Recent Advances in Nonlinear Filtering with a Financial Application to Derivatives Hedging under Incomplete Information Recent Advances in Nonlinear Filtering with a Financial Application to Derivatives Hedging under Incomplete Information

In this chapter, we present some recent results about nonlinear filtering for jump diffu- sion signal and observation driven by correlated Brownian motions having common jump times. We provide the Kushner-Stratonovich and the Zakai equation for the nor- malized and the unnormalized filter, respectively. Moreover, we give conditions under which pathwise uniqueness for the solutions of both equations holds. Finally, we study an application of nonlinear filtering to the financial problem of derivatives hedging in an incomplete market with partial observation. Precisely, we consider the risk-minimizing hedging approach. In this framework, we compute the optimal hedging strategy for an informed investor and a partially informed one and compare the total expected squared costs of the strategies.


Introduction
Bayesian inference and stochastic filtering are strictly related, since in both approaches, one wants to estimate quantities which are not directly observable. However, while in Bayesian inference, all uncertainty sources are considered as random variables, stochastic filtering refers to stochastic processes. It also covers many situations, from linear to nonlinear case, with various types of noises.
The objective of this chapter is to present nonlinear filtering results for Markovian partially observable systems where the state and the observation processes are described by jump diffusions with correlated Brownian motions and common jump times. We also aim at applying this theory to the financial problem of derivatives hedging for a trader who has limitative information on the market.
A filtering model is characterized by a signal process, denoted by X, which cannot be observed directly, and an observation process denoted by Y whose dynamics depends on X. The natural filtration of Y, F Y ¼ {F Y t , t ∈ ½0, T}, represents the available information. The goal of solving a filtering problem is to determine the best estimation of the signal X t from the knowledge of F Y t . Similar to optimal Bayesian filtering, we seek for the best estimation of the signal according to the minimum mean-squared error criterion, which corresponds to compute the posterior distribution of X t given the available observations up to time t.
Historically, the first example of continuous-time filtering problem is the well-known Kalman-Bucy filter which concerns the case where Y gives the observation of X in additional Gaussian noise and both processes X and Y are modeled by linear stochastic differential equations. In this case, one ends up with a filter having finite-dimensional realization. Since then, the problem has been extended in many directions. To start, a number of authors including Refs. [1][2][3] studied the nonlinear case in the setting of additional Gaussian noise. Other references in a similar framework are given, for instance, by Refs. [4][5][6][7][8]. Subsequently also the case of counting process or marked point process observation has been considered (see Refs. [9][10][11][12][13][14] and reference therein). A more recent literature contains the case of mixed-type observations (marked point processes and diffusions or jump-diffusion processes), see, for, example, Refs. [15][16][17][18].
There are two major approaches to nonlinear filtering problems: the innovations method and the reference probability method. The latter is usually employed when it is possible to find an equivalent probability measure that makes the state X and the observations Y independent. This technique may appear problematic when, for instance, signal and observation are correlated and present common jump times. Therefore, in this chapter, we use the innovations approach which allows circumventing the technical issues arising in the reference probability method. By characterizing the innovation process and applying a martingale representation theorem, we can derive the dynamics of the filter as the solution of the Kushner-Stratonovich equation, which is a nonlinear stochastic partial integral differential equation. By considering the unnormalized version of the filter, it is possible to simplify this equation and make it at least linear. The resulting equation is called the Zakai equation, and due to its linear nature, it is of particular interest in many applications. We also compute the dynamics of the unnormalized filter, and we investigate pathwise uniqueness for the solutions of both equations. Normalized and unnormalized filters are probability measure and finite measure-valued processes, respectively, and therefore in general infinite-dimensional. Due to this, various recursive algorithms for statistical inference have come in to address this intractability, such as extended Kalman filter, statistical linearization, or particle filters. These algorithms intend to estimate both state and parameters. For the parameter estimation, we also mention the expectation maximization (EM) algorithm which enables to estimate parameters in models with incomplete data, see, for example, Ref. [19].
The success of the filtering theory over the years is due to its use in a great variety of problems arising from many disciplines such as engineering, informational sciences and mathematical finance. Specifically, in this chapter, we have a financial application in view. In real financial markets, it is reasonable that investors cannot fully know all the stochastic factors that may influence the prices of negotiated assets, since these factors are usually associated with economic quantities which are hard to observe. Filtering theory represents a way to measure, in some sense, this uncertainty. A consistent part of the literature over the last years has considered stochastic factor models under partial information for analyzing various financial problems, as, for example, pricing and hedging of derivatives, optimal investment, credit risk, and insurance modeling. A list, definitely nonexhaustive, is given by Refs. [15,16,[20][21][22][23][24][25][26]).
In the following, we consider the problem of a trader who wants to determine the hedging strategy for a European-type contingent claim with maturity T in an incomplete financial market where the investment possibilities are given by a riskless asset, assumed to be the numéraire, and a risky asset with price dynamics given by a geometric jump diffusion, modeled by the process Y. We assume that the drift, as well as the intensity and the jump size distribution of the price process, is influenced by an unobservable stochastic factor X, modeled as a correlated jump diffusion with common jump times. By common jump times, we intend to take into account catastrophic events which affect both the asset price and the hidden state variable driving its dynamics. The agent knows the asset prices, since they are publicly available, and trades on the market by using the available information F Y .
Partial information easily leads to incomplete financial markets as clearly the number of random sources is larger than the number of tradeable risky asset. Therefore, the existence of a self-financing strategy that replicates the payoff of the given contingent claim at maturity is not guaranteed. Here, we assume that the risky asset price is modeled under a martingale measure, and we choose the risk-minimization approach as hedging criterion, see, for example, Refs. [27,28].
According to this method, the optimal hedging strategy is the one that perfectly replicates the claim at maturity and has minimum cost in the mean-square sense. Equivalently, we say that it minimizes the associated risk defined as the conditional expected value of the squared future costs, given the available information (see Refs. [28,29] and references therein).
The risk-minimizing hedging strategy under restricted information is strictly related to Galtchouk-Kunita-Watanabe decomposition of the random variable representing the payoff of the contingent claim in a partial information setting. Here, we provide a characterization of the risk-minimizing strategy under partial information via this orthogonal decomposition and obtain a representation in terms of the corresponding risk-minimizing hedging strategy under full information (see, e.g., Refs. [29,30]) via predictable projections on the available information flow by means of the filter. Finally, we investigate the difference of expected total risks associated with the optimal hedging strategies under full and partial information.
The chapter has the following structure. In Section 2, we introduce the general framework. In Section 3, we study the filtering equations. In particular, we derive the dynamics for both normalized and unnormalized filters, and we investigate uniqueness of the solutions of the Kushner-Stratonovich and the Zakai equation. In Section 4, we analyze a financial application to risk minimization by computing the optimal hedging strategies for a European-type contingent claim under full and partial information and providing a comparison between the corresponding expected squared total costs.

The setting
We consider a pair of stochastic processes (X,Y), with values on R Â R and càdlàg trajectories, on a complete filtered probability space ðΩ, F , F, PÞ, where F ¼ {F t , t ∈ ½0, T} is a filtration satisfying the usual condition of right continuity and completeness, and T is a fixed time horizon. The pair (X, Y) represents a partially observable system, where X is a signal process that describes a phenomenon which is not directly observable and Y gives the observation of X, and it is modeled by a process correlated with the signal, having possibly common jump times.
Remark 1. In view of the financial application discussed in Section 4, Y represents the price of some risky asset, while X is an unknown stochastic factor, which may describe the activity of other markets, macroeconomic factors or microstructure rules that influences the dynamics of the stock price process.
We define the observed history as the natural filtration of the observation process Y, that is, The σ-algebra F Y t can be interpreted as the information available from observations up to time t. We aim to compute the best estimate of the signal X from the available information, in the quadratic sense. In other terms, this corresponds to determine the filter which furnishes the conditional distribution of X t given F Y t , for every t ∈ [0, T].
Let MðRÞ be the space of finite measures over R and PðRÞ the subspace of the probability measures over R. Given μ ∈ MðRÞ, for any bounded measurable function f, we write Definition 2. The filter is the F Y -càdlàg process π taking values in PðRÞ defined by for all bounded and measurable functions f (t, x) on [0, T] Â R.
For the rest of the paper, we assume that strong existence and uniqueness for system Eq. (3) holds. Sufficient conditions are collected, for instance, in Ref. [18,Appendix]. These assumptions also imply Markovianity for the pair (X, Y).
Remark 3. Note that the quadratic variation process of Y defined by is F Y -adapted and ½Y t ¼ Therefore, it is natural to assume that the signal X does not affect the diffusion coefficient in the dynamics of Y. If Y describes the price of a risky asset, this implies that the volatility of the stock price does not depend on the stochastic factor X.
The jump component of Y can be described in terms of the following integer-valued random measure on [0, T] Â R: where δ a denotes the Dirac measure at point a. Note that the following equality holds: For all t ∈ [0, T], for all A ∈ BðRÞ, we define the following sets: Typically, we have D 0 t ∩ D t 6 ¼ Ø P À a.s., which means that state and observation may have common jump times. This characteristic is particularly meaningful in financial applications to model catastrophic events that produce jumps in both the stock price and the underlying stochastic factor that influences its dynamics.
To ensure existence of the first moment for the pair (X, Y) and non-explosiveness for the jump process governing the dynamics of X and Y, we make the following assumption: Denote by η P ðdt, dzÞ the ðF, PÞ compensator of mðdt, dzÞ (see, e.g., Refs. [9,31] for the definition).
Remark 5. Let us observe that both the local jump characteristics ðλðt, X tÀ , Y tÀ Þ, φðt, X tÀ , Y tÀ , dzÞÞ depend on X and, for all A ∈ BðRÞ, λðt, X tÀ , Y tÀ Þφðt, X tÀ , Y tÀ , AÞ ¼ νðD A t Þ provides the ðF, PÞ -intensity of the point process N t ðAÞ :¼ mðð0, t Â AÞ. According to this, the process λðt, X tÀ , Y tÀ Þ ¼ νðD t Þ is the ðF, PÞ -intensity of the point process N t ðRÞ which counts the total number of jumps of Y until time t.

The innovation process
To derive the filtering equation, we use the innovations approach. This method requires to introduce a pair ðI, m π Þ, called the innovation process, consisting of the ðF Y , PÞ-Brownian motion and the ðF Y , PÞ-compensated jump measure that drive the dynamics of the filter. The innovation also represents the building block of ðF Y , PÞ -martingales.
To introduce the first component of the innovation process, we assume that and define The process I is an ðF Y , PÞ-Brownian motion (see, e.g., Ref. [4]) and the ðF Y , PÞ-compensated jump martingale measure is given by See, e.g. Ref. [14]. The following theorem provides a characterization of the ðF Y , PÞ-martingale in terms of the innovation process.
Theorem 6 (A martingale representation theorem). Under Assumption 4 and the integrability condition Eq. (15), every ðF Y , PÞ-local martingale M admits the following decomposition: where wðzÞ ¼ {w t ðzÞ, t ∈ ½0, T} is an F Y -predictable process indexed by z, and h ¼ {h t , t ∈ ½0, T} is an F Y -adapted process such that Proof. The proof is given in Ref. [17,Proposition 2.4]. Note that here condition (15) implies that and also that the process L defined by for every t ∈ ½0, T, is an ðF, PÞ-martingale.

The filtering equations
Theorem 7 (The Kushner-Stratonovich equation). Under Assumptions 4 and condition (15), the filter π solves the following Kushner-Stratonovich equation, that is, for every f ∈ C 1;2 b ð½0, T Â RÞ : where Here, by dπtÀðλφf Þ dπtÀðλφÞ ðzÞ and dπtÀðLf Þ dπtÀðλφÞ ðzÞ, we mean the Radon-Nikodym derivatives of the measures π tÀ ðλf φðdzÞÞ and π tÀ ðLf ÞðdzÞ, with respect to π tÀ λφðdzÞ . Moreover, the operator L defined by L t f ðdzÞ :¼ Lf ð.; Y tÀ , dzÞ is such that for every A ∈ BðRÞ, takes into account common jump times between the signal X and the observation Y.
Finally, the operator L X given by denotes the generator of the Markov process X.
Example 8 (Observation dynamics driven by independent point processes with unobservable intensities). In the sequel, we provide an example where the Kushner-Stratonovich equation simplifies and the Radon-Nikodym derivatives appearing in the dynamics of π(f) reduce to ratios. Suppose that there exists a finite set of measurable functions K i 1 ðt, yÞ 6 ¼ 0 for all ðt, yÞ ∈ ½0, T Â R, for i ∈ {1,…; n}, such that the dynamics of Y is given by where N i are independent counting processes with ðF, PÞ intensities λ i ðt, X tÀ , Y tÀ Þ.
For simplicity, in this example, we assume that X and Y have no common jump times. Then, the filtering Eq. (21) reads as Note that Eq. (21) has an equivalent expression in terms of the operator L X 0 , given by where Moreover, the filter has a natural recursive structure. To show this, define the sequence {T n , Z n } n ∈ N of jump times and jump sizes of Y, that is, Z n ¼ Y Tn À Y T À n . These are observable data. Then, between two consecutive jump times the filter is governed by a diffusion process, that is, for t ∈ ðT n ∧ T, T nþ1 ∧ TÞ and at any jump time T n occurring before time T, it is given by which implies that π Tn ðf Þ is completely determined by the observed data (T n , Z n ) and the knowledge of π t (f) in the time interval ½T nÀ1 , T n Þ, since π T À n ðf Þ ¼ lim t!T À n π t ðf Þ. Note that the Kushner-Stratonovich equation is an infinite-dimensional nonlinear stochastic differential equation. Often, it is possible to characterize the filter in terms of a simpler equation, known as the Zakai equation which provides the dynamics of the unnormalized version of the filter. Although the Zakai equation is still infinite-dimensional, it has the advantage to be linear.
The idea for getting the dynamics of the unnormalized filter consists of performing an equivalent change of probability measure defined by dP 0 dP for a suitable strictly positive ðF, PÞ-martingale Z, in such a way that the so-called unnormalized filter p is the MðRÞ-valued process defined by Remark 9. By the Kallianpur-Striebel formula, we get that . This provides the relation between the filter and its unnormalized version.
In order to compute the Zakai equation, we make the following assumption.
Assumption 10. Suppose that there exists a transition function η 0 ðt, y, dzÞ such that the ðF Y , PÞpredictable measure η 0 ðt, Y tÀ , dzÞ is equivalent to λðt, X tÀ , Y tÀ Þφðt, X tÀ , Y tÀ , dzÞ and Remark 11. In Ref. [18], a weaker assumption is considered. That condition allows to introduce an equivalent probability measure on ðΩ, F Y T Þ which is not necessarily the restriction on F Y T of an equivalent probability measure on ðΩ, F T Þ.
Assumption 10 equivalently means that there exists an ðF Y , PÞ-predictable process Ψðt, X tÀ , Y tÀ , zÞ such that and 1 þ Ψðt, X t À , Y t À , zÞ > 0 P-a.s. for every t ∈ ½0, T, z ∈ R. Setting Uðt, zÞ : we also assume that the following integrability condition holds: The subsequent proposition provides a useful version of the Girsanov Theorem that fits to our setting. Uðs, zÞ mðds, dzÞ À λðs, X s À , Y s À Þφðs, X s À , Y s À , dzÞds , for every t ∈ ½0, T, where EðMÞ denotes the Doléans-Dade exponential of a martingale M. Then, Z is a strictly positive ðF, PÞ -martingale. Let P 0 be the probability measure equivalent to P given by Then, the process is an ðF, P 0 Þ-Brownian motion, and the ðF, P 0 Þ-predictable projection of the integer-valued random measure mðdt, dzÞ is given by η 0 ðt, Y t À , dzÞdt. Note that, by Eq. (16), we get that the process f W 1 can also be written as which implies that f W 1 is also an ðF Y , P 0 Þ-Brownian motion. Moreover, since η 0 ðt, Y t À , dzÞ is F Y predictable, it provides the ðF Y , P 0 Þ-predictable projection of the measure mðdt, dzÞ and the observation process Y satisfies dY t ¼ σ 1 ðt, Y t ÞdW   ii. Conversely, suppose that pathwise uniqueness for the solution of the Kushner-Stratonovich equation holds and let ξ be an MðRÞ-valued process which is a strong solution of the Zakai equation. Then ξ t ¼ p t P À a:s: for all t ∈ ½0, T.
Proof. The proof follows by Ref. [18,Theorems 4.5 and 4.6]. Here, note that Assumption 10 implies that the measures μ t À ðλφðdzÞÞ and π t À ðλφðdzÞÞ are equivalent.
Finally, strong uniqueness for the solution of both filtering equations is established in the subsequent theorems.

A financial application to risk minimization
In the current section, we focus on a financial application. We consider a simple financial market where agents may invest in a risky asset whose price is described by the process Y given in Eq. (3) and a riskless asset with price process B. Without loss of generality, we assume that B t = 1 for every t ∈ ½0, T. We also assume throughout the section the following dynamics for the process Y: for some functions σðt, yÞ and Kðt, x, y; ζÞ such that σðt, yÞ > 0 and Kðt, x, y; ζÞ > À1.
This choice for the dynamics of Y has a double advantage. On one side assuming a geometric form, together with the condition that Kðt, x, y; ζÞ > À1 guarantees nonnegativity which is desirable when talking about prices. On the other hand, we are modeling Y directly under a martingale measure, and by Assumption 18, it turns out to be a square integrable ðF, PÞmartingale.

Remark 19.
In the sequel, it might be useful to specify the dynamics of Y also in terms of the jump measure mðdt, dzÞ. Recalling Eqs. (6) and (14), we have The stochastic factor X which affects intensity and jump size distribution of Y may represent the state of the economy and is not directly observable by market agents. This is a typical situation arising in real financial markets.
We model by F Y the available information to investors. Since Y is F Y adapted, it is in particular an ðF Y , PÞ-martingale with the following decomposition: By Eqs. (14) and (45), in this setting the first component of the innovation process I defined in Eq. (16) is given by λðs, X s , Y s Þφðs, X s , Y s , dzÞ À π s ðλφðdzÞÞ ds.
Suppose that we are given a European-type contingent claim whose final payoff is a square The objective of the agent is to find the optimal hedging strategy for this derivative. Since the number of random sources exceeds the number of tradeable risky assets, the market is incomplete. It is well known that in this setting, perfect replication by self-financing strategies is not feasible. Then, we suppose that the investor intends to pursue the risk-minimization approach.
Risk minimization is a quadratic hedging method that allows determining a dynamic investment strategy that replicates perfectly the claim with minimal cost. Let us properly introduce the objects of interest. We start with the following notation. For any pair of F-adapted (respectively, F Y -adapted) processes Ψ 1 , Ψ 2 we refer to 〈Ψ 1 , Ψ 2 〉 F for the predictable covariation computed with respect to filtration F (respectively, 〈Ψ 1 , Ψ 2 〉 F Y for the predictable covariation computed with respect to filtration F Y ). Note that and since Y is also F Y adapted, we also have Definition 20. The space ΘðF Y Þ (respectively, ΘðFÞ) is the space of all F Y -predictable (respectively, F-predictable) processes θ such that We observe that for every θ ∈ ΘðF Y Þ, thanks to F Y -predictability, we have which implies that ΘðF Y Þ ⊆ ΘðFÞ.
Since we have two different levels of information represented by the filtrations F and F Y , we may define two classes of admissible strategies.
Definition 21. An F Y -strategy (respectively, F-strategy) is a pair ψ ¼ ðθ, ηÞ of stochastic processes, where θ represents the amount invested in the risky asset and η is the amount invested in the riskless asset, such that θ ∈ ΘðF Y Þ (respectively, θ ∈ ΘðFÞ) and η is F Y -adapted (respectively, F-adapted).
This definition reflects the fact that investor's choices should be adapted to her/his knowledge of the market. The value of a strategy ψ ¼ ðθ, ηÞ is given by and its cost is described by the process In other terms, the cost of a strategy is the difference between the value process and the gain process. For a self-financing strategy, the value and the gain processes coincide, up to the initial wealth V 0 , and therefore the cost is constant and equal to C t ¼ V 0 , for every t ∈ ½0, T. We continue by defining the risk process, in the partial information setting.

Definition 22.
Given an F Y -strategy (respectively, an F-strategy) ψ ¼ ðθ, ηÞ, we denote by R F Y ðψÞ (respectively, R F ðψÞ) the associated risk process defined as for every t ∈ ½0, T.
Then, we have the following definition of risk-minimizing strategy under partial information. ii.
for any other F Y -strategyψ we have R F Y t ðψÞ ≤ R F Y t ðψÞ, for every t ∈ ½0, T.
The corresponding definitions of risk process and risk-minimizing strategy under full information can be obtained replacing F Y and R F Y t with F and R F t in Definition 23. To differentiate, when it is necessary, we use the terms F Y -risk-minimizing strategy or F-risk-minimizing strategy. The criterion (ii) in Definition 23 can be also written as which intuitively means that a strategy is risk minimizing if it minimizes the variance of the cost. This equivalent definition allows to obtain a nice property of risk-minimizing strategies which turn out to be self-financing on average, that is, the cost process C is a martingale and therefore has constant expectation (see, e.g., Ref. [27,Lemma 2] or [28,Lemma 2.3]).
In the sequel, we aim to characterize the optimal hedging strategy for the contingent claim ξ under full and partial information, that is, the Fand the F Y -risk-minimizing strategies. To this, we introduce two orthogonal decompositions known as the Galtchouk-Kunita-Watanabe decompositions under full and partial information (see, e.g., [30]). To understand better the relevance of these decompositions, we assume for a moment completeness of the market and full information. Then, it is well known that for every European-type contingent claim with final payoff ξ, there exists a self-financing strategy ψ ¼ ðθ, ηÞ such that that is, a replicating portfolio is uniquely determined by the initial wealth and the investment in the risky asset. When the market is incomplete, decomposition Eq. (58) does not hold in general. Intuitively, this implies that we might expect additional terms in Eq. (58), and according to the risk-minimization criterion, this additional terms need to be such that the final cost does not deviate too much from the average cost, in the quadratic sense. Specifically, we have the following decomposition of the random variable ξ: where G T is the value at time T of a suitable process G. The minimality criterion requires that G is a martingale orthogonal to Y. We refer the reader to Ref. [28] for a detailed survey. Under suitable hypothesis, the above decomposition takes the name of Galtchouk-Kunita-Watanabe decomposition.
Now we wish to be more formal, and we introduce the following definitions: Consider a random variable ξ ∈ L 2 ðF Y T Þ. Since F Y T ⊆F T , we can define the following decompositions for ξ.
Definition 24. a. The Galtchouk-Kunita-Watanabe decomposition of ξ ∈ L 2 ðF Y T Þ with respect to Y and F is given by where U F 0 ∈ L 2 ðF 0 Þ, θ F ∈ ΘðFÞ and G F is a square integrable ðF, PÞ-martingale, with G F b. The Galtchouk-Kunita-Watanabe decomposition of ξ ∈ L 2 ðF Y T Þ with respect to Y and F Y is given by where In the sequel, we refer to Eqs. (60) and (61) as the Galtchouk-Kunita-Watanabe decompositions under full information and under partial information, respectively. Since Y is a square integrable martingale with respect to both filtrations F and F Y , decompositions Eqs. (60) and (61) exist.
Next proposition provides a relation between the integrands θ F and θ F Y of decompositions Eqs. (60) and (61) in terms of predictable projections. For any ðF, PÞ-predictable process A of finite variation, we denote by A p, F Y its ðF Y , PÞ-dual-predictable projection. 1 Proposition 25. The integrands in decompositions Eqs. (60) and (61) satisfy the following relation: Here, 〈Y〉 p, F Y denotes the ðF Y , PÞ-dual-predictable projection of 〈Y〉 F and it is given by 1 We call ðF Y , PÞdual predictable projection of a process A the F Y -predictable finite variation process A p, F Y such that for any Proof. First note that the ðF Y , PÞ-dual-predictable projection of the process 〈Y〉 F coincides with the predictable quadratic variation of the process Y itself, computed with respect to its internal filtration, given in Eq. (51), since for any ðF Y , PÞ-predictable-(bounded) process φ, we have that . This proves Eq. (63). Let By the Galtchouk-Kunita-Watanabe decomposition Eq. (60), we can write for every t ∈ ½0, T. We observe that for every F Y -predictable process φ the following holds: By choosing φ = θ and applying the Cauchy-Schwarz inequality, we obtain This implies that θ ∈ ΘðF Y Þ ⊆ ΘðFÞ and that e G is an ðF, PÞ-martingale. Taking the conditional expectation with respect to F Y T in Eq. (65) leads to is strongly orthogonal to Y, that is, if for any ðF Y , PÞ-predictable-(bounded) process φ the following holds: Note that orthogonality of the term follows by the orthogonality of G F and Y. Moreover, we have and by Eq. (64) which proves strong orthogonality.
Theorem 26 shows the relation between the Galtchouk-Kunita-Watanabe decompositions and the optimal strategies under full and partial information.
ii. Moreover, it also admits a unique F Y -risk-minimizing strategy ψ Ã, F ¼ ðθ Ã, F , η Ã, F Y Þ, explicitly given by for every t ∈ ½0, T, with minimal cost and θ F Y , U F Y 0 and G F Y are given in Definition 24 part b.
Proof. The proof of part i. is given, for example, in Ref. [28,Theorem 2.4]. For part ii., note that using the martingale representation of Y with respect to its inner filtration given in Eq. (48) and the fact that ξ ∈ L 2 ðF Y T Þ, it is possible to reduce the partial information case to full information and apply again [28,Theorem 2.4]. □ Proposition 25 helps us in the computation of the optimal strategy under partial information. Indeed, it is sufficient to compute the corresponding strategy θ Ã, F under full information and the Radon-Nikodym derivative given in Eq. (62). To get more explicit representations, we assume that the payoff of the contingent claim has the form ξ ¼ HðT, Y T Þ, for some function Let L X, Y denote the Markov generator of the pair (X, Y), that is Δf ðt, x, y; ζÞ :¼ f ðt, x þ K 0 ðt, x; ζÞ, yð1 þ Kðt, x, y; ζÞÞÞ À f ðt, x, yÞ: By the Markov property, we have that for any t ∈ ½0, T there exists a measurable function hðt, x, yÞ such that hðt, X t , Y t Þ ¼ E HðT, Y T ÞjF t ½ : If the function h is sufficiently regular, for instance h ∈ C 1;2;2 b ð½0, T Â R Â R þ Þ, we can apply Itô's formula and get that hðt, X t , Y t Þ ¼ hð0, X 0 , Y 0 Þ þ Δhðs, X s À , Y s À ; ζÞ Nðds, dζÞ À νðdζÞds : By Eq. (79), the process {hðt, X t , Y t Þ, t ∈ ½0, T} is an ðF, PÞ-martingale. Then, the finite variation term vanishes, which means that the function h satisfies L X, Y hðt, X t , Y t Þ ¼ 0, P-a.s. and for almost every t ∈ ½0, T. The next proposition provides the risk-minimizing strategy under partial information.
Our ultimate objective in this section is to investigate on the relation between costs of the F-optimal strategy and the F Y -optimal strategy, or equivalently the associated risk processes.
It clearly holds that θ Ã, F Y ∈ ΘðFÞ, and then the F Y -risk-minimizing strategy is also an F-strategy.
Considering the corresponding risks, we have Plugging in the expressions for the optimal strategies given in Eqs. (82) and (83), respectively, and denoting Σðt, X t , Y t Þ :¼ Y 2 t σ 2 ðt, Y t Þ þ ð Z z 2 λðt, X tÀ , Y tÀ Þφðt, X tÀ , Y tÀ , dzÞ , we have for some C > 0, where the inequality follows by Assumption 18, and in the last equality, we We can conclude by saying that we found an upper bound for the expected difference between the total risks taken by an informed investor and a partially informed one which is directly proportional to the mean-squared error between the process {gðt, X t , S t Þ, t ∈ ½0, T} and its filtered estimate πðgÞ ¼ {π t ðgÞ, t ∈ ½0, T}.