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Sensitivity analysis of the utility maximization problem with respect to model perturbations

We study the sensitivity of the expected utility maximization problem in a continuous semi-martingale market with respect to small changes in the market price of risk. Assuming that the preferences of a rational economic agent are modeled with a general utility function, we obtain a second-order expansion of the value function, a first-order approximation of the terminal wealth, and construct trading strategies that match the indirect utility function up to the second order. If a risk-tolerance wealth process exists, using it as a num\'eraire and under an appropriate change of measure, we reduce the approximation problem to a Kunita-Watanabe decomposition.


Introduction
It is well-known, see for example [DS06,HS10], that for a continuous stock price process, the no-arbitrage condition implies that the return of the stock price S has the following representation: where M is a continuous local martingale, and λ is a predictable process, i.e., that the quadratic variation of a stock price has to be absolutely continuous with respect to the quadratic variation of M. We analyze the effect of perturbations of the market price of risk λ, on the utility maximization problem.
In the setting of an incomplete model, where the preferences of a rational economic agent are modeled with a general utility function U with bounded (away from zero and infinity) relative risk-aversion and the stock prices process is continuous, we obtain a quadratic expansion of the value function, a first-order correction to the optimal terminal wealth, and a construction of the approximate trading strategies that match the value functions up to the second order. For the power-utility case, a first-order asymptotic expansion with respect to perturbations of the market price of risk is obtained in [CR16], whereas a second-order analysis is performed in [LMŽ14]. Mathematically, the results in the present paper rely on different techniques. We can summarize our contribution as three-fold: (1) We first need to increase dimensionality and look at the simultaneous perturbations of the market price of risk and the initial wealth. As the proofs show, the increase of dimensionality is a necessary way of getting the expansions of the value functions up to the second order 1 .
(2) Then, we formulate auxiliary quadratic stochastic control problems and relate the second-order approximations of both primal and dual value functions to these problems.
(3) Finally, if the risk-tolerance wealth process exists, we use it as a numéraire, and change the measure accordingly, to identify solutions to the general quadratic optimization problems above in terms of a Kunita-Watanabe decomposition (of a certain martingale) generated by the perturbation process.
To the best of our knowledge, the closest paper from the mathematical viewpoint is [KS06b], where the authors obtain a second-order expansion of the value function with respect to simultaneous perturbations of the initial wealth and the number of units of random endowment held in the portfolio. We would like to stress that unlike the present setting, in [KS06b], the value function is jointly concave (in both the initial wealth and the number of units of random endowment held in the portfolio), a fact that plays a significant role in the proofs there.
We combine here the increase of dimensionality described in item (1) with a similar change of measure and numéraire to [KS06b] relating them to general quadratic optimization problems. However, one of the main technical difficulties lies in the fact that our value function as a function of two variables is not concave or convex in the perturbation variable δ (in general). Despite this obstacle, our approach, which relies only partially on convex conjugacy, still produces a quadratic expansion via auxiliary quadratic problems and simultaneous expansions of u in (x, δ) and v in (y, δ). In addition to obtaining a quadratic expansion, we also get a relationship between the existence of such an approximation and the existence of the risk-tolerance wealth process, which was 1 In the constant relative risk aversion case considered in [LMŽ14], as the optimal terminal wealth depends on the initial wealth via a multiplicative constant, the increase of dimensionality is not needed for obtaining quadratic expansions.
introduced in [KS06b]. We show that the existence of the risk-tolerance wealth process allows for a more explicit form of the correction terms in our approximation coming from a Kunita-Watanabe decomposition under appropriate measure and numéraire that are specified in terms of the risk-tolerance wealth process. Another connection to [KS06b] is given in Lemma 6.1, where the perturbation to the market-price of risk plays the role of a multiplicative (and non-linear) random endowment.
To separate the financial aspects of the problem from the mathematical ones, we state and prove abstract versions of main theorems. After that we reduce the proofs of (some of) the main theorems to verification of the conditions in the abstract theorems.
As an application, we consider models, which admit closed form solutions in incomplete markets, see [KO96,Liu07,GR15] (we also refer to [LMŽ14] for more examples and a literature review). These models are sensitive to perturbations of the input parameters: ones they are perturbed even slightly, a close form solution typically ceases to exist. Our results show that even though we do not know how to obtain an exact solution for such perturbed problems, an approximation, which is accurate up to the second order, can still be constructed.
We prove our results under the assumption of no unbounded profit with bounded risk, the weakest no-arbitrage type condition, which allows for the utility maximization problem from terminal wealth to be non-degenerate, see [KK07,Proposition 4.19]. For the perturbation process, we formulate an assumption and give a counterexample, which shows the necessity of the assumption. In addition, we provide a set of sufficient conditions for the integrability assumption on the perturbation process to hold.
For the general utility function, we suppose that its relative risk-aversion is bounded away from zero and infinity. This condition is (essentially) necessary for twice differentiability with respect to the initial wealth to hold, see [KS06a] for counterexamples. On an even more technical side, as we consider perturbations of the initial wealth, we obtain as a by-product here the second-order derivatives of the primal and dual value functions with respect to the spatial variables (x and y, correspondingly). Note that, in [KS06a] this result was obtained for discontinuous stock prices, but under NFLVR.
The remainder of the paper is organized as follows: in section 2, we formulate the model and state the expansion theorems, section 4 contains the approximation of optimal trading strategies theorem, section 5 includes abstract versions of Theorems 3.7, 3.8, 3.10, and 3.12 with proofs, section 6 contains proofs of non-abstract theorems and Theorem 4.1, where a construction of corrections to the optimal trading strategies (accurate up to the second order of the value function) is specified. Section 7 includes a counterexample, which shows that without Assumption 3.2 on the perturbation process, the quadratic expansions of the value functions might not exist. In section 8 we relate the asymptotic expansions from previous sections to the existence of the risk-tolerance wealth process and a Kunita-Watanabe decomposition. We conclude the paper with an illustration of an application of our results to analysis of the perturbations of models that admit closed-form solutions.
2. Model 2.1. Parametrized family of stock prices processes. Let us consider a complete stochastic basis Ω, F , {F t } t∈[0,T ] , P , where T ∈ (0, ∞) is the time horizon, F satisfies the usual conditions, and F 0 is the completion of the trivial σ-algebra. We assume that there are two traded securities, a bank account with zero interest rate and a stock. Let M be a one-dimensional continuous local martingale and λ is a progressively measurable process, such that The stock price return process 2 for the unperturbed, or equivalently, 0-model is given by Here we consider a parametric family of semimartingales S δ , δ ∈ R, with the same martingale part M and where the market price of risk λ's are perturbed where for some progressively measurable process ν, such that we have 2.2. Primal problem. Let U be a utility function that satisfies Assumption 2.1 below.
Assumption 2.1. The utility function U is strictly increasing, strictly concave, two times differentiable on (0, ∞) and there exist positive constants c 1 and c 2 , such that i.e. the relative risk aversion of U is uniformly bounded away from zero and infinity. 2 We denote the return of the stock by S, since R is used for different purposes.
The family of primal feasible sets is defined as where H is a predictable and S δ -integrable process representing the amount invested in the stock. The corresponding family of the value functions is given by We use the convention where U − is the negative part of U.
2.3. Dual problem. The investigation of the primal problem (2.5) is conducted via the dual problem. First, let us define the dual domain as follows: We set the convex conjugate to utility function U as Note that for y = U ′ (x), we have Therefore, Assumption 2.1 implies that The parametrized family of dual value functions is given by We use the convention where V + is the positive part of V .

Technical assumptions
We recall the assumption that M is continuous. The absence of arbitrage opportunities in the 0-model in the sense of no unbounded profit with bounded risk follows from condition (2.1), which implies that Y(1, 0) = ∅. Note that (2.1) and (2.2) impliy no unbounded profit with bounded risk for every δ ∈ R, thus In order for the problem (2.5) to be non-degenerate, we also need to assume that Remark 3.1. Conditions (2.1) and (3.1) are necessary for the expected utility maximization problem to be non-degenerate. Note that we only impose them for δ = 0.
As in [KS06a,KS06b], an important role will be played by the probability measures R(x, δ), given by dR(x, δ) dP for x > 0 and y = u x (x, δ). As Example 7.1 below demonstrates, we need to impose an integrability condition. First, let us define Assumption 3.2. Let x > 0 be fixed. There exists c > 0, such that Remark 3.3. The stronger condition for some ε > 0 and c > 0, where B ε (x, 0) denotes the ball in R 2 of radius ε centered at (x, 0), implies local semiconcavity of the value function u(x, δ). Consequently, in the quadratic expansions of u and v given by (5.22) and (5.23), the matrices H u (x, 0) and H v (y, 0) defined in (5.20) and (5.21), respectively, are Hessian matrices, i.e. are derivatives of gradients. This will follow from Lemma 5.14. However, the very restrictive condition (3.3) is an assumption that depends on optimal solutions for δ = 0, and thus usually impossible to check.
Let us also set (3.4) L δ E −(δν) · S 0 T , δ ∈ R. Here and below E denotes the Doléans-Dade exponential. One can see that L δ is a terminal value of an element of X (1, 0) for every δ ∈ R.

Sufficient conditions for Assumption 3.2
Remark 3.4. A sufficient condition for Assumption 3.2 to hold is the existence a wealth process under the numéraire X(x, 0), X, and a constant c > 0, such that exp c(|ν · S 0 | + ν 2 · M ) T ≤ X T , a.s.
Remark 3.5. Let us assume that in (2.3), c 1 > 1, i.e. that relative-risk aversion of U is strictly greater than 1, (for example, this holds if U(x) = x p p with p < 0, note that for such a U, the conjugate function V (y) = y −q q for q ∈ (−1, 0)). In this case, a sufficient condition for Assumption 3.2 to hold is the existence of some positive exponential moments under This can be shown as follows. Let us set As c 2 ≥ c 1 > 1, we deduce that q i ∈ (−1, 0), i = 1, 2. Using Lemma 5.12, one can find a constant C > 0, such that (3.5) − V ′ (y)y ≤ C y −q 1 + y −q 2 , y > 0.
Remark 3.6 (On the relationship with existing literature). Assumption 3.2 is related to the condition on random endowment, Assumption 4 in [KS06b], via the following argument.
Assume that, for some x > 0 and c > 0, there exists a wealth process X ∈ X (x, 0), such where X(x, 0) is the optimal solution to (2.5). Then Assumption 3.2 is satisfied. The wealth process X X(x,0) under the numeraire X(x, 0) in condition (3.8) is local martingale under R(x, 0), i.e., X can be an arbitrary element of X (x, 0). In [KS06b] it is assumed that X X(x,0) is a square-integrable martingale under R(x, 0).

Expansion Theorems
In Theorem 3.7 we prove finiteness of the value functions and first-order derivatives with respect to δ.
In addition, u and v are jointly differentiable (and, consequently, continuous) at (x, 0) and (y, 0), respectively. We also have In order to characterize the second-order derivatives of the value functions, we will need the following notations. Let S X(x,0) be the price process of the traded securities under the numéraire X(x,0) x , i.e.
For every x > 0, let H 2 0 (R(x, 0)) denote the space of square integrable martingales under R(x, 0), such that

Auxiliary minimization problems
As in [KS06a], for x > 0 let us consider where A is the relative risk aversion and B is the relative risk tolerance of U, respectively. It is proven in [KS06a] 3 that (3.12) and (3.13) admit unique solutions M 0 (x, 0) and N 0 (y, 0), correspondingly, and (3.14) u xx (x, 0) = − y x a(x, x), v yy (y, 0) = x y b(y, y), a(x, x)b(y, y) = 1, T (x, 0)) = a(x, x)(1 + N 0 T (y, 0)). In order to characterize the derivatives of the value functions with respect to δ, with we consider the following minimization problems: 3 Under the assumption of NFLVR. Below we will show that the formulas (3.12), (3.13), and (3.14) can also be obtained in the present setting.
Denoting by M 1 (x, 0) and N 1 (y, 0) the unique solutions to (3.16) and (3.17) respectively, we also set Theorems 3.8, 3.10, and 3.12 contain the second-order expansions of the value functions, derivatives of the optimizers, and properties of such derivatives.
Remark 3.9. Although we only have second order expansions, we may abuse the language and call H u (x, 0) and H v (y, 0) the Hessians of u and v, without having twice differentiability. This causes no confusion, see the discussion e.g., in [LS02]. The meaning of partial derivatives u xx (x, 0), u xδ (x, 0) and so on then becomes apparent by identifying entries in the Hessian matrices.
Remark 3.11. Continuing the discussion in Remark 3.9, (3.24) implies that Theorem 3.12. Let x > 0 be fixed, the assumptions of Theorem 3.7 hold, and y = u x (x, 0). Then the terminal values of the wealth processes M 0 (x, 0) and M 1 (x, 0), which are the solutions to (3.12) and (3.16), respectively, satisfy where the convergence takes place in P-probability and L δ 's are defined in (5.3). Likewise, let N 0 T (y, 0) and N 1 T (y, 0), which are solutions to (3.13) and (3.17), correspondingly, satisfy where the convergence takes place in P-probability.
One can obtain the following corollary.
Corollary 3.13. Let x > 0 be fixed, the assumptions of Theorem 3.7 hold, and y = u x (x, 0). Then, if we define where the convergence takes place in P-probability.
Remark 3.14. Even though Corollary 3.13 gives a more explicit form of the derivatives of the terminal wealth, an approximation given in (3.27) turns out to be more useful in applications.

Approximation of the optimal trading strategies
Below in this section we will suppose that x > 0 is fixed. Let us denote where π(x, 0) = ( π t (x, 0)) t∈[0,T ] is the optimal proportion invested in stock corresponding the initial wealth x and δ = 0. Note that for every predictable pair of processes G 1 and are well-defined, by direct computations, we can find a process G, such that Let γ 0 and γ 1 be such that We need to define the following families of stopping times.
Remark 4.2. Theorem 4.1 shows how to correct the optimal proportion in order to match the primal value function up to the second order jointly in (∆x, δ).
Remark 4.3. Proportions have a nicer representation of the corrections to optimal trading strategies in terms of the quadratic optimization problems (3.12) and (3.16) because the optimal wealth process was used as numéraire, i.e.,X(x, 0)/x has a multiplicative structure. The result in Theorem 4.1 compliments the results in [KS06a] and (in a different additive random endowment framework) those in [KS06b] in the context of a one-dimensional and continuous stock model.

Abstract version
Abstract version for 0-model We begin with the formulation of the abstract version for 0-model. As in [Mos15], let (Ω, F , P) be a measure space and we define the sets C and D to be subsets of L 0 + that satisfy the following assumption. Note that Assumption 5.1 is the abstract version of no unbounded profit with bounded risk condition (2.1).
Assumption 5.1. Both C and D contain a stricly positive element and as well as We also set C(x, 0) xC and D(x, 0) xD, x > 0. Now we can state the abstract primal and dual problems as Under finiteness of both primal and dual value functions on R, existence and uniqueness of solutions to (5.1) and (5.2) follow from [Mos15, Theorem 3.2]. Likewise, with a deterministic utility function that has reasonable asymptotic elasticity, if u(x, 0) < ∞ for some x > 0, standard conclusions of the utility maximization theory also follow from the abstract theorems in [KS99] (see the discussion in [CCFM15, Remark 2.5]).

Abstract version for δ-models
For some random variables F and G ≥ 0, let us set Now, we can state the abstract versions of the perturbed optimization problems.
Under an appropriate integrability assumption specified below, existence and uniqueness of solutions to (5.5) and (5.6) as well as conjugacy relations between u(·, δ) and v(·, δ) for every δ sufficiently close to 0 will follow from [Mos15, Theorem 3.2].
Assumption 5.2. Let there exists c > 0, such that Note that, R(x, 0) is well-defined for every x > 0.

Expansion theorems
Auxiliary sets A and B As in [KS06a], for every x > 0 and δ ∈ R, we denote by A ∞ (x, δ) the family of bounded Likewise, for y > 0 and δ ∈ R, we set It follows from the Assumption 5.1 that for every One can see that A 2 (x, δ) and B 2 (y, δ) are closed orthogonal linear subspaces of L 2 (R(x, δ)). In order to make these sets related to the concrete versions of the expansion theorems, we need the following assumption.
The following theorem shows joint continuity, and differentiability, and is a consequence of the second-order expansion.
Theorem 5.4. Let x > 0 be fixed. Suppose that assumptions 2.1, 5.1, 5.2, and 5.3 hold, u(z, 0) < ∞ for some z > 0, and y = u x (x, 0), which is well-defined by the abstract theorems in [KS99]. Then there exists δ 0 > 0 such that for every δ ∈ (−δ 0 , δ 0 ), we have In addition, u and v are jointly differentiable (and, consequently, continuous) at (x, 0) and (y, 0), respectively. We also have Remark 5.5. It is possible to prove Theorem 5.4 without Assumption 5.3. We do not present such a proof for brevity of exposition.

Auxiliary minimization problems
As in [KS06a], for x > 0, let us consider where A is the relative risk aversion and B is the relative risk tolerance of U, respectively.
We are ready to state the following theorem.
Theorem 5.6. Let x > 0 be fixed, the conditions of Theorem 5.4 hold, and y = u x (x, 0). Define Moreover, and (5.26) Theorem 5.8. Let the conditions of Theorem 5.4 hold and x > 0 be fixed. Then the random variables α and α d , which are solutions to (5.13) and (5.16), respectively, are the partial derivatives of the solution ξ(x, 0) to (5.5) evaluated at (x, 0), that is where the convergence takes place in P-probability. Likewise, let β and β d , which are solutions to (5.14) and (5.17), correspondingly, are the partial derivatives of the solution η(y, 0) to (5.6) evaluated at (y, 0), where y = u x (x, 0), in the sense that where the convergence takes place in P-probability.
From Theorem 5.8, we obtain the following Corollary.
Corollary 5.9. Under the conditions of Theorem 5.8, (5.28) is equivalent to where the convergence takes place in P-probability.

Proofs
We begin the proofs with technical lemmas.
Corollary 5.11. Under the conditions of Lemma 5.10, for every k ∈ N, we have Below 1 E denotes the indicator function of a set E.
Lemma 5.12. Let Assumption 2.1 holds. Then for every z ∈ (0, 1] and x > 0, we have Proof. Let us fix an arbitrary d ∈ (exp(−1/c 2 ), 1). Using monotonicity of U ′ and Corollary 5.11, for every z ∈ (0, 1] and x > 0, we get (5.30) Let us set As a 1 (d) > 1 and for every k ∈ N we deduce that for every z ∈ (0, 1], we have Plugging (5.31) in (5.30), we get taking the limit in the latter inequality, we obtain that for every z ∈ (0, 1] and x > 0. The other assertion can be proven similarly. This completes the proof of the lemma.
Corollary 5.13. Under Assumption 2.1, for every z > 0 and x > 0, we have

Proof of the second-order expansion
Lemma 5.14. Let x > 0 be fixed and the conditions of Theorem 5.4 hold, and y = u x (x, 0). For arbitrary random variables α 0 and α 1 in A ∞ (x, 0), let us define where ξ = ξ(x, 0) is the solution to (5.5) corresponding to x > 0 and δ = 0. Then w admits the following second-order expansion at (0, 0).
By passing to a smaller ε, if necessary, and by applying Hölder's inequality, we deduce from Assumption 5.2 that the right-hand side of (5.40) integrable. As the right-hand side of (5.40) depends on ε (and not on (s, t)), the assertion of the lemma follows from the dominated convergence theorem.
Corollary 5.15. Let let x > 0 be fixed, the conditions of Theorem 5.4 hold, and y = u x (x, 0). Then, we have where H u (x, 0) is given by (5.20).
Proof. The result follows from Lemma 5.14 via the approximation of the solutions to (5.13) and (5.16), which are the elements of A 2 (x, 0), by the elements of A ∞ (x, 0).
Similarly to Lemma 5.14 and Corollary 5.15, we can establish the following results. where the second-order partial derivatives ofw at (0, 0) are given bȳ Lemma 5.17. Let let x > 0 be fixed, the conditions of Theorem 5.4 hold, and y = u x (x, 0). Then, we have where H v (y, 0) is given by (5.21).

Closing the duality gap
We begin from the proof of Theorem 5.7.
It follows from the second equation in (5.44) that where we have used (5.43) in the last equality. Multiplying by − x y , we obtain It follows from characterization of the unique solution to (5.16) given by (5.44) that From (5.45), we obtainα Plugging this back into the second equality in (5.44), we get Multiplying by x y A(ξ), we obtain Setting d ′ x y da(x, x), we claim that where a(x, d) is defined in (5.18). Multiplying both sides of (5.47) by (1 + α), taking expectation under R(x, 0), and using orthogonality of the elements of A 2 (x, 0) and B 2 (y, 0), we get Therefore, where in the last equality we have used (5.18). Thus, (5.48) holds. Now, (5.47) with It remains to prove (5.25). Let us set Then from (5.16) using (5.26), we get . Likewise, from (5.17) via (5.49), we obtain Let us define and Then, adding (5.49) and (5.50), we deduce that Let us rewrite T 2 as (5.52) as all the terms under the expectation cancel except for −2β d α d , which has still 0 expectation by orthogonality of A 2 (x, 0) and B 2 (y, 0). Let us consider T 1 . First, from (5.24), we get Therefore, we can rewrite T 1 as (5.54) On the other hand, from (5.19) we can express b(y, d) in terms ofβ,β d ,ᾱ, andᾱ d as follows.
where in the last equality we have used (5.43). Comparing (5.55) with (5.54), we get Plugging this into (5.51) and using (5.52), we deduce that i.e. (5.25) holds. This completes the proof of the lemma.
Proof. First, note that b(y, y) > 0 in (5.56). By direct computations, proving (5.57) is equivalent to establishing the following equality.
(5.58) − y xb(y, y) Now, let us consider the right-hand side or (5.58). By direct computations, it can be rewritten as follows.
Pluggin (5.60) into (5.59), we can rewrite the right-hand side of (5.59) as which is precisely the left-hand side of (5.58). We have just shown that (5.58) holds. By the argument preceding (5.58), this implies that (5.57) is valid as well. This completes the proof of the lemma.
Lemma 5.19. Let x > 0 be fixed, the conditions of Theorem 5.4 hold, and y = u x (x, 0). Then, we have where H u (x, 0) is given by (5.20). Likewise where H v (y, 0) is given by (5.21).

Derivatives of the optimizers
We begin with a technical lemma.
Lemma 5.20. Let x > 0 be fixed, the conditions of Theorem 5.4 hold, y = u x (x, 0), and let (δ n ) n≥1 be a sequence, which converges to 0. Then, we have Proof. The proof goes along the lines of the proof of Lemma 5.14, it is therefore skipped.
Proof. In view of Theorem 5.4, without loss of generality, we may assume that v(y n , δ n ) is finite for every n ∈ N. Let us assume by contradiction that (η n ) n∈N does not converge in probability to η. Then there exists ε > 0, such that lim sup n→∞ P [|η n − η| > ε] > ε.
From convexity of V , we have and from the strict convexity of V , we deduce the existence of a positive constant ε 0 , such that lim sup Therefore, using Lemma 5.20, we obtain where in the last equality we have also used continuity of v at (y, 0), which follows from Theorem 5.4. On the other hand, as h n ∈ D yn+y 2 , δ n , n ≥ 1, we get Combining (5.66) and (5.67) and using continuity of v at (y, 0) again, we get v(y, 0) = lim sup which is a contradiction as ε 0 = 0. Thus, (η n ) n∈N converges to η in probability. In turn, this and continuity of v at (y, 0) imply the other assertion of the lemma.
Proof of Theorem 5.8. We will only prove (5.29), as (5.28) can be shown similarly. In view of Theorem 5.4, without loss of generality we will assume that for every n ∈ N, u(·, δ n ) and v(·, δ n ) are finite-valued functions. The rest of the proof goes along the lines of the proof of Theorem 2 in [KS06a]. Let (y n , δ n ) n∈N be a sequence, which converges to (y, 0), where y = u x (x, 0) > 0. Let η n = η(y n , δ n ), n ∈ N, denote the corresponding dual optimizers and set φ 1 1 2 min η(y, 0), inf n∈N η n > 0, P − a.s.
Moreover, the bipolar construction of the sets C(x n , δ n ) and D(y n , δ n ) implies that E ξ(x n , δ n )η n ≤ x n y n .
Combining this with (5.69) and using the expansion for v from Theorem 5.6, we obtain (5.70) lim sup n→∞ 1 (∆y n ) 2 + (δ n ) 2 (E [η n ] − v(y n , δ n )) ≤ where for a vector a = a 1 a 2 and a two-by-two matrix A, we define their norms as a a 2 1 + a 2 2 and A sup a∈R 2 Aa a .
In view of Lemma 5.16 (by the choice of β 0 and β 1 ), we can make the right-hand side of (5.70) arbitrarily small. Combining this with (5.69), we deduce that can also be made arbitrarily small. As θ > 0, P − a.s., the assertion of the theorem follows.
6. Proofs of Theorems 3.7, 3.8, 3.10, 3.12, and 4.1 In order to link abstract theorems to their concrete counterparts, we will have to establish some structural properties of the perturbed primal and dual admissible sets first.

Characterization of primal and dual admissible sets
The following lemma gives a useful characterization of the primal and dual admissible sets after perturbations.
Proof of Theorem 3.7. Condition (2.1) implies that the respective closures of the convex solid hulls of {X T : X ∈ X (1, 0)} and {Y T : Y ∈ Y(1, 0)} satisfy (abstract) Assumption 5.1. In view of Lemma 6.1, we have Therefore, the respective closures of convex solid hulls of Proof of Theorem 3.12. As above, the affirmations of this theorem follow from (abstract) Theorem 5.8.
Lemma 6.2. Assume that x > 0 is fixed and the assumptions of Theorem 3.7 hold. Then, for f defined in (6.1), there exists a monotone function g, such that Proof. The proof goes along the lines of the proof of Lemma 5.14. We only outline the main steps for brevity of exposition. For a fixed ε > 0, let us define where M R is defined in (4.1). Let us first fix ε ′ > 0, then fix (∆x, δ) ∈ B ε ′ (0, 0), and set ψ(z) ψ(z∆x, zδ), z ∈ (−1, 1).

Counterexample
In the following example we show that even when 0-model is nice, but Assumption 3.2 fails, we might have u(z, δ) = v(z, δ) = ∞ f or every δ = 0 and z > 0.
Example 7.1. Consider the 0-model, where Let assume that B is a Brownian motion defined on a filtered probability space (Ω, F , P), where the filtration (F t ) t∈[0,T ] is generated by B. We recall that for the utility function Therefore by [KS03], the standard conclusions of the utility maximization theory hold.
The primal and dual optimizers are Now, let us consider a process ν such that (7.1) ν · B 1 = B 3 1 , P − a.s.

Relationship to the risk-tolerance wealth process
Following [KS06b], we recall that for an initial wealth x > 0 and δ ∈ R, the risktolerance wealth process is a maximal wealth process R(x, δ), such that i.e. it is a replication process for the random payoff given by the right-hand side of (8.1). In general the risk-tolerance wealth process R(x, δ) may not exist. It is shown in [KS06b] that the existence of the risk-tolerance wealth process is closely related to no changes are needed. Therefore, the proof of the following theorem is not presented.
In addition, if these assertions are valid, then the initial value of R(x) is given by where the limits in (8.4) and (8.5) take place in P-probability.
As in [KS06b], for x > 0 and with y = u x (x, 0), let us define and choose R(x,0) R 0 (x,0) as a numéraire, i.e., let us set We define the spaces of martingales and N 2 (y, 0) it the orthogonal complement in H 2 0 ( R(x, 0)). We start with the following simple lemma (stated without a proof) relating the change of numéraire to the structure of martingales: Lemma 8.2. Let x > 0 be fixed, assume that the conditions of Theorem 8.1 hold, and denote y = u x (x, 0). Then, we have (8.9) M ∈ M 2 (x, 0) if and only if M X T (x, 0) R T (x, 0) ∈ M 2 (x, 0), and N ∈ N 2 (y, 0) if and only if N ∈ N 2 (y, 0).
The following theorem describes the structural properties the approximations in Theorems 3.8, 3.10, and 3.12 under the assumption that the risk-tolerance process exists. In words, the second order approximation of the value function optimal strategies amounts to a Kunita-Watanabe decomposition under the changes of measure and numéraire described above.
Theorem 8.3. Let x > 0 be fixed, assume that the conditions of Theorem 8.1 hold, and denote y = u x (x, 0). Let us also assume that the risk-tolerance process R(x, 0) exists.
Consider the Kunita-Watanabe decomposition of the square integrable martingale given by Then, the optimal solutions M 1 (x, 0) and N 1 (y, 0) of the quadratic optimization problems In addition, the Hessian terms in the quadratic expansion of u and v can be identified as and (8.14) x y x 2 E R(y,0) M 1 T 2 + R 0 (x,0) x y x The cross terms in the Hessians of u and v are identified as a(x, d) = P 0 and b(y, d) is given by .
With these identifications, all the conclusions of Theorem 3.8 and Corollary 3.13 hold true.
Remark 8.4. Applying Itô formula, one can find expressions for the corrections of the optimal proportions in terms of the Kunita-Watanabe decomposition under risk-tolerance wealth process as numéraire, in the spirit of Theorem 4.1. However, in the general case when R(x,0) R 0 (x,0) = X ′ (x, 0) = X(x, 0)/x such a correction to proportions also contains the terms X(x, 0)/R(x, 0) and M 1 (x, 0).
Remark 8.5. Theorem 8.3 gives an interpretation of a(x, d) as an utility-based price. Let us start by observing that If there exists a wealth process X ′ such that and X ′ Y is a uniformly integrable martingale 4 , according to [HK04,HKS05], a(x, d) represents the marginal utility-based price of the "random endowment" ( X T (x,0)−R T (x,0)) R 0 (x,0) xF.

An extended remark
Below we will consider an application of our results. As was pointed out in the introduction, there is a number of models, or rather classes of models, which admit a closed form solution, see for example [Liu07] and references therein. Once we perturb the input parameters, the solution typically halts to exist in the closed form. However, Theorems 3.7, 3.8, 3.12, and 4.1 give approximations to the value function, the optimizer, and the optimal trading strategy. We will assume that U(x) = x p p , p ∈ (−∞, 0) ∪ (0, 1) and there are two traded securities, a money market account with zero interest rate and one traded stock that satisfies conditions of [Liu07]. In this case the optimal strategy can be obtained explicitly, see [Liu07,Proposition 2].
Using Theorem 3.10, we deduce that the Kunita-Watanabe decomposition of E R(x,0) p 1−p F |F t where F is specified in (3.15), is: where y = u x (1, 0) and N 1 (y, 0) is the solution to (3.17).