8) holds.Again, when r(•) is a strong solution of (1.3), say, it is unbounded in general.In such a case, even if σ(•) −1 is bounded, we do not have the boundedness of θ(•).Hence, one would like to know when (1.7) and (1.8) hold.The following result will be proved in a later section.Propsition 1.2.Let n = d = 1 and let b(•), σ(•), and σ(•) −1 be bounded.Let θ(•) be defined by (1.6) with r(•) being the strong solution of (1.3).Then (1.7) fails as long as δ ≥ 1/2, and for δ = 0, there exists a T 0 such that (1.7) fails as well if T > T 0 .Also, (1.8) fails when δ ≥ 1/2 together with some additional conditions.
The above is again a kind of negative result, which tells us that in the context involving equivalent martingale measures and/or arbitrage-freeness, one has to be careful to use (1.3) as the short-interest rate model.This remark also applies to the context when a multi-factor model is used (see [1]).The situation for more general models found in [5,6] is still under careful investigation and we will address the results elsewhere.
The above two motivations suggest we formulate a more general problem.Consider the following SDE on (Ω,Ᏺ,{Ᏺ t } t≥0 ,P): are given maps.Suppose (1.12) admits a strong solution X(•).Next, let ϕ,ψ : R n → [0,∞) and µ : R n → R d be given.We pose the following problem.
Problem (E).Find conditions on b(•), σ(•), ϕ(•), ψ(•), and µ(•) such that the following hold: ) It is standard that under some mild conditions, such as uniform Lipschitz condition on the coefficients b(t,x) and σ(t,x) in x, or even some weaker conditions (see [9,10]), (1.12) admits a unique strong solution X(•).Moreover, when b(t,x) and σ(t,x) grow at most linearly in x, the following estimate holds for the solution X(•): for any m > 0 and T > 0, for some constant C m,T > 0 depending on m, T, and b(•) and σ(•).It is then very natural to ask when (1.13) holds.From this point of view, one sees that our problem has its own interest, which is concerned with estimates of solutions to SDEs.We would like to point out that an interesting case for (1.13) is ϕ(x) = |x| γ for some γ > 0. This then indicates that our problem is also closely related to the topics discussed in [2].We would also like to mention the book [19] in which the exponential functional of Brownian motion (or even a Lévy process) has been systematically studied.If one replaces the Brownian motion by the strong solution of some (nonlinear) SDE, then one faces the problem that we are going to study in this paper.Hence, this work is closely related to that of [19] in a certain sense as well.
The rest of the paper is organized as follows.In Section 2, we present some preliminary results.Section 3 is devoted to some estimates for stochastic differential inequalities.In Section 4, we give precise statements and proofs of Propositions 1.1 and 1.2.Finally, general results for Problem (E) are presented in Section 5.

Some preliminaries
In this section, we present some preliminary results.First of all, we introduce some spaces.For any Euclidean space H (such as R n , R n×m , etc.) whose norm is denoted by (2.1) The spaces of the above types corresponding to p = ∞ and/or q = ∞ can be defined in an obvious way.We will simply denote L p Ᏺ (0,T;L p (Ω;H))=L p Ᏺ (Ω;L p (0,T;H))=L p Ᏺ (0,T;H), for any 1 ≤ p ≤ ∞.Also, we define (2.2) We now present the following result.
For the case γ = 2, we have Then we obtain the conclusions for γ = 2.
We make a remark here.For n = d = 1, it is known by Dambis-Dubins-Schwarz theorem [11,15,16] that when σ(•) ∈ L 2 Ᏺ (0,T;R), for each T > 0, we may define Then B(•) is a one-dimensional standard Brownian motion under filtration {Ᏻ s } s≥0 .Moreover, Consequently, for any T > 0, if we define provided that we have first proved (2.17), for which a careful estimate is no simpler than (2.10), (2.11), and (2.12).On the other hand, if n,d ≥ 2, a direct application of Dambis-Dubins-Schwarz theorem might still be possible, but it will be much messier.Further, to prove the conclusion for γ > 2, such an approach seems not any simpler than (2.14) and (2.15).Hence, we have taken a direct (and elementary) approach without using Dambis-Dubins-Schwarz theorem in the above.Note that (2.4) implies σ( In this case, one may actually take We point out that if σ(•) / ∈ L 2 Ᏺ (0,T;L ∞ (Ω;R n×d )), (2.8) might fail for γ ∈ (0,2).In fact, for n = d = 1, we take σ(t) = W(t).Then For such a case, by It ô's formula, we have provided T ≥ 1, by Corollary 2.2.This means that, in some sense, condition (2.7) is sharp.Also, we note that for part (i), we need only (2.4), and

Estimates for scalar stochastic differential inequalities
In this section, we establish some exponential estimates for solutions to some scalar stochastic differential inequalities which will be useful below.
(ii) Let ξ(•) be an {Ᏺ t } t≥0 -adapted process satisfying Further, In the above, (3.2) and (3.8) are understood as corresponding integral inequalities (involving Lebesgue and It ô integrals).Thus, the process ξ(•) involved in (3.2) or (3.8) is not necessarily an It ô process (i.e., it is not necessarily a solution to an SDE).The same convention applies below as well.
When all the coefficients are deterministic, condition (3.5) can be replaced by

Estimates involving interest rate term structure models
In this section, we will give precise statements as well as proofs of Propositions 1.1 and 1.2.To this end, we make the following assumption.
The following result is concerned with (1.8).

Higher-dimensional cases
In this section, we are going to present some results for higher dimensions.These results might be useful for problems involving multifactors (see [1,5,6]).We first make the following assumption.
In what follows, we assume that the SDE (1.12) admits a unique strong solution X(•).We now state and prove our main result of this paper.

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We now look at the case δ = 1.For this case, we only need to show that (5.4) and (5.6) hold if the first lines in (5.5) and (5.7) hold, respectively.To this end, we note that when δ = 1, instead of (5.15), one has (5.20) Thus, applying Itô's formula to ln X(t) ε , we have the following: where (5.22) Thus, by (5.1) (with δ = 1), we have (5.23) Also, we let This proves (i).
We now look at when (1.15) holds.This problem is a little more complicated.We present an example first.
The above example tells that not only the growth of x → |µ(t, x)| (as |x| → ∞) plays the role, but also the "direction" of µ(t,x) matters.The following result is concerned with (1.15) in a general manner.